Iterative Decoding of Non-Binary Cyclic Codes Using Minimum- Weight - - PowerPoint PPT Presentation

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Iterative Decoding of Non-Binary Cyclic Codes Using Minimum- Weight - - PowerPoint PPT Presentation

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Iterative Decoding of Non-Binary Cyclic Codes Using Minimum- Weight Dual Codewords Jiongyue Xing , Li Chen n School of EIT, Sun Yat-sen University , Guangzhou, China Martin Bossert, Sebastian Bitzer n Institute of CE, Ulm University , Ulm,

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n

Jiongyue Xing, Li Chen School of EIT, Sun Yat-sen University, Guangzhou, China

n

Martin Bossert, Sebastian Bitzer Institute of CE, Ulm University, Ulm, Germany

n

IEEE International Symposium on Information Theory 21st – 26th, June, 2020, Los Angeles, California, USA

Iterative Decoding of Non-Binary Cyclic Codes Using Minimum- Weight Dual Codewords

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Outline

n Introduction n Shift-Sum Decoding n Iterative Shift-Sum Decoding Algorithms

q Hard-decision ISS algorithm q Soft-decision ISS algorithm

n Simulation Results n Conclusions

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n Applications of cyclic codes

q

  • Eg. Reed-Solomon (RS) and BCH codes
  • I. Introduction

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n Cyclic codes

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Finite field: Fq = {σ0, σ1, …, σ!"#$}, its primitive element is α

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𝒟(2p; n, k, d), p ≤ s p = 1: 𝒟 is a binary BCH code p = s: 𝒟 is an RS code 1 < p < s: 𝒟 is a non-binary BCH (NB-BCH) code

q

Dual code: 𝒟⊥(2p; n, n – k, d⊥)

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Let c(x) = c0 + c1x + ∙ ∙ ∙ + cn-1xn-1 ∊ 𝒟, sup(c(x)) = { j | cj ≠ 0, ∀j }

n Cyclically different codewords

q

Let c1(x), c2(x) ∊ 𝒟, if c2(x) ≠ αjc1(x)x-h mod (xn – 1), ∀j and ∀h ∊ sup(c1(x)), they are cyclically different.

  • I. Introduction

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n Encoding of cyclic codes

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Message poly.: f(x) = f0 + f1x + ∙ ∙ ∙ + fk-1xk-1

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Generator poly.: g(x)

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Codeword: c(x) = f(x)g(x)

n

RS encoding

q

𝒟(2s; n, k, dRS), dRS = n – k + 1 gRS(x) = ∏ (x − α*)

,-.#$ */$

n NB-BCH encoding

q

Cyclotomic cosets Kj = {j･(2p)i mod n, i = 0, 1, …, s p – 1}

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Let K = ⋃Kj gBCH(x) = ∏ (x − α2)

  • 2∈5

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𝒟(2p; n, k, dBCH), k = n – deg gBCH(x), dBCH is the number of the consecutive roots over F!6 in gBCH(x)

  • I. Introduction

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n Bounded minimum-distance decoding

q

Berlekamp-Massey (BM) algorithm, Euclidean algorithm, etc.

n

Algebraic list decoding

q

Guruswami-Sudan (GS) algorithm, Koetter-Vardy (KV) algorithm, Wu’s decoding, etc.

n Reliability based decoding

q

Generalized minimum-distance decoding, Chase decoding, ordered statistics decoding, etc.

n Belief-propagation based decoding

q

Adaptive BP algorithm, multiple-bases BP algorithm, etc.

n Decoding with minimum-weight dual codewords

q

Reed-Muller codes [Santi_ISIT2018]

q

BCH codes [Bossert_SCC2019]

  • I. Introduction

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n Cyclic convolution

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For all c(x) ∊ 𝒟 and all b(x) ∊ 𝒟⊥ c(x)b(x) = 0 mod (xn – 1)

n Minimum-weight dual codeword (MWDC)

b(x) = β1xb1 + β2xb2 + ∙ ∙ ∙ + βd

⊥xbd⊥

n Channel

q

Error poly.: e(x) = ε1xe1 + ε2xe2 + ∙ ∙ ∙ + ετxeτ

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Received poly.: r(x) = c(x) + e(x)

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Check poly.: w(x) = r(x)b(x) = c(x)b(x) + e(x)b(x) = e(x)b(x) mod (xn – 1)

  • II. Shift-Sum Decoding

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With the choice of b1 = 0, β1 = 1

w(x) = e(x)b(x) mod (xn – 1)

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Eg., shift w(x) by

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Use d⊥ shifts

  • II. Shift-Sum Decoding

= e(x) + β2xb2e(x) + ⋯ + βd

⊥xbd⊥e(x) mod (xn – 1)

β2ε1xe1+b2 + β2ε2xe2+b2 + ⋯ + β2ετxeτ+b2+ = ε1xe1 + ε2xe2 + ⋯ + ετxeτ + βd

⊥ε1xe1+bd⊥ + βd ⊥ε2xe2+ bd⊥ + ⋯ + βd ⊥ετxeτ+bd⊥

⋮ x-bh βh w(x) mod (xn – 1), where h ∊ {1, 2, …, d⊥}

βd

⊥

β1

w(x) = e(x) + β7 β8xb2–b1e(x) + ⋯ + xbd⊥–b1e(x) mod (xn – 1) x-b1 β1 !! All nonzero coefficients of w(x) are errors or shifted scalar errors !!

