Full-Rank Perfect Codes over Finite Fields
Abstract
In this paper, we propose a construction of full-rank q-ary 1-perfect codes over finite fields. This construction is a generalization of the Etzion and Vardy construction of full-rank binary 1-perfect codes (1994). Properties of 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 are generalized for the q-ary case. We give a generalization of the concept of i-component of 1-perfect codes and introduce the concept of (i,{\sigma})-components of q-ary 1-perfect codes. We also present a generalization of the Lindstr\"om and Sch\"onheim construction of q-ary 1-perfect codes and provide a lower bound on the number of pairwise distinct q-ary 1-perfect codes of length n.
Keywords
Cite
@article{arxiv.1310.1174,
title = {Full-Rank Perfect Codes over Finite Fields},
author = {Alexander M. Romanov},
journal= {arXiv preprint arXiv:1310.1174},
year = {2016}
}
Comments
8 pages; submitted to IEEE Transactions on Information Theory