[Paper Review] Edge Coloring Technique to Remove Small Elementary Trapping Sets from Tanner Graph of QC-LDPC Codes with Column Weight 4

Mohammad‐Reza Sadeghi, Farzane Amirzade|arXiv (Cornell University)|Aug 15, 2018

Error Correcting Code Techniques10 references3 citations

TL;DR

This paper proposes an edge coloring technique applied to variable node (VN) graphs of quasi-cyclic LDPC (QC-LDPC) codes with column weight 4 to eliminate small elementary trapping sets (ETSs). By deriving sufficient conditions under which 4-edge coloring of VN graphs corresponding to ETSs becomes impossible, the method ensures that specific ETSs—(5,b) with b≤4, (6,b) with b≤2, and (7,4)—are absent in Tanner graphs of QC-LDPC codes with girth 6 or 8. The key contribution is a systematic design framework for constructing high-girth, error-floor-free QC-LDPC codes with column weight 4.

ABSTRACT

One of the phenomena that causes high decoding failure rates is trapping sets. Characterization of $(a,b)$ elementary trapping sets (ETSs), their graphical properties and the lower bounds on their size in variable regular LDPC codes with column weights 3, 4, 5 and 6, where $a$ is the size of the ETS and $b$ is the number of degree-one check nodes, have been an interesting subject among researchers. Although progressive-edge-growth method (PEG) to construct LDPC codes free of an specific ETS has been proposed in the literature, it is mostly applied to LDPC codes with column weight 3. In this paper, we focus on constructing QC-LDPC codes with column weight 4 whose Tanner graphs are free of small ETSs. Using coloring the edges of the variable node (VN) graph corresponding to an ETS, we provide the sufficient conditions to obtain QC-LDPC codes with column weight 4, girth 6 and free of $(5,b)$ ETSs, where $b\leq4$, and $(6,b)$ ETs, where $b\leq2$. Moreover, for $(4,n)$-regular QC-LDPC codes with girth 8, we present a method to remove $(7,4)$ ETSs from Tanner graphs.

Motivation & Objective

• To address the high decoding failure rates caused by small elementary trapping sets (ETSs) in QC-LDPC codes with column weight 4.
• To extend the edge coloring technique—previously used for column weight 3 codes—to column weight 4 codes for ETS avoidance.
• To derive sufficient conditions on exponent matrices that guarantee the absence of specific small ETSs in Tanner graphs of QC-LDPC codes.
• To provide a constructive method for designing (4,n)-regular QC-LDPC codes with girth 6 or 8 that are free of harmful ETSs such as (5,b), (6,b), and (7,4).

Proposed method

• The method uses edge coloring of the variable node (VN) graph corresponding to a given ETS to determine whether the ETS can exist in the Tanner graph.
• A 4-edge coloring is applied to the VN graph of an ETS; if such a coloring is impossible under specific constraints, the ETS cannot exist in the Tanner graph.
• For girth 6 codes, the method identifies that avoiding 6-cycles formed from the first three rows and from rows 1, 2, and 4 of the exponent matrix prevents (5,b) and (6,b) ETSs.
• For girth 8 codes, the method proves that avoiding 8-cycles formed by using the first row twice and the second row at least once in the exponent matrix prevents 4-edge coloring of the K_{3,4} VN graph of the (7,4) ETS.
• The approach relies on the equivalence between the non-existence of a valid 4-edge coloring and the absence of the corresponding ETS in the Tanner graph, as established in prior work.
• The construction uses exponent matrices with specific row and cycle constraints to ensure high girth and ETS-free properties.

Experimental results

Research questions

• RQ1 What conditions on the exponent matrix of a (4,n)-regular QC-LDPC code ensure the absence of (5,b) ETSs with b≤4 and (6,b) ETSs with b≤2 in its Tanner graph with girth 6?
• RQ2 Can the edge coloring technique be extended from column weight 3 to column weight 4 QC-LDPC codes to eliminate small ETSs?
• RQ3 What specific cycle patterns in the exponent matrix must be avoided to ensure that the Tanner graph of a (4,n)-regular QC-LDPC code with girth 8 is free of (7,4) ETSs?
• RQ4 Is the (7,4) ETS the smallest possible ETS in girth 8 QC-LDPC codes with column weight 4, and how can it be systematically removed?
• RQ5 What are the sufficient conditions on the exponent matrix to construct (4,n)-regular QC-LDPC codes with girth 8 and no (7,4) ETSs?

Key findings

• For (4,n)-regular QC-LDPC codes with girth 6, avoiding 6-cycles formed from the first three rows and from rows 1, 2, and 4 of the exponent matrix ensures the absence of (5,b) ETSs with b≤4 and (6,b) ETSs with b≤2.
• For (4,n)-regular QC-LDPC codes with girth 8, avoiding 8-cycles formed by using the first row twice and the second row at least once in the exponent matrix prevents 4-edge coloring of the K_{3,4} VN graph, thereby eliminating the (7,4) ETS.
• The (7,4) ETS is the only non-isomorphic ETS of its size in girth 8 QC-LDPC codes with column weight 4, confirming its significance as a target for removal.
• The paper provides explicit exponent matrices for (4,n)-regular QC-LDPC codes with n=5,6,7, achieving girth 6 and avoiding (5,b) and (6,b) ETSs, and girth 8 and avoiding (7,4) ETSs.
• The method reduces the search space for ETS-free codes by leveraging known results from column weight 3 codes and applying them to column weight 4 constructions.
• The proposed conditions are sufficient to guarantee that the Tanner graphs of constructed QC-LDPC codes are free of specified small ETSs, improving error-floor performance.