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Journal of Algebra Combinatorics Discrete Structures and Applications

The existence of optimal quaternary $[28,20,6]$ and quantum $[[28,12,6]]$ codes

Vladimir D. Tonchev

Published Sep 15, 2014• 2 citations
Publisher

Abstract

The existence of a quantum $[[28,12,6]]$ code was one of the few cases for codes of length $n\le 30$ that was left open in the seminal paper by Calderbank, Rains, Shor, and Sloane [2]. The main result of this paper is the construction of a new optimal linear quaternary $[28,20,6]$ code which contains its hermitian dual code and yields an optimal linear quantum $[[28,12,6]]$ code. Received: 31 May 2014 | Accepted: 3 August 2014

DOI
10.13069/jacodesmath.25090

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The existence of optimal quaternary [28,20,6][28,20,6][28,20,6] and quantum [[28,12,6]][[28,12,6]][[28,12,6]] codes
The existence of a quantum [[28,12,6]][[28,12,6]][[28,12,6]] code was one of the few cases for codes of length n≤30n\le 30n≤30 that was left open in the seminal paper by Calderbank, Rains, Shor, and Sloane [2]. The main result of this paper is the construction of a new optimal linear quaternary [28,20,6][28,20,6][28,20,6] code which contains its hermitian dual code and yields an optimal linear quantum [[28,12,6]][[28,12,6]][[28,12,6]] code. Received: 31 May 2014 | Accepted: 3 August 2014
A simple construction of quaternary hermitian self-orthogonal codes with parameters [2n+1,k+1][2n+1,k+1][2n+1,k+1] and [2n+2,k+2][2n+2,k+2][2n+2,k+2] from a given pair of self-orthogonal [n,k][n,k][n,k] codes, and its link to quantum codes is considered. As an application, an optimal quaternary linear [28,20,6][28,20,6][28,20,6] dual containing code is found that yields a new optimal [[28,12,6]][[28,12,6]][[28,12,6]] quantum code.
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 We introduce quaternary modified four μ\mu μ μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ\mu μ μ -circulant Hermitian self-dual codes. We also construct quaternary modified four μ\mu μ μ -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.
The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s + 5, 3, 16s + 3; 21s + 2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26, 3, 19] code.