STRUCTURE OF NON-BINARY REGULAR LDPC CYCLE CODES Jie Huang, Shengli Zhou, and Peter Willett Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT, 06269 ABSTRACT In this paper, we study non-binary regular LDPC cycle codes whose parity check matrix has fixed column weight 2 and fixed row weight d. We prove that the parity check ma- trix of any regular cycle code can be put into a concatenation form of row-permuted block-diagonal matrices after row and column permutations if d is even, or, if d is odd and the code’s associated graph contains at least one spanning subgraph that consists of disjoint edges. Utilizing this structure enables parallel processing in linear-time encoding, and parallel pro- cessing in sequential belief-propagation decoding, which in- creases the throughput without compromising performance or complexity. Numerical results are presented to compare the code performance and the decoding complexity. Index Terms— Nonbinary, LDPC, cycle code, Galois field, graph theory 1. INTRODUCTION Gallager’s binary low-density parity-check (LDPC) codes [1] are excellent error-correcting codes that achieve performance close to the benchmark predicted by the Shannon capacity [2]. The extension of LDPC to non-binary Galois field GF(q) was first investigated empirically by Davey and Mackay over the binary-input AWGN channel [3]. Since then, nonbinary LDPC codes have been actively studied. The LDPC codes with column weight j =2 in their parity check matrix H are termed as cycle codes [4]. Al- though the distance properties of binary cycle codes are not as good as the LDPC codes of column weight j ≥ 3 [1], it has been shown in [5] that cycle GF(q) codes can achieve near-Shannon-limit performance as q increases. Further, nu- merical results in [5] demonstrate that cycle GF(q) codes can outperform other LDPC codes, including degree-distribution- optimized binary irregular LDPC codes. For high order fields q ≥ 64, the best GF(q)-LDPC codes decoded by belief prop- agation (BP) should be ultra sparse [3], with a good exam- ple being the cycle codes that have j =2. Reduced com- plexity algorithms for decoding a general LDPC code over GF(q) have been proposed in [6], [7]. A universal linear- complexity encoding algorithm for any cycle GF(q) code is This work is supported by the ONR grant N00014-07-1-0429 and the NSF grant ECCS-0725562. available in [8]. With the performance and implementation advantages, cycle GF(q) codes are very promising for practi- cal applications. In this paper, we study LDPC cycle codes whose check matrix has fixed row weight d, termed as d-regular cycle codes. Using graph theory, we prove that through row and column permutations the parity check matrix H of d-regular cycle GF(q) codes can always be put into a concatenation form of row-permuted block-diagonal matrices if d is even, or, if d is odd and the code’s associated graph contains at least one spanning subgraph that consists of disjoint edges. This equivalent representation brings several benefits. First, encoding for regular cycle GF(q) codes can be performed in parallel in linear time. Second, it enables parallel pro- cessing in sequential belief propagation decoding for regular cycle GF(q) codes, which improves the decoding through- put considerably without compromising the performance and complexity. It also reduces the storage of the check matrix H for encoding and decoding, and facilitates code design [13]. Simulation results confirm very good performance and re- duced decoding complexity of regular cycle GF(q) codes. 2. MAIN RESULTS ON CODE STRUCTURE A cycle GF(q) code is an LDPC code whose m × n parity check matrix H has weight j =2 for each column. As such, it can be represented by an associated graph G =(V,E) with m vertices V = {v 1 ,...,v m } and n edges E = {e 1 ,...,e n }, where each vertex represents a constraint node correspond- ing to a row of H, and each edge represents a variable node corresponding to a column of H [8]. If the cycle GF(q) code also has a fixed row weight d in H, the graph G is d-regular in that each vertex is exactly linked to d edges [9]. We call this code as regular cycle GF(q) code. Obviously we have 2n = dm for regular cycle GF(q) codes. We first introduce two definitions from graph theory [9]. • k-factor:A k-regular spanning subgraph of G that con- tains all the vertices is called a k-factor of G. Obviously, a 1-factor is a spanning subgraph that con- sists of disjoint edges, while a 2-factor is a spanning subgraph that consists of disjoint cycles. • k-factorable: a graph G is k-factorable if there are edge-disjoint k-factors G 1 ,G 2 ,...,G L such that G = G 1 ∪ G 2 ···∪ G L . 2961 1-4244-1484-9/08/$25.00 ©2008 IEEE ICASSP 2008