On full-rank perfect codes over finite fields
A. M. Romanov · Journal of Applied and Industrial Mathematics · 2016
We propose a construction of full-rank q -ary 1-perfect codes. 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 to the q -ary case. We propose a generalization of the notion of an i -component of a 1-perfect code and introduce the concept of an ( i , σ )-component of a q -ary 1-perfect code. 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 .