Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound

TL;DR

This paper resolves a 75-year-old open problem by constructing infinite families of distance-optimal binary linear codes with arbitrarily large minimum distances relative to the sphere packing bound, and introduces new binary code families.

Contribution

It provides the first known infinite families of distance-optimal binary linear codes with arbitrarily large minimum distances, solving a longstanding open problem in coding theory.

Findings

Constructed infinite families of distance-optimal binary codes with large minimum distances.

Presented several infinite families of binary codes with small minimum distances.

Reported two infinite families of binary five-weight codes.

Abstract

R. W. Hamming published the Hamming codes and the sphere packing bound in 1950. In the past 75 years, infinite families of distance-optimal linear codes over finite fields with minimum distance at most 8 with respect to the sphere packing bound have been reported in the literature. However, it is a 75-year-old open problem in coding theory whether there is an infinite family of distance-optimal linear codes over finite fields with arbitrarily large minimum distance with respect to the sphere packing bound. This main objective of this paper is to settle this long-standing open problem in coding theory. As by-products, several infinite families of distance-optimal binary codes with small minimum distances are presented. Two infinite families of binary five-weight codes are reported. Some open problems are also proposed.