On the Generalized Concatenated Construction for Codes in $${L_1}$$ and Lee Metrics

Abstract

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L1 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 2m we construct codes over $$\mathbb{Z}_4$$ with Lee distance 4 which under the Gray mapping yield extended binary perfect codes of length 2m+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.