We investigate the asymptotic density of error-correcting codes with good distance properties and prescribed linearity degree, including (sub)linear and nonlinear codes. We focus on the general setting of finite translation-invariant metric spaces, and then specialize our results to the Hamming metric, to the rank metric, and to the sum-rank metric. Our results show that the asymptotic density of codes heavily depends on the imposed linearity degree and the chosen metric.

中文翻译：

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平移不变度量空间中不同线性度的码密度

我们研究了具有良好距离特性和规定线性度的纠错码的渐近密度，包括（子）线性和非线性码。我们专注于有限平移不变度量空间的一般设置，然后将我们的结果专门化为汉明度量、秩度量和秩和度量。我们的结果表明，代码的渐近密度在很大程度上取决于强加的线性度和所选择的度量。