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IEEE Access

New Quantum Codes Constructed by Quantum Caps in PG(3,9) and PG(4,9)

Husheng Li, Ruihu Li, Qiang Fu

Published Jan 1, 2020• 4 citations
Publisher

Abstract

In this paper, we present a computer-supported method of searching for quantum caps. By means of this method and relevant knowledge of combinatorics, many quantum caps in$PG(3,9)$and$PG(4,9)$are constructively proven to exist. Then, according to the theorem that each quantum cap corresponds to a quantum error-correcting code with$d=4$, we obtain 278 quantum error-correcting codes. Most of these results break the GV bound, and a number of them are optimal quantum codes or have improved parameters.

DOI
10.1109/access.2020.3045990

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Journal of Physics Conference Series•Jun 1, 2021•0 citations

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Abstract In this paper, we firstly give a computer-support method of searching for Hermitian self-orthogonal linear code. Then two linear codes with dual distance 5 over field F q2 for q = 2,3, respectively, are chosen as ingredients. By use of this searching method, we totally find 88 Hermitian self-orthogonal linear codes and construct 88 related quantum error-correcting codes with d = 5 . Most of these results are best known quantum codes and a number of them have better parameters than that of previous quantum codes.

Journal of Physics Conference Series•Nov 1, 2020•3 citations
In this paper, we present a computer-supported method of searching for quantum caps. By means of this method and relevant knowledge of combinatorics, many quantum caps inPG(3,9)PG(3,9)PG(3,9)andPG(4,9)PG(4,9)PG(4,9)are constructively proven to exist. Then, according to the theorem that each quantum cap corresponds to a quantum error-correcting code withd=4d=4d=4, we obtain 278 quantum error-correcting codes. Most of these results break the GV bound, and a number of them are optimal quantum codes or have improved parameters.
Aim To investigate the method of constructing quantum error-correcting code by self-orthogonal cap over PG(k-1,4).Methods Using random searching method to find self-orthogonal caps,and using combinatorial method to construct self-orthogonal codes and quantum error-correcting codes.Results A random searching algorithm for complete cap over PG(k-1,4) is designed,and some complete and self-orthogonal caps over PG(k-1,4) are found.By using of these self-orthogonal caps,nine new self-orthogonal codes of dual distance 4 are obtained.Conclusion Some good self-orthogonal codes over F4 and good quantum error-correcting codes are constructed,these codes are better than the previously known.
Calderbank, Rains, Shor and Sloane (see \cite{Sloane}) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in \cite{Bierbra}, where also the notion of quantum cap is introduced. Quantum caps correspond to the special case of quantum stabilizer codes of distance d=4d=4d=4 when the code is linear over GF(4). In the present paper we review the translation from quantum error-correction to symplectic geometry and study quantum codes in PG(4,4) where we construct complete quantum caps with 20, 29, 30, 32, 33, 34, 36 and 38 points and incomplete quantum caps with 10, 12, 13, 20, 23, 24, 25 and 26 points and we prove the non existence of 11-quantum caps. In particular the quantum caps of sizes 36 and 38 yield positive answers to the existence questions of quantum codes [[36,26,4]][[36,26,4]][[36,26,4]] and [[38,28,4]][[38,28,4]][[38,28,4]] that remained open in the data base \cite{codetable}.
Abstract Calderbank, Rains, Shor, and Sloane (see [10]) showed that quantum stabilizer codes correspond to additive quaternary codes in binary projective spaces, which are self‐orthogonal with respect to the symplectic form. A geometric description is given in [8, 19]. In [8] the notion of a quantum cap is introduced. Quantum caps are equivalent to quantum stabilizer codes of minimum distance when the code is linear over . In this paper, we determine the valuesksuch that there exists a quantumk‐cap in , corresponding to pure linear quantum codes, proving, by exhaustive search, that no 11, 37, 39‐quantum caps exist. Moreover we give examples of quantum caps in not already known in the literature.
Abstract In this paper, we firstly give a computer-support method of searching for Hermitian self-orthogonal linear code. Then two linear codes with dual distance 5 over field F q2 for q = 2,3, respectively, are chosen as ingredients. By use of this searching method, we totally find 88 Hermitian self-orthogonal linear codes and construct 88 related quantum error-correcting codes with d = 5 . Most of these results are best known quantum codes and a number of them have better parameters than that of previous quantum codes.