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Discrete Mathematics Algorithms and Applications

Optimal quantum codes from additive skew cyclic codes

Nuh Aydın, Taher Abualrub

Published Mar 30, 2016• 14 citations
Publisher

Abstract

Additive codes received much attention due to their connections with quantum codes. On the other hand, skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quaternary field [Formula: see text], obtaining some structural properties of these codes. Moreover, we also show that many best known and optimal quantum codes can be obtained from this class.

DOI
10.1142/s1793830916500373

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Additive codes received much attention due to their connections with quantum codes. On the other hand, skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quaternary field [Formula: see text], obtaining some structural properties of these codes. Moreover, we also show that many best known and optimal quantum codes can be obtained from this class.
FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes and their application to quantum codes
Let ppp be a prime and Fq\mathbb{F}_qFq​ be the finite field of order q=pmq=p^mq=pm. In this paper, we study FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes where R=Fq+uFq\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_qR=Fq​+uFq​ with u2=uu^2=uu2=u. To characterize FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes, we first establish their algebraic structure and then discuss the dual-containing properties by considering a non-degenerate inner product. Further, we define a Gray map over FqR\mathbb{F}_q\mathcal{R}Fq​R and obtain their Fq\mathbb{F}_qFq​-Gray images. As an application, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing FqR\mathbb{F}_q\mathcal{R}Fq​R-skew cyclic codes and obtain many quantum codes with better parameters than the best-known codes available in the literature.
Skew cyclic codes over Bk = 𝔽pr [v1, …, vk]/〈vi2 − vi, vivj – vjvi〉 have been studied by Irwansyah et al in 2018. In this paper we use B1 = 𝔽p2 [v]/〈v2 – v〉 and the Gray map φ : B1→𝔽p22 defined by φ(a + bv) = (a, a + b) for all a + bv ∈ B1 and extend φ to Φ : B1n→𝔽p22n. If C is a skew cyclic code over B1, we can get Φ(C) ⊆ 𝔽p22n a skew cyclic code if n is odd and a skew 2-quasicyclic code if n is even. Then by using the map S introduced by Ezerman et al in 2011, we get a (4n, p2k, 2d) additive code over 𝔽p2 and we use this to construct an additive asymmetric quantum code.
We introduce an additive but not $\mathbb F$4-linear map SSS from $\mathbb F$4nto $\mathbb F$42n and exhibit some of its interestingstructural properties. If CCC is a linear $[n,k,d]4−code,then4-code, then4−code,thenS(C)$ is an additive $(2n,2$2k,$2d)$4-code. If CCC is an additivecyclic code then S(C)S(C)S(C) is an additive quasi-cyclic code of index 222.Moreover, if CCC is a module θ\thetaθ-cyclic code, a recentlyintroduced type of code which will be explained below, then S(C)S(C)S(C) isequivalent to an additive cyclic code if nnn is odd andto an additive quasi-cyclic code of index 222 if nnn is even.Given any $(n,M,d)$4-code CCC, the code S(C)S(C)S(C) isself-orthogonal under the trace Hermitian inner product. Sincethe mapping SSS preserves nestedness, it can be used as a toolin constructing additive asymmetric quantum codes.
In this paper, we introduce a class of dualcontaining 2 -generator quasi-cyclic codes over GF(4)G F(4)GF(4). By Hermitian construction, we show computation constructions of binary quantum codes with good parameters from quaternary quasi-cyclic codes. Furthermore, 3 new quantum codes [[42, 10, 9]], [[74,,36, 9]] and [[78, 40, 9]] are derived from Hermitian dual-containing quasi-cyclic codes by an exhaustive search, along with 24 newly derived codes that enhance the previously best-known lower bounds on minimum distance.
The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codeshttps://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.
A method of describing quaternary cyclic codes with 4-cyclotomic cosets of modulo odd n and generator polynomials is presented.A necessary and sufficient condition under which a quaternary cyclic code containing its dual code is given,thus generalizing the related known results on self-orthogonal simple root cyclic codes and on BCH codes to al quaternary cyclic codes.Using the relation of simple root cyclic codes and repeat root cyclic codes,the linear quantum codes that can be constructed from quaternary cyclic codes of short lengths are determined.