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arXiv:2007.03963

Additive Conjucyclic Codes over $\F_{q^2}$: Trace Correspondence and Applications to Quantum Codes

Jingjie Lv, Ruihu Li, Ruihu Li, Hou, Hanxu

Published Jul 8, 2020• 0 citations
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Abstract

Additive conjucyclic codes over $\F_{q^2}$ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $\F_q$-linear isomorphism between $\F_{q^2}^n$ and $\F_q^{2n}$. This correspondence shows that additive conjucyclic codes of length $n$ correspond bijectively to $q$-ary linear cyclic codes of length $2n$, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $\F_{q^2}^n$, which is shown to be compatible with the symplectic inner product on $\F_q^{2n}$ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $q$-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $q$-ary QECCs from additive conjucyclic codes, together with an illustrative example.

DOI
10.48550/arxiv.2007.03963

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Additive conjucyclic codes over \Fq2\F_{q^2}\Fq2​ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based \Fq\F_q\Fq​-linear isomorphism between \Fq2n\F_{q^2}^n\Fq2n​ and \Fq2n\F_q^{2n}\Fq2n​. This correspondence shows that additive conjucyclic codes of length nnn correspond bijectively to qqq-ary linear cyclic codes of length 2n2n2n, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on \Fq2n\F_{q^2}^n\Fq2n​, which is shown to be compatible with the symplectic inner product on \Fq2n\F_q^{2n}\Fq2n​ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct qqq-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for qqq-ary QECCs from additive conjucyclic codes, together with an illustrative example.
Additive conjucyclic codes over $ \mathbb{F}_{q^2} $ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $ \mathbb{F}_q $-linear isomorphism between $ \mathbb{F}_{q^2}^n $ and $ \mathbb{F}_q^{2n} $. This correspondence shows that additive conjucyclic codes of length $ n $ correspond bijectively to $ q $-ary linear cyclic codes of length $ 2n $, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $ \mathbb{F}_{q^2}^n $, which is shown to be compatible with the symplectic inner product on $ \mathbb{F}_q^{2n} $ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $ q $-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $ q $-ary QECCs from additive conjucyclic codes, together with an illustrative example.
Abstract As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except Fq2{\mathbb {F}}_{q^2}Fq2​ Fq2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) , where r≥2r\ge 2r≥2 r≥2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of lengthnover GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) and the family of linear cyclic codes of length 2nover Z2r{\mathbb {Z}}_{2^r}Z2r​ Z2r . This correspondence helps to obtain additive conjucyclic codes over GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) via known linear cyclic codes over Z2r{\mathbb {Z}}_{2^r}Z2r​ Z2r . We prove that the trace dual CTr{\mathscr {C}}^{Tr}CTr CTr of an additive conjucyclic code C{\mathscr {C}}C C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z2r{\mathbb {Z}}_{2^r}Z2r​ Z2r contained in additive conjucyclic codes over GR(2r,2)GR(2^r,2)GR(2r,2) GR(2r,2) . Last but not least, we explicitly derive the generator matrices for these codes.
Conjucyclic codes are part of a family of codes that includes cyclic, constacyclic, and quasi-cyclic codes, among others. Despite their importance in quantum error correction, they have not received much attention in the literature. This paper focuses on additive conjucyclic (ACC) codes over $ \mathbb{F}_4 $ and investigates their properties. Specifically, we derive the duals of ACC codes by using a trace inner product and obtain the trace hull and its dimension. Also, we establish a necessary and sufficient condition for an additive code to have a complementary dual (ACD). Additionally, we identify a necessary condition for an additive conjucyclic complementary pair of codes over $ \mathbb{F}_4 $. Furthermore, we show that the trace code of an ACC code is cyclic and provide a condition for the trace code of an ACC code to be LCD. To demonstrate the practical application of our findings, we construct some good entanglement-assisted quantum error-correcting (EAQEC) codes using the trace code of ACC codes.
(Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic codes over Fql\mathbb{F}_q^lFql​ and their applications in the construction of quantum codes
In this article, for a finite field Fq\mathbb{F}_qFq​ and a natural number l,l,l, let R\mathcal{R}R denote the product ring Fql.\mathbb{F}_q^l.Fql​. Firstly, for an automorphism ΘΘΘ of R,\mathcal{R},R, a ΘΘΘ-derivation ΔΘΔ_ΘΔΘ​ of R\mathcal{R}R and for a unit a\mathbf{a}a in R,\mathcal{R},R, we study (Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic codes over R.\mathcal{R}.R. In this direction, we give an algebraic characterization of a (Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic code over R\mathcal{R}R, determine its generator polynomial, and find its decomposition over Fq.\mathbb{F}_q.Fq​. Secondly, we give a necessary and sufficient condition for a (Θ,0,a)(Θ, 0, \mathbf{a})(Θ,0,a)-cyclic code to be Euclidean dual-containing code over R.\mathcal{R}.R. Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over Fq\mathbb{F}_qFq​ as Gray images of (Θ,ΔΘ,a)(Θ, Δ_Θ, \mathbf{a})(Θ,ΔΘ​,a)-cyclic codes over R.\mathcal{R}.R. Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.
A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,one method for constructing quantum constacyclic codes is presented by finding self-orthogonal classical constacyclic codes over the field GF(4) under a trace inner product.Finally,some examples are given.As an application of this method,quantum Hamming codes are constructed.This class of quantum codes includes well-known quantum cyclic codes.
In this article, for a finite field $ \mathbb{F}_q $ and a natural number $ l, $ let $ \mathcal{R} $ denote the product ring $ \mathbb{F}_q^l. $ Firstly, for an automorphism $ \Theta $ of $ \mathcal{R}, $ a $ \Theta $-derivation $ \Delta_\Theta $ of $ \mathcal{R} $ and for a unit $ \mathit{\boldsymbol{a}} $ in $ \mathcal{R}, $ we study $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic codes over $ \mathcal{R}. $ In this direction, we give an algebraic characterization of a $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic code over $ \mathcal{R} $, determine its generator polynomial, and find its decomposition over $ \mathbb{F}_q. $ Secondly, we give a necessary and sufficient condition for a $ (\Theta, 0, \mathit{\boldsymbol{a}}) $-cyclic code to be Euclidean dual-containing code over $ \mathcal{R}. $ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over $ \mathbb{F}_q $ as Gray images of $ (\Theta, \Delta_\Theta, \mathit{\boldsymbol{a}}) $-cyclic codes over $ \mathcal{R}. $ Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.