Maximum distance separable (MDS) codes have the highest possible error-correcting capability among codes with the same length and size. Let \(\gamma \) be nonzero in \(\mathbb {F}_{2^m}.\) We consider all cyclic and \((1+u\gamma )\)-constacyclic codes of length \(2^s\) over \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) with their Lee distance and investigate all the cases whether the corresponding Gray images are MDS by giving an analogue of the Singleton bound for codes over \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) with the Lee distance through Gray map.

中文翻译：

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$$\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$$ F 2 m + u F 2 m 上关于 Lee 的最大距离可分重复根常常循环码距离

最大距离可分（MDS）码在相同长度和大小的码中具有最高可能的纠错能力。令\(\gamma \)在\(\mathbb {F}_{2^m}.\ )中不为零。我们考虑所有长度为\(2 ^s\)对\(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\)及其李距离，并调查所有情况下对应的灰度图像是否为 MDS给出具有通过 Gray 映射的 Lee 距离的\(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\)上的代码的单例界的类似物。