The Treewidth of MDS and Reed-Muller Codes

TL;DR

This paper establishes that for MDS and Reed-Muller codes, the treewidth equals trelliswidth, providing explicit formulas for their constraint complexity, which aids in understanding their decoding complexity.

Contribution

It proves that for MDS and Reed-Muller codes, treewidth equals trelliswidth, leading to explicit formulas for their constraint complexity.

Findings

Treewidth equals trelliswidth for MDS and Reed-Muller codes.

Exact expressions for the treewidth of these codes are derived.

These are the only known explicit formulas for algebraic code treewidth.

Abstract

The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parametrization of the maximum-likelihood decoding complexity for linear codes. In this paper, we prove the surprising fact that for maximum distance separable codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code, is defined to be the least constraint complexity (or branch complexity) of any of its trellis realizations. From this, we obtain exact expressions for the treewidth of these codes, which constitute the only known explicit expressions for the treewidth of algebraic codes.