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 $n =(q^{s_1} - 1) / (q - 1)$ 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

The existence of perfect codes in Doob graphs

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