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.

翻译：R. W. Hamming于1950年发表了汉明码及球填充界。在过去的75年中，文献中已报道了关于球填充界、最小距离至多为8的有限域上距离最优线性码的无限族。然而，是否存在关于球填充界、具有任意大最小距离的有限域上距离最优线性码的无限族，这是编码理论中一个悬而未决75年的公开问题。本文的主要目标正是解决编码理论中这一长期存在的公开问题。作为副产品，本文提出了若干具有较小最小距离的距离最优二进制码的无限族。报道了两个二进制五重量码的无限族。同时提出了一些待解决的公开问题。