Xing-Ling Codes, Duals of their Subcodes, and Good Asymmetric Quantum Codes

A class of powerful $q$-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are $q$-ary block codes that encode $k$ qudits of quantum information into $n$ qudits and correct up to \flr{(d_{x}-1)/2} bit-flip errors and up to \flr{(d_{z}-1)/2} phase-flip errors.. In many cases where the length $(q^{2}-q)/2 \leq n \leq (q^{2}+q)/2$ and the field size $q$ are fixed and for chosen values of $d_{x} \in \{2,3,4,5\}$ and $d_{z} \ge \delta$, where $\delta$ is the designed distance of the Xing-Ling (XL) codes, the derived pure $q$-ary asymmetric quantum CSS codes possess the best possible size given the current state of the art knowledge on the best classical linear block codes.Comment: To appear in Designs, Codes and Cryptography (accepted Sep. 27, 2013

Quantum codes from a new construction of self-orthogonal algebraic geometry codes

[EN] We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.G. McGuire was partially supported by Science Foundation Ireland Grant 13/IA/1914. The remainder authors were partially supported by the Spanish Government and the EU funding program FEDER, Grants MTM2015-65764-C3-2-P and PGC2018-096446-B-C22. F. Hernando and J. J. Moyano-Fernandez are also partially supported by Universitat Jaume I, Grant UJI-B2018-10.Hernando, F.; Mcguire, G.; Monserrat Delpalillo, FJ.; Moyano-Fernández, JJ. (2020). Quantum codes from a new construction of self-orthogonal algebraic geometry codes. 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Low-complexity quantum codes designed via codeword-stabilized framework

We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or non-additive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl

Application of Constacyclic codes to Quantum MDS Codes

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on the existence condition for nontrivial dual-containing constacyclic codes. This would help us to avoid unnecessary attempts and provide effective ideas in order to construct dual-containing codes. Several classes of dual-containing MDS constacyclic codes are constructed and their parameters are computed. Consequently, new quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.Comment: 16 page