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

11/01/2017

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by   Alexander M. Romanov, et al.

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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.