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q

Use L cyclically different MWDC b(𝑚)(x)= β$

(:)xb1 + β! (:)xb2 + ⋯ + β

d⊥

(:)xbd⊥, 𝑚 = 1, 2, …, L

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Calculate w(𝑚)(x) = r(x)b(𝑚)(x) mod (xn – 1)

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Use d⊥ cyclic shifts of w;

(:)(x), where ℎ ∊ {1, 2, …, d⊥}

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Coefficients of w;

(:)(x)

w;,*

(:) = $

β?

(@) ∑

βB

(:)𝑠 *D;#B EFG I

  • B∈JKL(M @ (N))

q

Indicator function 𝑈 𝑚, 𝑗, 𝑘, ℎ = R 1, if w;,*

(:) = 𝜏2

0, otherwise

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Reliability measure 𝜚2,* = ∑ ∑ 𝑈(𝑚, 𝑗, 𝑘, ℎ)

  • ;∈JKL(M @ (N))

` :/$

  • II. Shift-Sum Decoding

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n

Example: (7, 3) RS code defined over F8

Codeword c(x) = 5 + 6x + 6x2 + 3x3 + 5x5 + 3x6 Received r(x) = 5 + 7x + 6x2 + 3x3 + 3x6 Cyclically different minimum-weight dual codewords b(1)(x) = 1 + 2x + 7x2 + 4x6 b(2)(x) = 1 + x + x3 + x6 b(3)(x) = 1 + 4x2 + 7x3 + 2x6 b(4)(x) = 1 + 4x + 6x2 + 3x5 b(5)(x) = 1 + 3x2 + 7x4 + 5x6 Error poly. e(x) = x + 5x5 Update c a(x) = r(x) – e(x) = 5 + 6x + 6x2 + 3x3 + 5x5 + 3x6

  • II. Shift-Sum Decoding

w;

(:)(x) = xb?

β?

(@) r(x)b(𝑚)(x)

1 2 3 4 5 6 7

  • F8 is an extension field of F2, defined by p(x) = x3 + x + 1
  • In F8, {σ0, σ1, σ2, σ3, σ4, σ5, σ6, σ7} = {0, 1, 2, 4, 3, 6, 7, 5}

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n Hard-decision iterative shift-sum (HISS) algorithm

q

Sort 𝜚c,* in an ascending order as 𝜚c,*d

(8) < 𝜚c,*8 (8) < ⋯ < 𝜚c,*gb8 (8)

Let 𝒦($) = {𝑘c

($), 𝑘$ ($), … , 𝑘k#$ ($) }

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Let 𝜔* = max{𝜚2,*|𝑗 ≠ 0} and 𝑓̂* is the corresponding field elements. Sort 𝜔* in a descending order as 𝜔*d

(7) < 𝜔*8 (7) < ⋯ < 𝜔*gb8 (7)

Let 𝒦(!) = {𝑘c

(!), 𝑘$ (!), … , 𝑘k#$ (!) }

q

Updated positions set 𝒦 = 𝒦($) ∩ 𝒦(!)

q

Updated polynomial 𝑓̂ x = ∑ 𝑓̂*

  • *∈𝒦

x*

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Refine polynomial r(x) ← r(x) – 𝑓̂ x

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If r(x) ∊ 𝒟(2p; n, k, d), output r(x). Otherwise, recalculate 𝜚2,*

  • III. Iterative Shift-Sum Decoding Algorithms

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  • III. Iterative Shift-Sum Decoding Algorithms

n Reliability matrix

0.02 0.00 0.01 0.01 0.71 0.56 0.01 0 0.02 0.10 0.02 0.11 0.05 0.02 0.08 1 0.01 0.05 0.01 0.05 0.03 0.01 0.29 2 0.00 0.03 0.11 0.68 0.09 0.00 0.46 3 0.12 0.02 0.02 0.00 0.00 0.38 0.11 4 0.80 0.01 0.00 0.10 0.10 0.00 0.01 5 0.00 0.05 0.83 0.03 0.00 0.02 0.01 6 0.30 0.74 0.00 0.02 0.02 0.01 0.03 7 r0 r1 r2 r3 r4 r5 r6

Π =

n Reliability of received symbol rj

𝛿* = max{𝜌2,*|∀𝑗} 1 − max{𝜌2,*|∀𝑗} Hard-decision 5 7 6 3 0 3 Reliability 4.00 2.85 4.88 2.13 2.45 1.27 0.85

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n Soft-decision iterative shift-sum (SISS) algorithm

q

Coefficient w;,*

(:) = $

β?

