On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q2-ary linear codes. This result can be generalized to q2m-ary linear codes, m>1. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with q≠2, improving others in the literature.Funding for open access charge: CRUE-Universitat Jaume

Some constructions of quantum MDS codes

We construct quantum MDS codes with parameters $[\![ q^2+1,q^2+3-2d,d ]\!] _q$ for all $d \leqslant q+1$, $d \neq q$. These codes are shown to exist by proving that there are classical generalised Reed-Solomon codes which contain their Hermitian dual. These constructions include many constructions which were previously known but in some cases these codes are new. We go on to prove that if $d\geqslant q+2$ then there is no generalised Reed-Solomon $[n,n-d+1,d]_{q^2}$ code which contains its Hermitian dual. We also construct an $[\![ 18,0,10 ]\!] _5$ quantum MDS code, an $[\![ 18,0,10 ]\!] _7$ quantum MDS code and a $[\![ 14,0,8 ]\!] _5$ quantum MDS code, which are the first quantum MDS codes discovered for which $d \geqslant q+3$, apart from the $[\![ 10,0,6 ]\!] _3$ quantum MDS code derived from Glynn's code

Classical and Quantum Evaluation Codesat the Trace Roots

We introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field F 4 which are records at http://codetables.de/

Classical and quantum evaluation codes at the trace roots

Producción CientíficaWe introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field with 4 elements which are records at http://codetables.de/.This work was supported in part by the Spanish MINECO/FEDER (Grants No. MTM2015-65764-C3-2-P and MTM2015-69138-REDT), in part by the University Jaume I (Grant No. P1-1B2015-02), in part by The Danish Council for Independent Research (Grant No. DFF--4002-00367), and in part by RYC-2016-20208 (AEI/FSE/UE)

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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