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.

中文翻译：

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$ [46，9，20] _2 $代码是唯一的

尺寸最大为8的所有二进制线性代码的最小距离是已知的。维度9的最小开放情况为长度$ n = 46 $，已知边界为$ 19 \ le d \ le 20 $。在这里，我们提供$ [46,9,20] _2 $代码并显示其唯一性。有趣的是，这种独特的最优代码是不对称的，即，它具有一个琐碎的自同构群。此外，我们显示了不存在$ [47,10,20] _2 $和$ [85,9,40] _2 $代码。