TY - JOUR
T1 - On the constructions of ZpZp2-linear generalized Hadamard codes.
AU - Bhunia, Dipak K.
AU - Fernández-Córdoba, Cristina
AU - Villanueva, Mercè
N1 - Publisher Copyright:
© 2022 The Author(s)
PY - 2022/10
Y1 - 2022/10
N2 - The ZpZp2-additive codes are subgroups of Zpα1×Zp2α2, and can be seen as linear codes over Zp when α2=0, Zp2-additive codes when α1=0, or Z2Z4-additive codes when p=2. A ZpZp2-linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2-additive code. In this paper, we generalize some known results for ZpZp2-linear GH codes with p=2 to any p≥3 prime when α1≠0. First, we give a recursive construction of ZpZp2-additive GH codes of type (α1,α2;t1,t2) with t1,t2≥1. We also present many different recursive constructions of ZpZp2-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z4-linear GH codes, when p≥3 prime, the Zp2-linear GH codes are not included in the family of ZpZp2-linear GH codes with α1≠0. Indeed, we observe that the constructed codes are not equivalent to the Zps-linear GH codes for any s≥2.
AB - The ZpZp2-additive codes are subgroups of Zpα1×Zp2α2, and can be seen as linear codes over Zp when α2=0, Zp2-additive codes when α1=0, or Z2Z4-additive codes when p=2. A ZpZp2-linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2-additive code. In this paper, we generalize some known results for ZpZp2-linear GH codes with p=2 to any p≥3 prime when α1≠0. First, we give a recursive construction of ZpZp2-additive GH codes of type (α1,α2;t1,t2) with t1,t2≥1. We also present many different recursive constructions of ZpZp2-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z4-linear GH codes, when p≥3 prime, the Zp2-linear GH codes are not included in the family of ZpZp2-linear GH codes with α1≠0. Indeed, we observe that the constructed codes are not equivalent to the Zps-linear GH codes for any s≥2.
KW - Generalized Hadamard code
KW - Gray map
KW - ZZ-linear code
UR - http://www.scopus.com/inward/record.url?scp=85134834061&partnerID=8YFLogxK
U2 - 10.1016/j.ffa.2022.102093
DO - 10.1016/j.ffa.2022.102093
M3 - Article
AN - SCOPUS:85134834061
SN - 1071-5797
VL - 83
JO - Finite Fields and Their Applications
JF - Finite Fields and Their Applications
M1 - 102093
ER -