Characterization of Graph-cover Pseudocodewords of Codes over $F_3$

TL;DR

This paper characterizes the fundamental cone of graph-cover pseudocodewords for codes over the finite field F_3, enhancing understanding of LP decoding pseudocodewords beyond binary codes.

Contribution

It provides a new characterization of the fundamental cone for ternary codes, extending previous binary code results to larger alphabets.

Findings

Characterization of the fundamental cone for codes over F_3

Insights into the structure of graph-cover pseudocodewords

Potential implications for decoding performance analysis

Abstract

Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becom is more involved. In this work, a characterization of the fundamental cone for codes over $F_3$ is discussed.