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Journal of Combinatorial Designs

New Quantum Caps in PG(4, 4)

Daniele Bartoli, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco

Published Jun 18, 2012• 17 citations
Publisher

Abstract

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.

DOI
10.1002/jcd.21321

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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.
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}.
The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.
For a given quaternary self-orthogonal (SO) code, using information of weights of codewords in the dual of the binary code generated by the supports of this SO code, one can construct new quaternary SO code by shortening method. Two methods of determining weights of codewords in the dual of the binary support code of a given large length SO code are presented. Using these methods, we construct many SO codes from a quantum 286-cap in PG(6, 4), and deduce existence of many quantum caps in PG(6, 4) and good quantum codes of distance 4.
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.