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arXiv:1311.2549

A doubling construction for self-orthogonal codes

Vladimir D. Tonchev

Published Nov 11, 2013• 0 citations
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Abstract

A simple construction of quaternary hermitian self-orthogonal codes with parameters $[2n+1,k+1]$ and $[2n+2,k+2]$ from a given pair of self-orthogonal $[n,k]$ codes, and its link to quantum codes is considered. As an application, an optimal quaternary linear $[28,20,6]$ dual containing code is found that yields a new optimal $[[28,12,6]]$ quantum code.

DOI
10.48550/arxiv.1311.2549

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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.
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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
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.
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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.
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