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Performance of SPC product codes under the erasure A. Lpez Martn - - PowerPoint PPT Presentation

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SPC product codes under the BEC S. D. Cardell, J. J. Climent, Performance of SPC product codes under the erasure A. Lpez Martn channel Sara D. Cardell 1 Joan-Josep Climent 1 Alberto Lpez Martn 2 1 Universitat dAlacant, Spain 2

slide-1
SLIDE 1

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín

Performance of SPC product codes under the erasure channel

Sara D. Cardell1 Joan-Josep Climent1 Alberto López Martín2

1 Universitat d’Alacant, Spain 2 Instituto Nacional de Matemática Pura e Aplicada, Brazil

ALCOMA 2015

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-2
SLIDE 2

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Outline

1

Preliminaries SPC product code Kotska Numbers

2

Counting patterns

3

Conclusions

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-3
SLIDE 3

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Outline

1

Preliminaries SPC product code Kotska Numbers

2

Counting patterns

3

Conclusions

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-4
SLIDE 4

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure channel

Elias, P .: Coding for noisy channels. In: IRE International Convention Record, part 4, pp. 37–46 (1955) Properties ◮ Each sent symbol is either correctly received or considered as erased. ◮ Each codeword symbol is lost with a fixed independent probability. ◮ An [n, k, d]-code can recover up to d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-5
SLIDE 5

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure channel

Elias, P .: Coding for noisy channels. In: IRE International Convention Record, part 4, pp. 37–46 (1955) Properties ◮ Each sent symbol is either correctly received or considered as erased. ◮ Each codeword symbol is lost with a fixed independent probability. ◮ An [n, k, d]-code can recover up to d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-6
SLIDE 6

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure channel

Elias, P .: Coding for noisy channels. In: IRE International Convention Record, part 4, pp. 37–46 (1955) Properties ◮ Each sent symbol is either correctly received or considered as erased. ◮ Each codeword symbol is lost with a fixed independent probability. ◮ An [n, k, d]-code can recover up to d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-7
SLIDE 7

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure channel

Elias, P .: Coding for noisy channels. In: IRE International Convention Record, part 4, pp. 37–46 (1955) Properties ◮ Each sent symbol is either correctly received or considered as erased. ◮ Each codeword symbol is lost with a fixed independent probability. ◮ An [n, k, d]-code can recover up to d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-8
SLIDE 8

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Product code

Let Fq be the Galois field with q elements. Definition A linear product code C over Fq is formed from two other linear codes C− and C with parameters [n−, k −, d −] and [n, k , d] over Fq, respectively. The product code C will have parameters [n−n, k −k , d −d] over Fq. Over the erasure channel, the product code corrects up to d −d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-9
SLIDE 9

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Product code

Let Fq be the Galois field with q elements. Definition A linear product code C over Fq is formed from two other linear codes C− and C with parameters [n−, k −, d −] and [n, k , d] over Fq, respectively. The product code C will have parameters [n−n, k −k , d −d] over Fq. Over the erasure channel, the product code corrects up to d −d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-10
SLIDE 10

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Product code

Let Fq be the Galois field with q elements. Definition A linear product code C over Fq is formed from two other linear codes C− and C with parameters [n−, k −, d −] and [n, k , d] over Fq, respectively. The product code C will have parameters [n−n, k −k , d −d] over Fq. Over the erasure channel, the product code corrects up to d −d − 1 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-11
SLIDE 11

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Product code

The codewords of length n−n can be seen as arrays with size n− × n in a way that the columns are codewords of C and the rows are codewords

  • f C−.

C parity checks C− parity checks Information bits k − k n n−

Figure: Codeword of a product code, with systematic encoding

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-12
SLIDE 12

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC code

Definition A single parity-check code is a linear binary code with parameters [n, n − 1, 2]. ◮ The single parity-check (SPC) code is a very popular error detection code, since it is very easy to implement. ◮ This codes can correct one single erasure over the erasure channel.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-13
SLIDE 13

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC code

Definition A single parity-check code is a linear binary code with parameters [n, n − 1, 2]. ◮ The single parity-check (SPC) code is a very popular error detection code, since it is very easy to implement. ◮ This codes can correct one single erasure over the erasure channel.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-14
SLIDE 14

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC code

Definition A single parity-check code is a linear binary code with parameters [n, n − 1, 2]. ◮ The single parity-check (SPC) code is a very popular error detection code, since it is very easy to implement. ◮ This codes can correct one single erasure over the erasure channel.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-15
SLIDE 15

