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Optimal binary linear codes of length \(\leq 30\). (English) Zbl 0976.94031

Summary: For \(n\leq 30\), 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 \([n-1,k,d]\), \([n-1,k,d]\), \([n+1,k+1,d]\), 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 S. N. Kapralov [Proc. Fifth Workshop on Algebraic and Combinatorial Coding Theory, Unicorn, Shumen, Bulgaria, 1996, 151-156 (1996; Zbl 0907.94030)[, 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.

MSC:

94B05 Linear codes (general theory)
51E22 Linear codes and caps in Galois spaces

Citations:

Zbl 0907.94030
Full Text: DOI