Generalized Singleton Type Upper Bounds

TL;DR

This paper introduces new upper bounds on the sizes and lengths of various error-correcting codes, including list-decodable, Singleton defect, insertion-deletion, and locally recoverable codes, often surpassing existing bounds for large code lengths.

Contribution

It provides generalized Singleton type upper bounds for multiple code classes, improving upon classical bounds and extending their applicability to insertion-deletion and locally recoverable codes.

Findings

Stronger bounds than Griesmer for large code lengths

New bounds for insertion-deletion codes surpass Singleton bounds

Upper bounds on lengths of optimal locally recoverable codes

Abstract

In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new Singleton type upper bounds on the sizes of codes with a given minimum Hamming distance. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes or list-decodable codes attaining the generalized Singleton bound are given. As an application of our generalized Singleton type upper bounds on Hamming metric error-correcting codes, the generalized Singleton type upper bounds on insertion-deletion codes are given, which are much stronger than the direct Singleton bound for insertion-deletion codes when the lengths are large. We also give upper bounds on the lengths of small dimension optimal locally recoverable codes and small dimension optimal $(r, \delta)$ locally recoverable codes with any fixed given minimum distance.