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC product code

◮ C− = C is a linear binary code with parameters [n, n − 1, 2]. ◮ We consider the product code C = C− ⊗ C. ◮ The parameters of the product code are [n2, (n − 1)2, 4]. ◮ The code C corrects only 3 erasures. In some special cases this code can correct more than 3 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-16
SLIDE 16

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC product code

◮ C− = C is a linear binary code with parameters [n, n − 1, 2]. ◮ We consider the product code C = C− ⊗ C. ◮ The parameters of the product code are [n2, (n − 1)2, 4]. ◮ The code C corrects only 3 erasures. In some special cases this code can correct more than 3 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-17
SLIDE 17

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC product code

◮ C− = C is a linear binary code with parameters [n, n − 1, 2]. ◮ We consider the product code C = C− ⊗ C. ◮ The parameters of the product code are [n2, (n − 1)2, 4]. ◮ The code C corrects only 3 erasures. In some special cases this code can correct more than 3 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-18
SLIDE 18

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC product code

◮ C− = C is a linear binary code with parameters [n, n − 1, 2]. ◮ We consider the product code C = C− ⊗ C. ◮ The parameters of the product code are [n2, (n − 1)2, 4]. ◮ The code C corrects only 3 erasures. In some special cases this code can correct more than 3 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-19
SLIDE 19

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC product code

◮ C− = C is a linear binary code with parameters [n, n − 1, 2]. ◮ We consider the product code C = C− ⊗ C. ◮ The parameters of the product code are [n2, (n − 1)2, 4]. ◮ The code C corrects only 3 erasures. In some special cases this code can correct more than 3 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-20
SLIDE 20

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

SPC product code

◮ C− = C is a linear binary code with parameters [n, n − 1, 2]. ◮ We consider the product code C = C− ⊗ C. ◮ The parameters of the product code are [n2, (n − 1)2, 4]. ◮ The code C corrects only 3 erasures. In some special cases this code can correct more than 3 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-21
SLIDE 21

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure pattern

Definition An erasure pattern of size m × m, with t erasures, where 0 ≤ t ≤ m2 and 1 ≤ m ≤ n, is an array of size m × m where t of the entries correspond to the position of the erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-22
SLIDE 22

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ An erasure pattern of size n × n corresponds to words of size n × n, where the position of the erasures is the unique information we consider. ◮ Given a received word with t erasures, the decoder will perform iterative row-wise and column-wise decoding to recover the erased bits. ◮ When a single bit is erased in a row or column, it can be recovered. ◮ If more than one bit is erased in a row (column), it is skipped. ◮ Decoding is performed until no further recovery is possible.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-23
SLIDE 23

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ An erasure pattern of size n × n corresponds to words of size n × n, where the position of the erasures is the unique information we consider. ◮ Given a received word with t erasures, the decoder will perform iterative row-wise and column-wise decoding to recover the erased bits. ◮ When a single bit is erased in a row or column, it can be recovered. ◮ If more than one bit is erased in a row (column), it is skipped. ◮ Decoding is performed until no further recovery is possible.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-24
SLIDE 24

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ An erasure pattern of size n × n corresponds to words of size n × n, where the position of the erasures is the unique information we consider. ◮ Given a received word with t erasures, the decoder will perform iterative row-wise and column-wise decoding to recover the erased bits. ◮ When a single bit is erased in a row or column, it can be recovered. ◮ If more than one bit is erased in a row (column), it is skipped. ◮ Decoding is performed until no further recovery is possible.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-25
SLIDE 25

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ An erasure pattern of size n × n corresponds to words of size n × n, where the position of the erasures is the unique information we consider. ◮ Given a received word with t erasures, the decoder will perform iterative row-wise and column-wise decoding to recover the erased bits. ◮ When a single bit is erased in a row or column, it can be recovered. ◮ If more than one bit is erased in a row (column), it is skipped. ◮ Decoding is performed until no further recovery is possible.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-26
SLIDE 26

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ An erasure pattern of size n × n corresponds to words of size n × n, where the position of the erasures is the unique information we consider. ◮ Given a received word with t erasures, the decoder will perform iterative row-wise and column-wise decoding to recover the erased bits. ◮ When a single bit is erased in a row or column, it can be recovered. ◮ If more than one bit is erased in a row (column), it is skipped. ◮ Decoding is performed until no further recovery is possible.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-27
SLIDE 27

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4]. ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-28
SLIDE 28

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4]. ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-29
SLIDE 29

