The $[46,9,20{]}_2$ code is unique

Abstract

The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case for dimension nine is length $n=46$ with known bounds $19\le d\le 20$. Here we present a $[46,9,20{]}_2$ code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Additionally, we show the non-existence of $[47,10,20{]}_2$ and $[85,9,40{]}_2$ codes.

7 pages, 3 tables, typos corrected

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