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

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 = (q1 − 1)/(q − 1) and m = (q2 − 1)/(q − 1), where s1, s2 are natural numbers not less than two. This construction allows us to also construct q-ary codes with parameters (qs1+s2 , q s+s−(s1+s2)−1, 3)q and can be regarded as a q-ary analogue of the well-known Phelps construction.