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-30
SLIDE 30

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-31
SLIDE 31

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-32
SLIDE 32

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-33
SLIDE 33

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-34
SLIDE 34

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-35
SLIDE 35

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-36
SLIDE 36

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-37
SLIDE 37

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-38
SLIDE 38

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-39
SLIDE 39

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-40
SLIDE 40

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-41
SLIDE 41

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-42
SLIDE 42

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-43
SLIDE 43

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-44
SLIDE 44

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-45
SLIDE 45

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× ×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-46
SLIDE 46

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-47
SLIDE 47

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

×

◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-48
SLIDE 48

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4]. ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-49
SLIDE 49

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4]. ◮ The code is supposed to correct 3 erasures. ◮ We have corrected 8 erasures.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 50

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Definition An erasure pattern of size m × m is said to be uncorrectable if and only if it contains a subpattern of size l × l, l ≤ m, such that each row and each column have two or more erasures. Example

× × × × × × ×

(a) Correctable erasure pattern of size 4 × 4 with 7 erasures

× × × × × × ×

(b) Uncorrectable erasure pattern

  • f size 4 × 4 with 7 erasures
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 51

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure pattern

Example

× × × × × × × ×

Figure: Uncorrectable erasure pattern of size 4 × 4 with 7 erasures

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 52

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure pattern

Example

× × × × × × × ×

Figure: Uncorrectable erasure pattern of size 4 × 4 with 7 erasures

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 53

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ Erasure patterns of size n × n with 3 erasures or less are always correctable. ◮ Erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2n − 1 may or may not be correctable.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 54

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

◮ Erasure patterns of size n × n with 3 erasures or less are always correctable. ◮ Erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2n − 1 may or may not be correctable.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 55

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Decoding

Example (cont.) Consider the SPC code C with parameters [6, 5, 2]. We can construct the binary product code C = C ⊗ C with parameters [36, 25, 4].

× × × × × × × × × × × × × × × × × × × × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 56

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Classification

Definition An uncorrectable erasure pattern is said to be strict if none of the erasures can be corrected. Equally, an uncorrectable erasure pattern is said to be partial if it can be partially corrected. × × × × × × × × × × × × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 57

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Strict uncorrectable erasure patterns

Lemma An strict uncorrectable erasure pattern contains two or more erasures in each row and column in error. × × × × × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 58

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Partial uncorrectable erasure pattern

Lemma A partial uncorrectable erasure pattern always contains an strict uncorrectable erasure pattern. × × × × × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 59

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Idea

Purpose We would like to count the number of uncorrectable erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2n − 1. In this work, we count the number of strict uncorrectable erasure patterns.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-60
SLIDE 60

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Idea

Purpose We would like to count the number of uncorrectable erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2n − 1. In this work, we count the number of strict uncorrectable erasure patterns.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-61
SLIDE 61

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Partition of an integer

Definition If t is a positive integer, then a partition of t is a non-increasing sequence of positive integers (λ1, λ2, λ3, . . . , λp) such that p

i=1 λi = t.

We denote by Pt the set of possible partitions of the integer t. Example For example, the set of partitions of 6 is given by P6 = {(6), (5, 1), (4, 2), (4, 1, 1), (3, 3), (3, 2, 1), (3, 1, 1, 1), (2, 2, 2), (2, 2, 1, 1), (2, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1)}.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 62

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Partition of an integer

Definition If t is a positive integer, then a partition of t is a non-increasing sequence of positive integers (λ1, λ2, λ3, . . . , λp) such that p

i=1 λi = t.

We denote by Pt the set of possible partitions of the integer t. Example For example, the set of partitions of 6 is given by P6 = {(6), (5, 1), (4, 2), (4, 1, 1), (3, 3), (3, 2, 1), (3, 1, 1, 1), (2, 2, 2), (2, 2, 1, 1), (2, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1)}.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-63
SLIDE 63

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Partition of an integer

Definition If t is a positive integer, then a partition of t is a non-increasing sequence of positive integers (λ1, λ2, λ3, . . . , λp) such that p

i=1 λi = t.

We denote by Pt the set of possible partitions of the integer t. Example For example, the set of partitions of 6 is given by P6 = {(6), (5, 1), (4, 2), (4, 1, 1), (3, 3), (3, 2, 1), (3, 1, 1, 1), (2, 2, 2), (2, 2, 1, 1), (2, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1)}.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 64

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conjugate partition

Definition Let us consider a partition λ = (λ1, λ2, . . . , λp) of t. The conjugate of λ is defined as the vector λ∗ = (λ∗

1, λ∗ 2, . . . , λ∗ p′) where

λ∗

j = |{i | 1 ≤ i ≤ p, λi ≥ j}| .

