2 research outputs found

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

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    We consider perfect 1-error correcting codes over a finite field with qq elements (briefly qq-ary 1-perfect codes). In this paper, a generalized concatenation construction for qq-ary 1-perfect codes is presented that allows us to construct qq-ary 1-perfect codes of length (qβˆ’1)nm+n+m(q - 1)nm + n + m from the given qq-ary 1-perfect codes of length n=(qs1βˆ’1)/(qβˆ’1)n =(q^{s_1} - 1) / (q - 1) and m=(qs2βˆ’1)/(qβˆ’1)m = (q^{s_2} - 1) / (q - 1), where s1,s2s_1, s_2 are natural numbers not less than two. This construction allows us to also construct qq-ary codes with parameters (qs1+s2,qqs1+s2βˆ’(s1+s2)βˆ’1,3)q(q^{s_1 + s_2}, q^{q^{s_1 + s_2} - (s_1 + s_2) - 1}, 3)_q and can be regarded as a qq-ary analogue of the well-known Phelps construction

    The existence of perfect codes in Doob graphs

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    We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices
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