Distance Properties of Short LDPC Codes and Their Impact on the BP, ML and Near-ML Decoding Performance

Transcription

1 Distance Properties of Short LDPC Codes and Their Impact on the BP, ML and Near-ML Decoding Performance Irina E. Bocharova 1,2, Boris D. Kudryashov 1, Vitaly Skachek 2, Yauhen Yakimenka 2 1 St. Petersburg University of Information Technologies, Mechanics and Optics (Russia) 2 University of Tartu (Estonia)

2 Acknowledgements Norwegian-Estonian Research Cooperation Programme (grant EMP133) Estonian Research Council (grant PUT405) University of Tartu ASTRA project PER ASPERA Doctoral School of Information and Communication Technologies High Performance Computing Centre (University of Tartu) 1

3 Outline 1. Code parameters 2. Stopping redundancy hierarchy 3. Considered codes 4. Simulations: FER performance 5. Spectra and bounds 6. But what about BAWGN channel? 2

4 Code parameters

5 Code parameters influence on decoding success (BEC) Decoding problem on BEC Solve linear system: H x = 0 for x = (x 1, x 2, x 3, x 4,?, x 6, x 7,?, x 9,?) 1 BP with extended parity-check matrix 3

6 Code parameters influence on decoding success (BEC) Decoding problem on BEC Solve linear system: H x = 0 for x = (x 1, x 2, x 3, x 4,?, x 6, x 7,?, x 9,?) Table 1: Linear [n, k, d min ] code and its parameters parameter Decoding algorithm BP (belief propagation) Near-ML 1 Maximum-likelihood (ML) d min, distance spectrum d stop, stopping spectrum d dual girth g SR hierarchy 1 BP with extended parity-check matrix 3

7 Stopping redundancy hierarchy

8 Stopping sets and d stop H = x 1 x 2 x 3 x 4? x 6 x 7? x 9?

9 Stopping sets and d stop H = x 1 x 2 x 3 x 4? x 6 x 7? x 9? Stopping distance, d stop Size of the smallest stopping set. 4

10 Stopping redundancy hierarchy (by example) Aim By adding redundant rows, remove small stopping sets (up to size l) 5

11 Stopping redundancy hierarchy (by example) Aim By adding redundant rows, remove small stopping sets (up to size l) H = x 1 x 2 x 3 x 4? x 6 x 7? x 9? c c c c c c c

12 Stopping redundancy hierarchy (by example) Aim By adding redundant rows, remove small stopping sets (up to size l) H = x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 c c c c c c c c 1 + c 2 + c c 2 + c

13 Stopping redundancy hierarchy (by example) Aim By adding redundant rows, remove small stopping sets (up to size l) H = x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 c c c c c c c c 1 + c 2 + c c 2 + c l-th stopping redundancy Minimum number of rows ρ l, s.t. there are no stopping sets of size up to l (except codewords). 5

14 Achieving ML performance Corrollary from definition It s possible to build ρ n k n extended parity-check matrix H, s.t. BP decoder achieves ML performance. 6

15 Stopping redundancy upper bound (i) Intuition/main observation H stopping set 2 n k rows Pr = i 2n k i 2 n k 7

16 Stopping redundancy upper bound (i) Intuition/main observation H stopping set 2 n k rows Pr = i 2n k i 2 n k 7

17 Stopping redundancy upper bound (i) Intuition/main observation H stopping set 2 n k rows Pr = i 2n k i 2 n k 7

18 Stopping redundancy upper bound (i) Intuition/main observation H stopping set 2 n k rows Pr = i 2n k i 2 n k 7

19 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

20 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

21 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

22 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

23 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

24 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

25 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

26 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

27 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

28 Stopping redundancy upper bound (ii) Upper bound construction H ssets profile {u 1, u 2,..., u l } # of ssets D t t rows κ t rows 8

29 Stopping redundancy upper bound (ii) Upper bound 2,3 on stopping redundancy ρ l n k + min t N {t + κ t} where D t = l i=1 x u i ( 1 n k+t j=n k+1 (1 i ) 2n k i 2 n k j ) l 2 n k l P t,j (x) = 2 n k (n k + t + j) κ t = min { j N : P t,j (P t,j 1 (... P t,1 ( D t )... )) = 0 } 2 Han, Siegel, Vardy. (2008). Improved probabilistic bounds on stopping redundancy. 3 Yakimenka, Skachek. (2015). Refined upper bounds on stopping redundancy of binary linear codes. 9

30 Considered codes

31 Parameters of studied [48, 24]-codes d min A dmin,n d stop d dual g (J, K) ρ dmin, ρ dmin+1 ρ r Type (6, 12) 6240, L (6, 12) 261, RU (4, 8) 83, RU (3, 6) 58, QC (3, 6) 355, NB 10

32 Simulations: FER performance

33 FER QC (3, 6)-regular code BP, (3,6)-QC RPC-32, (3,6)-QC RPC-64, (3,6)-QC RPC-128, (3,6)-QC RPC-256, (3,6)-QC ML, (3,6)-QC ML, (48,24) erasure probability Good convergence to ML, but ML performance is poor 11

34 FER Linear [48, 24, 12] code BP, (48,24) RPC-16, (48,24) RPC-64, (48,24) RPC-256, (48,24) RPC-1024, (48,24) ML, (48,24) erasure probability RPC is efficient enough, but convergence to ML is slow 12

35 FER Binary image of non-binary (3, 6)-code ML:(3,6)-NB BP:(3,6)-NB RPC-16: (3,6)-NB RPC-64: (3,6)-NB RPC-256: (3,6)-NB channel erasure probability Both convergence and ML performance are good 13