Notice that both, λ and λ∗, are partitions of the same integer t.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 65

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conjugate partition

Definition Let us consider a partition λ = (λ1, λ2, . . . , λp) of t. The conjugate of λ is defined as the vector λ∗ = (λ∗

1, λ∗ 2, . . . , λ∗ p′) where

λ∗

j = |{i | 1 ≤ i ≤ p, λi ≥ j}| .

Notice that both, λ and λ∗, are partitions of the same integer t.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 66

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conjugate partition

Example λ = (3, 1) and λ∗ = (2, 1, 1) are conjugate partitions of t = 4.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 67

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Young diagram and Young tableaux

Let λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) be two partitions of the same integer t. Definition A Young diagram of shape λ is an arrangement of t boxes in p rows where there are λi boxes in row i, with i = 1, 2, . . . , p, and these boxes are left justified.

λ = (4, 3, 2, 2)

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 68

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Young diagram and Young tableaux

Let λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) be two partitions of the same integer t. Definition A Young tableau of shape λ and content µ is obtained from a Young diagram of shape λ by inserting in each box one of the integers 1, 2, . . . , q in such a way that the following conditions hold: i) the elements in each row are non-decreasing, ii) the elements in each column are strictly increasing, iii) the integer j occurs µj times, with j = 1, 2, . . . , q.

λ = (4, 3, 2, 2)

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 69

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Young diagram and Young tableaux

Let λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) be two partitions of the same integer t. Definition A Young tableau of shape λ and content µ is obtained from a Young diagram of shape λ by inserting in each box one of the integers 1, 2, . . . , q in such a way that the following conditions hold: i) the elements in each row are non-decreasing, ii) the elements in each column are strictly increasing, iii) the integer j occurs µj times, with j = 1, 2, . . . , q.

λ = (4, 3, 2, 2) µ = (3, 3, 2, 2, 1)

1 1 1 2 2 2 3 3 4 4 5

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 70

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Young diagram and Young tableaux

This is not the only option; there are 5 more Young tableaux with these properties.

1 1 1 2 2 2 3 3 4 4 5 1 1 1 2 2 2 4 3 3 4 5 1 1 1 2 2 2 5 3 3 4 4 1 1 1 3 2 2 2 3 4 4 5 1 1 1 4 2 2 2 3 3 4 5 1 1 1 5 2 2 2 3 3 4 4

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 71

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Kotska numbers

Definition If λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) are two partitions of the same integer t, then the Kostka number denoted by κλ,µ, is the number of Young tableaux of shape λ and content µ. Example We want to compute κ(3,2,1),(3,2,1). We have to count the number of Young tableaux of shape λ = (3, 2, 1) and content µ = (3, 2, 1).

1 1 1 2 2 3

Thus, κ(3,2,1),(3,2,1) = 1.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 72

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Kotska numbers

Definition If λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) are two partitions of the same integer t, then the Kostka number denoted by κλ,µ, is the number of Young tableaux of shape λ and content µ. Example We want to compute κ(3,2,1),(3,2,1). We have to count the number of Young tableaux of shape λ = (3, 2, 1) and content µ = (3, 2, 1).

1 1 1 2 2 3

Thus, κ(3,2,1),(3,2,1) = 1.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 73

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Kotska numbers

Definition If λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) are two partitions of the same integer t, then the Kostka number denoted by κλ,µ, is the number of Young tableaux of shape λ and content µ. Example We want to compute κ(3,2,1),(3,2,1). We have to count the number of Young tableaux of shape λ = (3, 2, 1) and content µ = (3, 2, 1).

1 1 1 2 2 3

Thus, κ(3,2,1),(3,2,1) = 1.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 74

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Kotska numbers

Definition If λ = (λ1, λ2, . . . , λp) and µ = (µ1, µ2, . . . , µq) are two partitions of the same integer t, then the Kostka number denoted by κλ,µ, is the number of Young tableaux of shape λ and content µ. Example We want to compute κ(3,2,1),(3,2,1). We have to count the number of Young tableaux of shape λ = (3, 2, 1) and content µ = (3, 2, 1).