(@) ∑

βB

(:)𝑠 *D;#B EFG I

  • B∈JKL(M @ (N))

q

Reliability measure for w;,*

(:)

𝜊;,*

(:) = min{𝛿 *D;#B EFG I|𝑤 ∈ sup 𝑐 𝑦

and 𝑤 ≠ ℎ}

q

Indicator function 𝑈 𝑚, 𝑗, 𝑘, ℎ = R𝜊;,*

(:), if w;,* (:) = 𝜏2

0 , otherwise

q

Reliability measure 𝜚2,* = ∑ ∑ 𝑈(𝑚, 𝑗, 𝑘, ℎ)

  • ;∈JKL(M @ (N))

` :/$

q

Determine 𝒦 = 𝒦($) ∩ 𝒦(!) and 𝑓̂ 𝑦 = ∑ 𝑓̂*

  • *∈𝒦

𝑦*

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Refine r(x) = r(x) – 𝑓̂ 𝑦 . If r(x) is not a valid codeword, continue the decoding

  • III. Iterative Shift-Sum Decoding Algorithms

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5.95 1.27 4.68 5.00 5.95 2.45 8.84 3.39 13.94 1.70 2.12 0.85 0.85 8.39 2.12 1.27 3.83 2.12 3.40 0.85 1.27 4.25 0.85 1.27 0.85 2.98 3.30 3.40 1.27 2.54 6.26 6.26 2.97 4.98 4.12 1.70 2.12 2.12 3.70 3.70 13.94 2.54 2.13 2.12 4.25 1.70 3.39 2.13 3.72 4.15 0.85 0.85 4.57 2.12 2.98 3.40

n

Example: (7, 3) RS code defined over F8

Codeword c(x) = 5 + 6x + 6x2 + 3x3 + 5x5 + 3x6 Received r(x) = 5 + 7x + 6x2 + 3x3 + 3x6 Symbols reliability: 4.00, 2.85, 4.88, 2.13, 2.45, 1.27, 0.85 Error poly. e(x) = x + 5x5 Update c a(x) = r(x) – e(x) = 5 + 6x + 6x2 + 3x3 + 5x5 + 3x6

  • III. Iterative Shift-Sum Decoding Algorithms

1 2 3 4 5 6 7

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  • IV. Simulation Results

n 𝒟(16; 15, 5, 11) RS code, QSC, L = 335, d⊥ = 6

WER(𝜍) = ∑

€ • ‚ • I

  • 𝜍• 1 − 𝜍 I#•

I

  • /$

1.E-9 1.E-8 1.E-7 1.E-6 1.E-5 1.E-4 1.E-3 1.E-2 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 WER ρ BM GS HISS (10)

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  • IV. Simulation Results

n 𝒟(4; 63, 27, 21) NB-BCH code, QSC, L = 183, d⊥ = 14

WER(𝜍) = ∑

€ • ‚ • I

  • 𝜍• 1 − 𝜍 I#•

I

  • /$

1.E-9 1.E-8 1.E-7 1.E-6 1.E-5 1.E-4 1.E-3 1.E-2 1.E-1 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 WER ρ BM HISS (10)

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  • IV. Simulation Results

n 𝒟(16; 15, 5, 11) RS code, AWGN, L = 335, d⊥ = 6 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 2 3 4 5 6 7 8 9 10 WER SNR (dB) BM KV (𝑚 = 4) KV (𝑚 = 8) MBBP (10) HISS (3) HISS (5) HISS (10) SISS (3) SISS (5) SISS (10) MLLB MLUB

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  • IV. Simulation Results

n 𝒟(16; 15, 9, 7) RS code, AWGN, L = 201, d⊥ = 10 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 3 4 5 6 7 8 WER SNR (dB) BM / GS KV (𝑚 = 4) KV (𝑚 = 8) HISS (3) SISS (3) MLLB MLUB

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  • IV. Simulation Results

n 𝒟(4; 63, 27, 21) NB-BCH code, AWGN, L = 183, d⊥ = 14 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 2 3 4 5 6 7 8 WER SNR (dB) BM HISS (5) HISS (10) HISS (20) SISS (5) SISS (10) SISS (20) MBBP (10)

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  • IV. Simulation Results

n 𝒟(16; 15, 5, 11) RS code, AWGN, L = 335, d⊥ = 6

q

Decoding complexity of each iteration: Ln(d⊥)2

q

GS-Koetter complexity: 9.72 × 106

q

GS-BR complexity: 2.37 × 106

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  • V. Conclusions

n The shift-sum decoding for non-binary cyclic codes has

been introduced.

n Two iterative decoding algorithms based on the shift-

sum decoding have been proposed.

n Simulation results show the decoding performance and

complexity potentials of the proposed algorithms. Thank you!

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