Minimizing the alphabet size in codes with restricted error sets

TL;DR

This paper investigates how to design error-correcting codes with minimal alphabet size for specific error patterns, linking combinatorial hypergraph properties to code parameters, and demonstrating advantages of non-linear codes.

Contribution

It establishes a novel connection between minimum alphabet size and hypergraph properties, providing bounds and showing non-linear codes can outperform linear ones in this context.

Findings

Bounds on minimum alphabet size based on hypergraph properties

Non-linear codes can have advantages over linear codes

Relation between decoding error probability and hypergraph coloring

Abstract

This paper focuses on error-correcting codes that can handle a predefined set of specific error patterns. The need for such codes arises in many settings of practical interest, including wireless communication and flash memory systems. In many such settings, a smaller field size is achievable than that offered by MDS and other standard codes. We establish a connection between the minimum alphabet size for this generalized setting and the combinatorial properties of a hypergraph that represents the prespecified collection of error patterns. We also show a connection between error and erasure correcting codes in this specialized setting. This allows us to establish bounds on the minimum alphabet size and show an advantage of non-linear codes over linear codes in a generalized setting. We also consider a variation of the problem which allows a small probability of decoding error and relate it to an approximate version of hypergraph coloring.