1 1 1 2 2 3

Thus, κ(3,2,1),(3,2,1) = 1.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-75
SLIDE 75

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Counting binary matrices

◮ Let A be a binary matrix of size m1 × m2. ◮ Let R = (r1, r2, . . . , rm1) be the vector where ri is the sum of the elements in row i of matrix A. ◮ Let C = (c1, c2, . . . , cm2) be the vector where cj is the sum of the elements in row j of matrix A. ◮ Note that r1 + r2 + · · · + rm1 = c1 + c2 + · · · + cm2 and call this number t.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 76

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Counting binary matrices

  • R. A. Brualdi, Algorithms for constructing (0,1)-matrices with prescribed

row and column sum vectors, Discrete Mathematics 306 (23) (2006) 3054–3062. Theorem The number of binary matrices with R and C as the row sum and the column sum, respectively, is given by

  • λ∈Pt

κλ,R κλ∗,C.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-77
SLIDE 77

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Counting binary matrices

  • R. A. Brualdi, Algorithms for constructing (0,1)-matrices with prescribed

row and column sum vectors, Discrete Mathematics 306 (23) (2006) 3054–3062. Theorem The number of binary matrices with R and C as the row sum and the column sum, respectively, is given by

  • λ∈Pt

κλ,R κλ∗,C.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

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SLIDE 78

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Outline

1

Preliminaries SPC product code Kotska Numbers

2

Counting patterns

3

Conclusions

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-79
SLIDE 79

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Counting patterns

Purpose We would like to count the number of uncorrectable erasure patterns of size n × n with t erasures, 4 ≤ t ≤ 2n − 1.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-80
SLIDE 80

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with four erasures

Assume we have a codeword of size n × n and that 4 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 4 erasures is formed by a square: The total number of uncorrectable erasure patterns with 4 erasures is n

2

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-81
SLIDE 81

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with four erasures

Assume we have a codeword of size n × n and that 4 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 4 erasures is formed by a square: The total number of uncorrectable erasure patterns with 4 erasures is n

2

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-82
SLIDE 82

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with four erasures

Assume we have a codeword of size n × n and that 4 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 4 erasures is formed by a square:

× × × ×

The total number of uncorrectable erasure patterns with 4 erasures is n

2

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-83
SLIDE 83

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with four erasures

Assume we have a codeword of size n × n and that 4 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 4 erasures is formed by a square:

× × × ×

The total number of uncorrectable erasure patterns with 4 erasures is n

2

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-84
SLIDE 84

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with four erasures

Assume we have a codeword of size n × n and that 4 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 4 erasures is formed by a square:

× × × ×

The total number of uncorrectable erasure patterns with 4 erasures is n

2

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-85
SLIDE 85

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with four erasures

Assume we have a codeword of size n × n and that 4 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 4 erasures is formed by a square:

× × × ×

The total number of uncorrectable erasure patterns with 4 erasures is n

2

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-86
SLIDE 86

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with five erasures

Assume we have a codeword of size n × n and that 5 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 5 erasures is formed by a square (uncorrectable pattern of size 2 × 2) and one extra erasure: The total number of uncorrectable erasure patterns with 5 erasures is

n

2

2n2−4

1

  • .
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-87
SLIDE 87

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with five erasures

Assume we have a codeword of size n × n and that 5 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 5 erasures is formed by a square (uncorrectable pattern of size 2 × 2) and one extra erasure: The total number of uncorrectable erasure patterns with 5 erasures is

n

2

2n2−4

1

  • .
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-88
SLIDE 88

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with five erasures

Assume we have a codeword of size n × n and that 5 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 5 erasures is formed by a square (uncorrectable pattern of size 2 × 2) and one extra erasure:

× × × × ×

The total number of uncorrectable erasure patterns with 5 erasures is

n

2

2n2−4

1

  • .
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-89
SLIDE 89

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with five erasures

Assume we have a codeword of size n × n and that 5 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 5 erasures is formed by a square (uncorrectable pattern of size 2 × 2) and one extra erasure:

× × × × ×

The total number of uncorrectable erasure patterns with 5 erasures is

n

2

2n2−4

1

  • .
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-90
SLIDE 90

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with five erasures

Assume we have a codeword of size n × n and that 5 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 5 erasures is formed by a square (uncorrectable pattern of size 2 × 2) and one extra erasure:

× × × × ×

The total number of uncorrectable erasure patterns with 5 erasures is

n

2

2n2−4

1

  • .
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-91
SLIDE 91

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with five erasures

Assume we have a codeword of size n × n and that 5 erasures have

  • ccurred.

