We consider the problem of describing the typical (possibly) non-linear code of minimum distance bounded from below over a large alphabet. We concentrate on block codes with the Hamming metric and on subspace codes with the injection metric. In sharp contrast with the behavior of linear block codes, we show that the typical non-linear code in the Hamming metric of cardinality $q^{n-d+1}$ is far from having minimum distance $d$, i.e., from being MDS. We also give more precise results about the asymptotic proportion of block codes with good distance properties within the set of codes having a certain cardinality. We then establish the analogous results for subspace codes with the injection metric, showing also an application to the theory of partial spreads in finite geometry.
翻译:我们考虑了描述典型(可能)非线性最低距离代码的问题,该代码与下方以大字母分隔。我们集中关注哈明度的区块代码和注入度的子空间代码。与线性区块代码的行为形成鲜明对比的是,我们显示,哈明度基度$q ⁇ n-d+1美元中的典型非线性代码远非最低距离,即远非最低距离$d$,即不是MDS。我们还给出了一套具有一定基本特征的代码中具有良好距离属性的区块代码的无症状比例的更精确结果。我们随后为亚空间代码与注入度度代码建立了类似的结果,也显示了对有限几何测量中部分扩展理论的应用。