Woven Graph Codes: Asymptotic Performances and Examples

TL;DR

This paper investigates woven graph codes constructed from block and convolutional codes, demonstrating their asymptotic performance, existence of codes meeting bounds, and providing examples and encoding methods.

Contribution

It introduces a new class of woven graph codes, analyzes their asymptotic properties, and presents explicit examples and encoding procedures.

Findings

Codes satisfying VG and Costello bounds exist in the ensemble.

A connection between bipartite graphs and tailbiting codes is established.

An example of a woven graph code with specific parameters is provided.

Abstract

Constructions of woven graph codes based on constituent block and convolutional codes are studied. It is shown that within the random ensemble of such codes based on $s$-partite, $s$-uniform hypergraphs, where $s$ depends only on the code rate, there exist codes satisfying the Varshamov-Gilbert (VG) and the Costello lower bound on the minimum distance and the free distance, respectively. A connection between regular bipartite graphs and tailbiting codes is shown. Some examples of woven graph codes are presented. Among them an example of a rate $R_{\rm wg}=1/3$ woven graph code with $d_{\rm free}=32$ based on Heawood's bipartite graph and containing $n=7$ constituent rate $R^{c}=2/3$ convolutional codes with overall constraint lengths $\nu^{c}=5$ is given. An encoding procedure for woven graph codes with complexity proportional to the number of constituent codes and their overall constraint length $\nu^{c}$ is presented.