Band Splitting Permutations for Spatially Coupled LDPC Codes Achieving Asymptotically Optimal Burst Erasure Immunity

Abstract

It is well known that spatially coupled (SC) codes with erasure-BP decoding have powerful error correcting capability over memoryless erasure channels. However, the decoding performance of SC-codes significantly degrades when they are used over burst erasure channels. In this paper, we propose band splitting permutations (BSP) suitable for (l, r, L) SC-codes. The BSP splits a diagonal band in a base matrix into multiple bands in order to enhance the span of the stopping sets in the base matrix. As theoretical performance guarantees, lower and upper bounds on the maximal burst correctable length of the permuted (l, r, L) SC-codes are presented. Those bounds indicate that the maximal correctable burst ratio of the permuted SC-codes is given by λmax ≃ 1/k where k = r/l. This implies the asymptotic optimality of the permuted SC-codes in terms of burst erasure correction.

Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

“…To verify, consider the C R (3, 6, 3, 100, M = 64) SC-LDPC ensemble. By averaging over all the SPs of 1000 random code instances of the ensemble, the empirical average number of size-2 stopping sets is obtained as (Lλ 0 , (L−1)λ 1 , (L−2)λ 2 ) ≈ (0.876, 0.488, 0.060), which is reasonably close to (0.829, 0.494, 0.061) computed using (15), even for small M = 64.…”

“…where λ k is the average number of size-2 stopping sets given by (15). (ii) SPBC and BEC at SP z 0 : The VNs of SP z 0 are all erased.…”

“…This situation has been studied in [14], where the authors additionally provide protograph constructions that avoid this situation and also maximize the correctable burst length given some structural constraints of the code. To correct bursts encompassing an SP, the authors in [15] apply interleaving (therein denoted band splitting) to a protograph-based SC-LDPC code. Interleaving however increases system latency and if windowed decoding is used, this approach results in an increased required window length and thus complexity.…”

Spatially Coupled LDPC codes affected by a single random burst of erasures

“…These figures show that for b < b BP (ε = 0), the error floor is well estimated by (6) even for small M = 100. It implies that the size-2 stopping sets are the main cause of decoding error.…”

“…The authors give asymptotic lower and upper bounds for the bit and block erasure probabilities obtained from density evolution. Protograph-based codes that maximize the correctable burst lengths are constructed in [5], while interleaving (therein denoted band splitting) is applied to a protograph-based SC-LDPC code in [6] to increase the correctable burst length. If windowed decoding is used, this approach results however Parts of this work were conducted while N. Rengaswamy was visiting Bell Labs as a research intern funded by a scholarship of the DAAD-RisePro programme.…”