Selective max‐min algorithm for low‐density parity‐check decoding

Abstract

With the growing importance of error correction in different communication systems, using an efficient and easily implementable code is always appreciated. One of the most important codes is the low-density parity-check (LDPC) code. Two main iterative decoding algorithms are usually used, namely the sum-product (SP) algorithm (also referred to as belief propagation) and the min-sum (MS). The SP algorithm is more accurate but suffers from very high complexity. On the other hand, the MS algorithm has a much lower complexity at the expense of some performance degradation. To handle this performance degradation, many algorithms were presented in the literature as improvements for the MS, like the scaled MS and the offset MS. However, all those improved algorithms are more complex than the traditional MS. In this study, an efficient and low complexity LDPC decoding algorithm, called selective max-min (SMM), is proposed. The SMM performance is closer to SP than to MS as long as the average number of ones per column in the parity check matrix is around or less than 4 (which is the case for most of the communication systems using LDPC). On the other hand, the SMM exhibits only a minor complexity increase over traditional MS making it suitable for practical implementation.

Binary Wyner–Ziv code design based on compound LDGM–LDPC structures

“…The distortion values d 1 , d 2 , and D t are obtained from the calculation of the average Hamming distance between the source and the decoded sequences according to (2). In fact, two types of gaps can be defined to compare the results from rate and distortion point of view.…”

A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage

“…Thus, N (W cm h ) = {0}, which means dim (N (W cm h )) > 0, and proves the first condition in the lemma. Moreover, if a GAST is removed successfully, this implies that each WCM in the set W associated with that GAST has broken weight 4 Note that Lemma 4 also applies to any WCM W cm h that has G cm h with δ h > 1 and at least one of the disconnected components having a short adjacency matrix (a very rare case). conditions.…”

“…In the simulations of this subsection, Code 9 is an NB-PB-LDPC code defined over GF (4), with block length = 3996 bits, rate ≈ 0.89, and γ = 3. Code 10 is the result of optimizing Code 9 for the asymmetric NLM channel (with 6 reads this time) by attempting to remove the dominant GASTs (4, 2, 2, 5, 0), (4, 3, 2, 5, 0), (5, 2, 2, 5, 1), (6, 0, 0, 9, 0), (6, 1, 0, 9, 0), (6, 1, 1, 7, 1), (6,2,2,5,2), and (6, 2, 2, 8, 0) using the WCM framework.…”

“…We also reuse Code 6 in this subsection (its parameters are stated in the previous subsection). Code 13 is the result of optimizing Code 6 for the asymmetric CHMM channel (with 6 reads) by attempting to remove the dominant GASTs (4, 4, 4, 6, 0), (6,4,4,11,0), and (7, 4, 3, 11, 1) in addition to the dominant OSTs (6, 5, 2, 11, 0), (7, 5, 3, 11, 1), (7,5,4,9,2), (7,6,6,8,2), (8,6,2,15,0), and (10,7,5,11,4) using the WCM framework.…”

“…11, 0) GAST/OST,(6,4,2,11, 0) GAST/OST,(6,5,2,11, 0) OST,(6,6,2,11, 0) OST, (6, 7, 2, 11, 0) OST} (this is because d 1 = 2 andb o_max = d 1 + b o_ut = 7).For a given OST, define a matrix W z o to be the matrix obtained by removing b o , d 1 ≤ b o ≤ b o_max , rows corresponding to CNs ∈ (O ∪ T ) from the matrix A, which is the OST adjacency matrix. These b o CNs can simultaneously be unsatisfied under some edge labeling that produces a GAST/an OST which has the same unlabeled GAST/OST as the given OST.…”