TY - JOUR
T1 - Equivalences among Z2^s-linear Hadamard codes
AU - Fernández-Córdoba, Cristina
AU - Vela, Carlos
AU - Villanueva, Mercè
N1 - Funding Information:
This work has been partially supported by the Spanish MINECO under Grant TIN2016-77918-P (AEI/FEDER, UE) , and by the Catalan AGAUR, Spain under Grant 2017SGR-00463 . The material in this paper was presented in part at the th International Workshop on Algebraic and Combinatorial Coding Theory in Svetlogorsk (Kaliningrad region), Russia, 2018 [7] .
Publisher Copyright:
© 2019 Elsevier B.V.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2020/3
Y1 - 2020/3
N2 - The Z2s -additive codes are subgroups of Z2s n, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s -linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s -linear Hadamard codes of length 2t is established by giving the exact number of such codes.
AB - The Z2s -additive codes are subgroups of Z2s n, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s -linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s -linear Hadamard codes of length 2t is established by giving the exact number of such codes.
KW - Classification
KW - Gray map
KW - Hadamard code
KW - Kernel
KW - Rank
KW - Z-additive code
UR - http://www.scopus.com/inward/record.url?scp=85075281433&partnerID=8YFLogxK
U2 - https://doi.org/10.1016/j.disc.2019.111721
DO - https://doi.org/10.1016/j.disc.2019.111721
M3 - Article
AN - SCOPUS:85075281433
VL - 343
JO - Discrete Mathematics
JF - Discrete Mathematics
SN - 0012-365X
IS - 3
M1 - 111721
ER -