An efficient algorithm for determining the girth and ACE distributions in LDPC code Tanner graphs

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We propose a simple and efficient algorithm for evaluating the girth and approximate cycle extrinsic message degree (ACE) of all short cycles through symbol nodes in short-to-medium length low-density parity-check (LDPC) code Tanner graphs. The algorithm uses single-edge tree-apex (SETA) subgraph expansions to find all the short cycles in a code graph in a direct manner, unlike existing algorithms which use message-passing algorithms or trellis representations of code graphs. The complexity of the algorithm, when applied to a parity-check matrix H of dimension m × n, is O(n m). Due to the fact that both girth and ACE are features of cycles, we adopt parameters similar to those used for girth descriptions for ACE descriptions. Accordingly, we define the ‘local ACE’ at a symbol node or the ‘symbol node ACE’, as the minimum ACE of the short cycles through the symbol node in the graph. This basic parameter enables the local ACE distribution, minimum and maximum local ACE, modal ACE, and the average ACE of an LDPC code graph to be determined using SETA subgraph expansions. The average ACE of an LDPC code graph is a concise single-valued measure of the overall connectivity of cycles in the graph. The proposed algorithm determines the local girth and ACE distributions in irregular short length LDPC codes in less time than existing algorithms. The algorithm is not intended for quasi-cyclic (QC) LDPC codes.