The only uncorrectable erasure pattern of 5 erasures is formed by a square (uncorrectable pattern of size 2 × 2) and one extra erasure:

× × × × ×

The total number of uncorrectable erasure patterns with 5 erasures is

n

2

2n2−4

1

  • .
  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-92
SLIDE 92

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable patterns with six erasures

The total number of uncorrectable erasure patterns with 6 erasures is

n

2

2n2−4

2

  • − 4n

2

n

3

  • + 6n

3

2.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-93
SLIDE 93

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure patterns/ binary matrices

Let us represent an erasure pattern of size n × n by a binary matrix of size n × n where there is 1 in the erasure positions and 0 otherwise. × × × × ×

  

0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0

  

In this work, our purpose is to count the number of strict uncorrectable erasure patterns of size n × n with t erasures. Equivalently, we want to find all matrices of size n × n with t ones (and 2 or more ones in each non-zero row and non-zero column).

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-94
SLIDE 94

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Erasure patterns/ binary matrices

Let us represent an erasure pattern of size n × n by a binary matrix of size n × n where there is 1 in the erasure positions and 0 otherwise. × × × × ×

  

0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0

  

In this work, our purpose is to count the number of strict uncorrectable erasure patterns of size n × n with t erasures. Equivalently, we want to find all matrices of size n × n with t ones (and 2 or more ones in each non-zero row and non-zero column).

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-95
SLIDE 95

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable strict erasure patterns

Notation ◮ The partitions that we will consider will have all length n. ◮ If a partition has length r < n, it will be filled in with n − r zeros. ◮ For example, (6, 1) is a partition of 7 with length 2, but if we are considering partitions of length 4, we write (6, 1, 0, 0).

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-96
SLIDE 96

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Uncorrectable strict erasure patterns

Notation ◮ The partitions that we will consider will have all length n. ◮ If a partition has length r < n, it will be filled in with n − r zeros. ◮ For example, (6, 1) is a partition of 7 with length 2, but if we are considering partitions of length 4, we write (6, 1, 0, 0).

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-97
SLIDE 97

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-98
SLIDE 98

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P6 = {(6), (5, 1), (4, 2), (4, 1, 1), (3, 3), (3, 2, 1), (3, 1, 1, 1), (2, 2, 2), (2, 2, 1, 1), (2, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1)}. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-99
SLIDE 99

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P6 = {(6), (5, 1), (4, 2), (4, 1, 1), (3, 3), (3, 2, 1),✘✘✘✘

✘ ❳❳❳❳ ❳

(3, 1, 1, 1), (2, 2, 2),

✘✘✘✘ ✘ ❳❳❳❳ ❳

(2, 2, 1, 1),✭✭✭✭✭

❤❤❤❤❤

(2, 1, 1, 1, 1),✭✭✭✭✭✭

❤❤❤❤❤❤

(1, 1, 1, 1, 1, 1)}. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-100
SLIDE 100

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P6 = {(6, 0, 0), (5, 1, 0), (4, 2, 0), (4, 1, 1), (3, 3, 0), (3, 2, 1),✘✘✘✘

✘ ❳❳❳❳ ❳

(3, 1, 1, 1), (2, 2, 2),✘✘✘✘

✘ ❳❳❳❳ ❳

(2, 2, 1, 1),✭✭✭✭✭

❤❤❤❤❤

(2, 1, 1, 1, 1),✭✭✭✭✭✭

❤❤❤❤❤❤

(1, 1, 1, 1, 1, 1)}. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-101
SLIDE 101

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P6 = {(6, 0, 0),✘✘✘

✘ ❳❳❳ ❳

(5, 1, 0), (4, 2, 0),✘✘✘

✘ ❳❳❳ ❳

(4, 1, 1), (3, 3, 0),✘✘✘

✘ ❳❳❳ ❳

(3, 2, 1),✘✘✘✘

✘ ❳❳❳❳ ❳

(3, 1, 1, 1), (2, 2, 2),✘✘✘✘

✘ ❳❳❳❳ ❳

(2, 2, 1, 1),✭✭✭✭✭

❤❤❤❤❤

(2, 1, 1, 1, 1),✭✭✭✭✭✭

❤❤❤❤❤❤

(1, 1, 1, 1, 1, 1)}. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-102
SLIDE 102

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P6 = {✘✘✘

✘ ❳❳❳ ❳

(6, 0, 0),✘✘✘

✘ ❳❳❳ ❳

(5, 1, 0),✘✘✘

✘ ❳❳❳ ❳

(4, 2, 0),✘✘✘

✘ ❳❳❳ ❳

(4, 1, 1), (3, 3, 0),✘✘✘

✘ ❳❳❳ ❳

(3, 2, 1),✘✘✘✘

✘ ❳❳❳❳ ❳

(3, 1, 1, 1), (2, 2, 2),✘✘✘✘

✘ ❳❳❳❳ ❳

(2, 2, 1, 1),✭✭✭✭✭

❤❤❤❤❤

(2, 1, 1, 1, 1),✭✭✭✭✭✭

❤❤❤❤❤❤

(1, 1, 1, 1, 1, 1)}. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-103
SLIDE 103

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Definition Given two positive integers n and t, the set Pn

t is a subset of the set of

partitions of t of length n defined as Pn

t = {λ ∈ Pt | λi = 1, λi ≤ n, i = 1, 2, . . . , n}.

