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

TL;DR

This paper investigates how the structure of proto- and Tanner-graphs influences the upper bounds on the minimum Hamming distance of quasi-cyclic LDPC codes, providing explicit constructions and extending results to convolutional codes.

Contribution

It establishes new upper bounds on the minimum Hamming distance of QC LDPC codes based on graph parameters and offers explicit constructions that approach these bounds, also relating to convolutional codes.

Findings

Upper bounds depend on graph parameters like girth and degrees.

Explicit QC LDPC code constructions achieve near the upper bounds.

Results extend to the free Hamming distance of convolutional codes.

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.