On the Minimal Pseudo-Codewords of Codes from Finite Geometries

TL;DR

This paper investigates minimal pseudo-codewords in codes from finite geometries, revealing that non-multiple minimal pseudo-codewords have higher pseudo-weight, which benefits decoding performance.

Contribution

It provides the first analysis of minimal pseudo-codewords in finite geometry codes, highlighting their impact on LP and iterative decoding.

Findings

Minimal pseudo-codewords not multiples of codewords have higher pseudo-weight.

Finite geometry codes exhibit a property beneficial for decoding.

Numerical results support the theoretical observations.

Abstract

In order to understand the performance of a code under maximum-likelihood (ML) decoding, it is crucial to know the minimal codewords. In the context of linear programming (LP) decoding, it turns out to be necessary to know the minimal pseudo-codewords. This paper studies the minimal codewords and minimal pseudo-codewords of some families of codes derived from projective and Euclidean planes. Although our numerical results are only for codes of very modest length, they suggest that these code families exhibit an interesting property. Namely, all minimal pseudo-codewords that are not multiples of a minimal codeword have an AWGNC pseudo-weight that is strictly larger than the minimum Hamming weight of the code. This observation has positive consequences not only for LP decoding but also for iterative decoding.