$${\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]$$ Z p Z p [ v ] -additive cyclic codes are asymptotical
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Z pZ p [v]-additive cyclic codes are asymptotically good Xiaotong Hou1 · Jian Gao1 Received: 9 August 2020 / Revised: 15 November 2020 / Accepted: 17 November 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract We construct a class of Z p Z p [v]-additive cyclic codes, where p is a prime number and v 2 = v. We determine the asymptotic properties of the relative minimum distance and rate of this class of codes. We prove that, for any positive real number 0 < δ < 1 1 such that the p-ary entropy at k+l 2 δ is less than 2 , the relative minimum distance of the random code is convergent to δ and the rate of the random code is convergent to 1 k+l , where p, k, l are pairwise coprime positive integers. Keywords Z p Z p [v]-additive cyclic codes · Relative minimum distance · Rate · Asymptotically good codes Mathematics Subject Classification 94B05 · 94B65
1 Introduction Additive codes are important error-correcting codes in coding theory. In 1998, Delsarte firstly gave the definition of additive codes in [9]. Afterwards, many coding scientists paid their attentions on additive codes. Recently, Z2 Z4 -additive cyclic codes were studied impressed [1,6–8] including generator matrix, minimum generating sets, codes construction and so on. From then on, there are many papers on additive codes. Aydogdu et al. studied properties of Z2 Z2 [u]-additive cyclic codes and Z pr Z ps additive cyclic codes in [3,4], respectively. Diao et al. studied Z p Z p [v]-additive cyclic codes in [10]. Many good linear codes and quantum codes were constructed by Z p Z p [v]-additive cyclic codes.
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Jian Gao [email protected] Xiaotong Hou [email protected]
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School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, People’s Republic of China
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X. Hou, J. Gao
The asymptotic property is an important index of good codes. A class of codes is said to be asymptotically good if there exist a sequence of codes C1 , C2 , C3 , . . . with length n i , when n i → ∞, both the relative minimum distance and the rate of Ci are positively bounded from below. Assmus et al. had already studied the problem of the asymptotic property of cyclic codes in [2]. Afterwards, Kasami proved that quasi-cyclic codes of index 2 are asymptotically good in [15]. Bazzi et al. proved that random binary quasi-abelian codes of index 2 and random binary dihedral group codes are asymptotically good [5]. Martínez-Pérez et al. proved that self-dual doubly even 2quasi-cyclic transitive codes are asymptotically good [16]. Fan and Liu proved that the quasi-cyclic codes of fractional index between 1 and 2 are asymptotically good in [12]. Mi et al. proved that quasi-cyclic codes of fractional index are also asymptotically good [17]. In [14], we proved that Z4 -double cyclic codes are asymptotically good. In recent years, the asymptotic property of additive cyclic codes has been studied more widely. In [18], Shi et al. proved the existence of asymptotically good additive cyclic codes. Fan and Liu proved that Z2
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