A generalized concatenation construction for q-ary 1-perfect codes

Abstract

We consider perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). In this paper, a generalized concatenation construction for $q$-ary 1-perfect codes is presented that allows us to construct $q$-ary 1-perfect codes of length $(q-1)nm+n+m$ from the given $q$-ary 1-perfect codes of length $n=(q^{s_1}-1)/(q-1)$ and $m=(q^{s_2}-1)/(q-1)$, where $s_1,s_2$ are natural numbers not less than two. This construction allows us to also construct $q$-ary codes with parameters $(q^{s_1+s_2},q^{q^{s_1+s_2}-(s_1+s_2)-1},3{)}_q$ and can be regarded as a $q$-ary analogue of the well-known Phelps construction.

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