On completely regular and completely transitive codes derived from Hamming codes
Abstract
Given a parity-check matrix of a -ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the supplementary code of the other one. We obtain that if one of these codes is a Hamming code, then the supplementary code is completely regular and completely transitive. If one of the codes is completely regular with covering radius , then the supplementary code is also completely regular with covering radius at most . Moreover, in this case, either both codes are completely transitive, or both are not. With this technique, we obtain infinite families of completely regular and completely transitive codes which are quasi-perfect uniformly packed.
Cite
@article{arxiv.1902.07628,
title = {On completely regular and completely transitive codes derived from Hamming codes},
author = {J. Borges and J. Rifà and V. A. Zinoviev},
journal= {arXiv preprint arXiv:1902.07628},
year = {2019}
}