On non-full-rank perfect codes over finite fields
Information Theory
2017-04-11 v1 math.IT
Abstract
The paper deals with the perfect 1-error correcting codes over a finite field with elements (briefly -ary 1-perfect codes). We show that the orthogonal code to the -ary non-full-rank 1-perfect code of length is a -ary constant-weight code with Hamming weight equals to where is any natural number not less than two. We derive necessary and sufficient conditions for -ary 1-perfect codes of non-full rank. We suggest a generalization of the concatenation construction to the -ary case and construct the ternary 1-perfect codes of length 13 and rank 12.
Keywords
Cite
@article{arxiv.1704.02627,
title = {On non-full-rank perfect codes over finite fields},
author = {Alexander M. Romanov},
journal= {arXiv preprint arXiv:1704.02627},
year = {2017}
}