Abstract
Aydogdu et al. studied the standard forms of generator and parity-check matrices of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, and presented generators of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes (Finite Fields Appl. 48: 241-260, 2017). In this paper, we investigate some other useful properties of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, including asymptotically good $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive complementary dual codes. The present paper can be viewed as a necessary complementary part of Aydogdu’s work.
Highlights
In recent years, coding scholars proposed a class of codes over additive structures, which are called additive codes [1]–[10]
[2], it is interesting to study the asymptotic properties of Z2Z2[u3]-additive cyclic codes and the structural properties of Z2Z2[u3]-additive complementary duality (ACD) codes
We construct a class of Z2Z2[u3]-additive cyclic codes. We prove that this class of Z2Z2[u3]-additive cyclic codes are asymptotically good
Summary
In recent years, coding scholars proposed a class of codes over additive structures, which are called additive codes [1]–[10]. To be the necessary complementary part of the work [2], it is interesting to study the asymptotic properties of Z2Z2[u3]-additive cyclic codes and the structural properties of Z2Z2[u3]-ACD codes. The relative minimum distance and the rate of Z2Z2[u3]-additive cyclic code C are denoted by (C ). A. A CLASS OF Z2Z2[u3]-ADDITIVE CYCLIC CODES Let Rkm = Z2[x]/ xkm − 1 , Rlm = Z2[x]/ xlm − 1 , Rlm = Z2[u3][x]/ xlm − 1 , where m, k, l are positive integers such that gcd(m, 2) = 1 and 2, k, l are pairwise prime. In terms of the pairwise coordinate addition and the scalar multiplication by the elements of Rklm = Z2[x]/ xklm − 1 , the Rkm × Rlm forms an Rklmmodule.
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