Example Consider t = 6 and n = 3. P6 = {✘✘✘

✘ ❳❳❳ ❳

(6, 0, 0),✘✘✘

✘ ❳❳❳ ❳

(5, 1, 0),✘✘✘

✘ ❳❳❳ ❳

(4, 2, 0),✘✘✘

✘ ❳❳❳ ❳

(4, 1, 1), (3, 3, 0),✘✘✘

✘ ❳❳❳ ❳

(3, 2, 1),✘✘✘✘

✘ ❳❳❳❳ ❳

(3, 1, 1, 1), (2, 2, 2),✘✘✘✘

✘ ❳❳❳❳ ❳

(2, 2, 1, 1),✭✭✭✭✭

❤❤❤❤❤

(2, 1, 1, 1, 1),✭✭✭✭✭✭

❤❤❤❤❤❤

(1, 1, 1, 1, 1, 1)}. P3

6 = {(2, 2, 2), (3, 3, 0)}

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-104
SLIDE 104

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Lemma For a partition λ = (λ1, λ2, . . . , λn) of a positive integer t and length n, the number of possible combinations of the elements in λ is given by δλ = n! η1!η2! · · · ηn!, where ηi = |{j | λj = i}|, for i = 1, 2, . . . , n. Example λ = (3, 1, 1, 0) ∈ P4

5

δλ = 4! 1!2!1! = 12.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-105
SLIDE 105

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Notation

Lemma For a partition λ = (λ1, λ2, . . . , λn) of a positive integer t and length n, the number of possible combinations of the elements in λ is given by δλ = n! η1!η2! · · · ηn!, where ηi = |{j | λj = i}|, for i = 1, 2, . . . , n. Example λ = (3, 1, 1, 0) ∈ P4

5

δλ = 4! 1!2!1! = 12.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-106
SLIDE 106

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Theorem The number of strict uncorrectable erasure patterns of size n × n with t erasures is given by

  • R,C∈Pn

t

δRδC

  • λ∈Pt

κλ,R κλ∗,C. (1)

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-107
SLIDE 107

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4. P4 = {(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)} P3

4 = {(2, 2, 0)} −

→ C = R = (2, 2, 0) δR = δC =

3! 2!1! = 3

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-108
SLIDE 108

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4. P4 = {(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)} P3

4 = {(2, 2, 0)} −

→ C = R = (2, 2, 0) δR = δC =

3! 2!1! = 3

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-109
SLIDE 109

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4. P4 = {(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)} P3

4 = {(2, 2, 0)} −

→ C = R = (2, 2, 0) δR = δC =

3! 2!1! = 3

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-110
SLIDE 110

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4. P4 = {(4), (3, 1), (2, 2), (2, 1, 1), (1, 1, 1, 1)} P3

4 = {(2, 2, 0)} −

→ C = R = (2, 2, 0) δR = δC =

3! 2!1! = 3

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-111
SLIDE 111

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example λ κλ,R λ∗ κλ∗,C κλ,R · κλ∗,C 4 1 1, 1, 1, 1 3, 1 1 2, 1, 1 2, 2 1 2, 2 1 1 2, 1, 1 3, 1 1 1, 1, 1, 1 4 1 1

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-112
SLIDE 112

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-113
SLIDE 113

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-114
SLIDE 114

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-115
SLIDE 115

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-116
SLIDE 116

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-117
SLIDE 117

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-118
SLIDE 118

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-119
SLIDE 119

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-120
SLIDE 120

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-121
SLIDE 121

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Main result

Example Consider n = 3 and t = 4.

