Decoding F q -linear codes over erasure channels Sara D. Cardell - - PowerPoint PPT Presentation

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Decoding F q -linear codes over erasure channels Sara D. Cardell Universidad de Alicante SPCoding School S.D. Cardell Decoding F q -linear codes over erasure channels Erasure channel We consider as model of errors the erasure channel

SLIDE 1

Decoding Fq -linear codes over erasure channels

Sara D. Cardell

Universidad de Alicante SPCoding School

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 2

Erasure channel

We consider as model of errors the erasure channel introduced by Elias.

P. ELIAS.

Coding for noisy channels. IRE International Convention Record, pt. 4, 37–46 (1955).

1 − pe pe pe 1 − pe 1 e ? 1

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 3

Codes over extension alphabets

Consider the code CF2 whose (systematic) generator matrix is: G =

  

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  

Here [N, K, D] = [8, 4, 3] and CF2 is not an MDS code. We can consider the block matrix: G =

  

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  

Over F2

2, the code CF2

2 has parameters [n, k, d] = [4, 2, 3]. Therefore, it is an

MDS F2-linear code.

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 4

Codes over extension alphabets

Consider the code CF2 whose (systematic) generator matrix is: G =

  

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  

Here [N, K, D] = [8, 4, 3] and CF2 is not an MDS code. We can consider the block matrix: G =

  

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  

Over F2

2, the code CF2

2 has parameters [n, k, d] = [4, 2, 3]. Therefore, it is an

MDS F2-linear code.

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 5

Construction

C is the companion matrix of a primitive polynomial of degree b over Fq[x]. Using the isomorphism ψ : Fqb − → Fq[C] α − → C, we construct a family of MDS Fq-linear codes over Fb

q.

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 6

Decoding

The codeword: v = uG = [

information

v 1 v 2 . . . vk | v k+1 . . . v n
redundancy

] If the number of erased symbols is t ≤ n − k, then we propose a decoding algorithm based on solving linear reduced systems.

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 7

Conclusions

So far ◮ Construction of MDS Fq−linear codes, based on the isomorphism ψ : Fqb − → Fq[C]. ◮ Decoding algorithm for these codes over the erasure channel, based on solving a linear system over Fq. In the future ◮ Analyze the complexity. ◮ Try to adapt this algorithm for other codes. ◮ More applications.

S.D. Cardell Decoding Fq -linear codes over erasure channels

SLIDE 8

Conclusions

So far ◮ Construction of MDS Fq−linear codes, based on the isomorphism ψ : Fqb − → Fq[C]. ◮ Decoding algorithm for these codes over the erasure channel, based on solving a linear system over Fq. In the future ◮ Analyze the complexity. ◮ Try to adapt this algorithm for other codes. ◮ More applications.

S.D. Cardell Decoding Fq -linear codes over erasure channels