Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds

Abstract

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close to) the respective minimum Hamming distance upper bounds. Because of the tight algebraic connection between QC codes and convolutional codes, we can state similar results for the free Hamming distance of convolutional codes. In fact, some QC code statements are established by first proving the corresponding convolutional code statements and then using a result by Tanner that says that the minimum Hamming distance of a QC code is upper bounded by the free Hamming distance of the convolutional code that is obtained by "unwrapping" the QC code.Comment: To appear in IEEE Transactions on Information Theory. Changes compared to v1: some convolutional code results have been added; some incompleteness issues with some of the proofs have been corrected; a typo in one of the parity-check matrices has been corrected (i.e., an entry of H"(x) in Example 28 of v1 needs to be changed so that d_min=56 as written there, cf. \hat H(x) in Example 29 of v2