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 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.

Journal of Combinatorial DesignsJun 18, 201217 citations

Self-orthogonal caps in PG(k-1,4) and construction quantum codes

Youqian Feng

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.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.

Journal of Northwest UniversityJan 1, 20080 citations

Quantum Error-Correcting Codes

Dan Cristian Marinescu, Gabriela M. Marinescu

Elsevier eBooksFeb 15, 20117 citations

QUANTUM ERROR-CORRECTING CODES

Keqin Feng

Lecture notes series, Institute For Mathematical SciencesDec 1, 200229 citations

New quantum caps in PG(4,4)

Daniele Bartoli, Stefano Marcugini, Fernanda Pambianco

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 $d=4$ 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 $[[36,26,4]]$ and $[[38,28,4]]$ that remained open in the data base \cite{codetable}.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 $d=4$ 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 $[[36,26,4]]$ and $[[38,28,4]]$ that remained open in the data base \cite{codetable}.

arXiv:0905.1059May 7, 20092 citations

d=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]]

and

[[38,28,4]]

that remained open in the data base \cite{codetable}.