Maximum distance separable repeated-root constacyclic codes over \(\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}\) with respect to the Lee distance

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.