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

Hao Chen; Conghui Xie; Cunsheng Ding

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arXiv · 2510.22259

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

Hao Chen·Conghui Xie·Cunsheng Ding

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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.

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Hao Chen, Conghui Xie, Cunsheng Ding. 2025-10-25. Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound. https://arxiv.org/abs/2510.22259

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