TY - JOUR
T1 - On completely regular and completely transitive supplementary codes
AU - Borges, J.
AU - Rifà, J.
AU - Zinoviev, V. A.
N1 - Funding Information:
This work has been partially supported by the Spanish grant TIN2016-77918-P, (AEI/FEDER, UE). The research of the third author of the paper was carried out at the IITP RAS at the expense of the Russian Fundamental Research Foundation (project No. 19-01-00364).
Funding Information:
This work has been partially supported by the Spanish grant TIN2016-77918-P , (AEI/ FEDER , UE). The research of the third author of the paper was carried out at the IITP RAS at the expense of the Russian Fundamental Research Foundation (project No. 19-01-00364 ).
Publisher Copyright:
© 2019 Elsevier B.V.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2020/3
Y1 - 2020/3
N2 - Given a parity-check matrix Hm of a q-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 2, then the supplementary code is also completely regular with covering radius at most 2. 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.
AB - Given a parity-check matrix Hm of a q-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 2, then the supplementary code is also completely regular with covering radius at most 2. 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.
KW - Completely regular codes
KW - Completely transitive codes
KW - Hamming codes
UR - http://www.scopus.com/inward/record.url?scp=85075218966&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2019.111732
DO - 10.1016/j.disc.2019.111732
M3 - Article
AN - SCOPUS:85075218966
SN - 0012-365X
VL - 343
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 3
M1 - 111732
ER -