36 Spectra and bounds

37 Spectra We consider the following spectra for ensembles: Distance spectra Stopping sets spectra Stopping sets spectra for binary images of non-binary codes (gave us bounds for BP decoding) 14

38 FER QC (3, 6)-regular code BP, (3,6)-QC RPC-32, (3,6)-QC RPC-64, (3,6)-QC RPC-128, (3,6)-QC RPC-256, (3,6)-QC ML, (3,6)-QC Lower bound, ML Random coding, ML Upper bound, ML, J=3 ML, (48,24) erasure probability 15

39 FER Linear [48, 24, 12] code BP, (48,24) RPC-16, (48,24) RPC-64, (48,24) RPC-256, (48,24) RPC-1024, (48,24) ML, (48,24) Lower bound, ML Random coding, ML erasure probability 16

40 FER Binary image of non-binary (3, 6)-code ML: [48,24,12]-code ML S-bound (3,6) NB GF(2 4 ) BP S-bound (3,6) NB GF(2 4 ) ML:(3,6)-NB BP:(3,6)-NB RPC-16: (3,6)-NB RPC-64: (3,6)-NB RPC-256: (3,6)-NB channel erasure probability 17

41 But what about BAWGN channel?

42 FER FER BEC vs BAWGNC Lower bound Random coding Upper bound, J=3 ML, (48,24) BP (3,6)-QC RPC-256 (3,6)-QC erasure probability Lower bound Random coding Upper bound, J=3 ML, (48,24) BP (3,6)-QC RPC-256 (3,6)-QC SNR, db 18

43 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) 19

44 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) However, some improvement can be achieved even with a relatively small number of redundant rows 19

45 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) However, some improvement can be achieved even with a relatively small number of redundant rows There is a soft threshold to overcome it one needs plenty of redundant rows 19

46 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) However, some improvement can be achieved even with a relatively small number of redundant rows There is a soft threshold to overcome it one needs plenty of redundant rows NB codes are a good compromise: 19

47 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) However, some improvement can be achieved even with a relatively small number of redundant rows There is a soft threshold to overcome it one needs plenty of redundant rows NB codes are a good compromise: 1. ML performance close to the ML performance of best linear codes; 19

48 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) However, some improvement can be achieved even with a relatively small number of redundant rows There is a soft threshold to overcome it one needs plenty of redundant rows NB codes are a good compromise: 1. ML performance close to the ML performance of best linear codes; 2. BP performance converges rather fast (due to their suitability for iterative decoding?) 19

49 Conclusion Near-ML decoding converges to ML decoding with increasing number of redundant rows (but requires exponential number of rows) However, some improvement can be achieved even with a relatively small number of redundant rows There is a soft threshold to overcome it one needs plenty of redundant rows NB codes are a good compromise: 1. ML performance close to the ML performance of best linear codes; 2. BP performance converges rather fast (due to their suitability for iterative decoding?) Adding redundant rows works on BAWGNC too! 19

50 Open problem What code we want to contstruct for RPC: large d min large d stop small d dual 20

51 Thank you 20

52 Just in case

53 Ensemble-Average Spectra

54 (J, K)-regular Gallager ensemble Ensemble of (J, K)-regular parity-check matrices of LDPC [n, k]-codes H = Each M n strip is a permutation of columns of the first strip.

55 Weight-generating functions G n (s) = Recurrent coefficient calculation Let n A n,w s w w=0 f(s) = l 0 f l s l F L (s) = (f(s)) L = l 0 F l,l s l then F l,l = { fl, L = 1 l i=0 f if l i,l 1, L > 1

56 Average weight spectrum One row of H g(s) = i even ( K )s i = (1 + s)k + (1 s) K i 2 One strip of H n G(s) = N n,w s w = (g(s)) M w=0 Ensemble-average spectrum coefficients ( ) n E{A n,w } = (p(w)) J = w ( n w ) 1 J N J n,w

57 Other spectra Stopping set spectrum g(s) = w=0,2,3,...,k ( ) K s w = (1 + s) K Ks w Weight spectrum ϕ(s) = 1 q 1 m w=1 ( ) m s w = (1 + s)m 1 w q 1 g(ϕ) = (1 + (q 1)ϕ)K + (q 1)(1 ϕ) K q Can calculate fast!

58 Calculation of d min, d stop Minimum distance Stopping distance d min 1 w=0 d stop 1 w=0 A n,w < 1 B n,w < 1

59 Numerical Results

60 Observations Random regular LDPC codes (esp. non-binary) with optimised J have minimum distance close to random linear codes

61 Observations Random regular LDPC codes (esp. non-binary) with optimised J have minimum distance close to random linear codes and gap decreases with decrease of R

62 Observations Random regular LDPC codes (esp. non-binary) with optimised J have minimum distance close to random linear codes and gap decreases with decrease of R for LDPC, stopping distances are about half of min distances (for best LDPC equal)

63 Example Table 2: Examples of codes from the Gallager (J, 2J) ensemble (n, k, d) J ˆd dgv d L d stop ˆdstop ρ ˆρ (40,24,6) (60,35,8) (90,49,10)

64 Asymptotic Analysis

65 For binary images (q = 2 m ) Asymptotics for binary images of Gallager ensemble of nonbinary LDPC codes over F q = F 2 m (following Gallager) E{A n,w } ( ) 1 J nm (g(ϕ(s))) MJ s wj, s w Replace s by e ρ, find critical value δ = w/(mn), where { ln E{A nm,w } lim = min (1 J)h e (δ) + J } n nm ρ Km ln(g(ϕ(eρ ))) ρδj = 0

66 Normalised minimum distances Table 3: Normalized minimum distances for binary and nonbinary LDPC code ensembles. Numbers in parentheses are typical asymptotic normalized stopping distances. m J (0.0180) (0.0454) (0.0580) (0.0619)