  • R,C∈P3

4

δRδC

  • λ∈P4

κλ,R κλ∗,C = 9

  • λ∈P4

κλ,R κλ∗,C = 9

× × × ×

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-122
SLIDE 122

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Outline

1

Preliminaries SPC product code Kotska Numbers

2

Counting patterns

3

Conclusions

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-123
SLIDE 123

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conclusions

So far... ◮ We know the number of uncorrectable erasure patterns of size n × n with t erasures when t = 4, 5, 6, 7, 8. ◮ Using Kotska numbers and Young tableaux, we can compute the number of strict uncorrectable erasure patterns of size n × n. Future work ◮ An erasure pattern of size n × n can be represented by a bipartite graph with 2n vertices (n in each vertex class). ◮ For 9 ≤ t ≤ 2n − 1, the number of uncorrectable patterns can be computed taking partitions of 2n and considering connected components of the corresponding graph in each case.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-124
SLIDE 124

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conclusions

So far... ◮ We know the number of uncorrectable erasure patterns of size n × n with t erasures when t = 4, 5, 6, 7, 8. ◮ Using Kotska numbers and Young tableaux, we can compute the number of strict uncorrectable erasure patterns of size n × n. Future work ◮ An erasure pattern of size n × n can be represented by a bipartite graph with 2n vertices (n in each vertex class). ◮ For 9 ≤ t ≤ 2n − 1, the number of uncorrectable patterns can be computed taking partitions of 2n and considering connected components of the corresponding graph in each case.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-125
SLIDE 125

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conclusions

So far... ◮ We know the number of uncorrectable erasure patterns of size n × n with t erasures when t = 4, 5, 6, 7, 8. ◮ Using Kotska numbers and Young tableaux, we can compute the number of strict uncorrectable erasure patterns of size n × n. Future work ◮ An erasure pattern of size n × n can be represented by a bipartite graph with 2n vertices (n in each vertex class). ◮ For 9 ≤ t ≤ 2n − 1, the number of uncorrectable patterns can be computed taking partitions of 2n and considering connected components of the corresponding graph in each case.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-126
SLIDE 126

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conclusions

So far... ◮ We know the number of uncorrectable erasure patterns of size n × n with t erasures when t = 4, 5, 6, 7, 8. ◮ Using Kotska numbers and Young tableaux, we can compute the number of strict uncorrectable erasure patterns of size n × n. Future work ◮ An erasure pattern of size n × n can be represented by a bipartite graph with 2n vertices (n in each vertex class). ◮ For 9 ≤ t ≤ 2n − 1, the number of uncorrectable patterns can be computed taking partitions of 2n and considering connected components of the corresponding graph in each case.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-127
SLIDE 127

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín Prelimina- ries

SPC product code Kotska Numbers

Counting patterns Conclusions

Conclusions

So far... ◮ We know the number of uncorrectable erasure patterns of size n × n with t erasures when t = 4, 5, 6, 7, 8. ◮ Using Kotska numbers and Young tableaux, we can compute the number of strict uncorrectable erasure patterns of size n × n. Future work ◮ An erasure pattern of size n × n can be represented by a bipartite graph with 2n vertices (n in each vertex class). ◮ For 9 ≤ t ≤ 2n − 1, the number of uncorrectable patterns can be computed taking partitions of 2n and considering connected components of the corresponding graph in each case.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-128
SLIDE 128

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-129
SLIDE 129

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín

Performance of SPC product codes under the erasure channel

Sara D. Cardell1 Joan-Josep Climent1 Alberto López Martín2

1 Universitat d’Alacant, Spain 2 Instituto Nacional de Matemática Pura e Aplicada, Brazil

ALCOMA 2015

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC

slide-130
SLIDE 130

SPC product codes under the BEC

  • S. D.

Cardell, J.

  • J. Climent,
  • A. López

Martín

References

P . Elias, Coding for noisy channels, in: IRE International Convention Record, pt. 4, 1955, pp. 37–46.

  • M. A. Kousa, A novel approach for evaluating the performance of SPC product codes under erasure

decoding 50 (1) (2002) 7–11.

  • M. A. Kousa, A. H. Mugaibel, Cell loss recovery using two-dimensional erasure correction for ATM networks,

in: Proceedings of the Seventh International Conference on Telecommunication Systems, 1999, pp. 85–89.

  • J. M. Simmons, R. G. Gallager, Design of error detection scheme for class C service in ATM, IEEE/ACM

Transactions on Networking 2 (1) (1994) 80–88.

  • D. M. Rankin, T. A. Gulliver, Single parity check product codes 49 (8) (2001) 1354–1362.
  • R. A. Brualdi, Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors,

Discrete Mathematics 306 (23) (2006) 3054–3062.

  • D. E. Knuth, Permutations, matrices, and generalized young tableaux, Pacific Journal of Mathematics 34 (3)

(1970) 707–727.

  • S. D. Cardell, J. J. Climent, A. López Martín

SPC product codes under the BEC