Iterative List Decoding of LDPC Codes

Abstract

In the last decade two old methods for decoding linear block codes have gained considerable interest, iterative decoding as first described by Gallager in [1] and list decoding as introduced by Elias [2]. In particular iterative decoding of low-density parity-check (LDPC) codes, has been an important subject of research, see e.g. [3] and the references therein. "Good" LDPC codes are often randomly generated by computer, but recently codes with an algebraic or geometric structure have also been considered e.g [3] and [4]. The performance of the iterative decoder is typically studied by simulations and a theoretical analysis is more difficult. In this paper we combine the two decoding methods and present an iterative list decoding algorithm. In particular we apply this decoder to a class of LDPC codes from finite geometries and show that the (73, 45, 10) projective geometry code can be maximum likelihood decoded with low complexity. Moreover the list decoding approach enables us to give a complete analysis of the performance in this case. We also discuss the performance of the list bit-flipping algorithm for longer LDPC codes. We consider hard-decision iterative decoding of a binary (n, k, d) code. For a received vector, y, we calculate an extended syndrome s = Hy′, where H is a parity check matrix, but usually has more than n - k rows. Let r denote the length of the syndrome. The idea of using extended syndromes was also used in [5]. Our approach is based on one of the common versions of bit flipping (BF) [3], where the schedule is such that the syndrome is updated after each flip. In each step we flip a symbol chosen among those positions that reduce the weight of the extended syndrome, which we refer to briefly as the syndrome weight, u. A decoded word is reached when u = 0. In this paper we consider a variation of the common algorithm in the form of a tree-structured search. Whenever there is a choice between several bits, all possibilities are tried in succession. The result of the decoding algorithm is, in general, a list of codewords, obtained as leaves of the search tree. This form of the bit flipping algorithm leads naturally to a solution in the form of a list of codewords at the same smallest distance from y [6]. This list decoding concept is somewhat different from list decoding in the usual sense of all codewords within a certain distance from y. The paper is a continuation of [7] including results on long codes from [8].