Optimal binary linear codes of length ⩽30

Abstract

For n⩽30, we determine when an [n,k,d] (binary linear) code exists, and we classify optimal [n,k,d] codes, where by optimal we mean that no [n−1,k,d], [n+1,k+1,d] , or [n+1,k,d+1] code exists. Subsumed therein are the following nontrivial new results: there are exactly six [24,7,10] codes (discovered independently by Kapralov, Enumeration of the binary linear [24,7,10] codes, in: Proceedings of the Fifth International Workshop on Algebraic and Combinatorial Coding Theory, Unicorn, Shumen, Bulgaria, 1996, pp. 151–156.), exactly 11 [28,10,10] codes, no [29,11,10] code, exactly one [28,14,8] code, and no [29,15,8] code. We also show that there are exactly two [32,11,12] codes. All the results, new and old, are presented as a proof in the author's computer language Split, whose execution takes about 11 h on a 1996-era desktop computer, exclusive of a single line in the [28,10,10] classification, which takes 115 h.