Relaxation Bounds on the Minimum Pseudo-Weight of Linear Block Codes

TL;DR

This paper investigates the pseudo-weight spectrum of linear block codes, providing new bounds on the minimum pseudo-weight using polyhedral cone properties and optimization techniques, which enhance understanding of LP decoder performance.

Contribution

It introduces two novel lower bounds on the minimum pseudo-weight, one based on column weight and another via an optimization problem, applicable to longer codes.

Findings

Derived the pseudo-weight spectrum for some short codes

Established two lower bounds on minimum pseudo-weight

Bounds are computationally more tractable for longer codes

Abstract

Just as the Hamming weight spectrum of a linear block code sheds light on the performance of a maximum likelihood decoder, the pseudo-weight spectrum provides insight into the performance of a linear programming decoder. Using properties of polyhedral cones, we find the pseudo-weight spectrum of some short codes. We also present two general lower bounds on the minimum pseudo-weight. The first bound is based on the column weight of the parity-check matrix. The second bound is computed by solving an optimization problem. In some cases, this bound is more tractable to compute than previously known bounds and thus can be applied to longer codes.