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Discrete Mathematics Algorithms and Applications

On quantum codes via cyclic codes of arbitrary length over 𝔽4 + u𝔽4

Amit Sharma, Ramakrishna Bandi, Maheshanand Bhaintwal

Published Mar 2, 2018• 3 citations
Publisher

Abstract

In this paper, we study cyclic codes over [Formula: see text]. A necessary and sufficient condition for a cyclic code over [Formula: see text] to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over [Formula: see text] containing its dual is a linear code over [Formula: see text] which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over [Formula: see text]. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over [Formula: see text]. Some examples have been given to illustrate the results.

DOI
10.1142/s1793830918500337

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In this paper, we study cyclic codes over [Formula: see text]. A necessary and sufficient condition for a cyclic code over [Formula: see text] to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over [Formula: see text] containing its dual is a linear code over [Formula: see text] which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over [Formula: see text]. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over [Formula: see text]. Some examples have been given to illustrate the results.
In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.
In this paper, we extend the results given in [3] to a nonchain ring Rp=Fp+vFp+⋯+vp−1FpR_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_pRp​=Fp​+vFp​+⋯+vp−1Fp​ , where vp=vv^p=vvp=v and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring RpR_pRp​ and study the structure of their duals. We classify cyclic codes containing their duals over RpR_pRp​ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map π{\pi}π defined in [4], we give the parameters of the quantum codes of length pn over Fp{\mathbb{F}}_pFp​ which are obtained from cyclic codes over RpR_pRp​ . Finally, we illustrate the results by giving some examples.
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In this paper, we use 2-cyclotomic cosets of modulo n and generator polynomials to describe binary cyclic codes of length N=2αn with n odd. We discuss the conditions under which two cyclic codes [Formula: see text] and [Formula: see text] can be used to construct quantum codes by CSS construction or Steane's construction. Using the results of Chen, Promhouse and Tavares, and Castagnoli et al., we study the quantum codes that can be constructed from binary cyclic codes of length N=2αn with n odd and n≤99, and α≤2. We find that except the quantum codes constructed by Steane, there are also some very interesting quantum codes constructed from repeated-root cyclic codes, and some of the quantum codes constructed by Steane can be improved.