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

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arxiv 1906.02621 v2 pith:TCVTKVBJ submitted 2019-06-06 math.CO cs.ITmath.IT

classification math.COcs.ITmath.IT
keywords codecodesdimensionknownuniqueadditionallyasymmetricautomorphism
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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.

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