LP Pseudocodewords of Cycle Codes are Half-Integral

TL;DR

This paper investigates the structure of LP pseudocodewords in cycle codes, showing they are constrained to half-integral values, which enhances understanding of LP decoding performance.

Contribution

It establishes necessary conditions for LP pseudocodewords in cycle codes, proving they are limited to values of 0, 0.5, or 1, revealing their half-integral nature.

Findings

LP pseudocodewords in cycle codes are half-integral

Components of pseudocodewords are only 0, 0.5, or 1

Provides conditions characterizing pseudocodewords in cycle codes

Abstract

In his Ph.D. disseration, Feldman and his collaborators define the linear programming decoder for binary linear codes, which is a linear programming relaxation of the maximum-likelihood decoding problem. This decoder does not, in general, attain maximum-likelihood performance; however, the source of this discrepancy is known to be the presence of non-integral extreme points (vertices) within the fundamental polytope, vectors which are also called nontrivial linear programming pseudocodewords. Restricting to the class of cycle codes, we provide necessary conditions for a vector to be a linear programming pseudocodeword. In particular, the components of any such pseudocodeword can only assume values of zero, one-half, or one.