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arXiv:2406.13943

New QEC codes and EAQEC codes from repeated-root cyclic codes of length $2^rp^s$

Lanqiang Li, Ziwen Cao, Tingting Wu, Li Liu

Published Jun 20, 2024• 0 citations
arXivHTMLPDF

Abstract

Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length $2^rp^s$, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.

DOI
10.48550/arxiv.2406.13943

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New QEC codes and EAQEC codes from repeated-root cyclic codes of length 2rps2^rp^s2rps
Let ppp be an odd prime and r,s,mr,s,mr,s,m be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of 2rps2^rp^s2rps over the finite field Fpm\mathbb{F}_{p^m}Fpm​. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length 2rps2^rp^s2rps, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.
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Let p be an odd prime, and Fp(m)is the finite field of pmelements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2psover Fp(m)are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2psover finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2psover Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.
Entanglement-assisted quantum error-correcting codes, which can be seen as a generalization of quantum error-correcting codes, can be constructed from arbitrary classical linear codes by relaxing the self-orthogonality properties and using pre-shared entangled states between the sender and the receiver, and can also improve the performance of quantum error-correcting codes. In this paper, we construct some families of entanglement-assisted quantum maximum-distance-separable codes with parameters [[q2−1a,q2−1a−2d+2+c,d;c]]q[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q[[aq2−1​,aq2−1​−2d+2+c,d;c]]q​, where qqq is a prime power with the form q=am±ℓq = am \pm \ellq=am±ℓ, a=ℓ2−13a = \frac{{\ell^2} - 1}{3}a=3ℓ2−1​ is an odd integer, ℓ≡2 ( mod  6)\ell \equiv 2\ (\bmod\ 6)ℓ≡2 (mod 6) or ℓ≡4 ( mod  6)\ell \equiv 4\ (\bmod\ 6)ℓ≡4 (mod 6), and mmm is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.
The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing preshared entanglement between the sender and receiver. In this paper, a new family of EAQECCs is constructed from negacyclic codes of length n=q2+1an=\frac{q^2+1}{a}n=aq2+1​, where qqq is an odd prime power, a=m2+12a=\frac{m^2+1}{2}a=2m2+1​ and mmm is an odd integer. Some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes are obtained in the sense that their parameters are not covered by the previously known ones.