On the generalized concatenated construction for codes in $L_1$ and Lee metrics

We consider a generalized concatenated construction for error-correcting codes over the $q$-ary alphabet in the modulus metric $L_1$ and Lee metric $L$. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length $2^m$ we construct codes over $\mathbb{Z}_4$ with Lee distance $4$ which under the Gray mapping yield extended binary perfect codes of length $2^{m+1}$ (with code distance $4$). We construct codes over $\mathbb{Z}_4$ of length $n$ with Lee distance $n$ which under the Gray mapping yield Hadamard matrices of order $2n$ (under the additional condition that an Hadamard matrix of order $n$ exists). The constructed new codes in the Lee metric are often better in their parameters than previously known ones; in particular, they are essentially better than previously constructed Astola codes.