On full-rank perfect codes over finite fields

We propose a construction of full-rank $q$-ary $1$-perfect codes over finite fields. This is a generalization of the construction of full-rank binary $1$-perfect codes by Etzion and Vardy (1994). The properties of the $i$-components of q-ary Hamming codes are investigated and the construction of full-rank $q$-ary $1$-perfect codes is based on these properties. The switching construction of $1$-perfect codes is generalized for the $q$-ary case. We propose a generalization of the notion of $i$-component of a $1$-perfect code and introduce the concept of an $(i,\sigma)$-component of $q$-ary $1$-perfect codes. We also present a generalization of the Lindström–Schönheim construction of $q$-ary $1$-perfect codes and provide a lower bound for the number of pairwise distinct $q$-ary $1$-perfect codes of length $n$. Bibliogr. 16.