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

Year 2014, , 13 - 17, 01.03.2014

Vladimir D. Tonchev

https://doi.org/10.13069/jacodesmath.25090

Abstract

The existence of a quantum [[28, 12, 6]] code was one of the few cases for codes of length n ≤ 30 thatwas left open in the seminal paper by Calderbank, Rains, Shor, and Sloane [2]. The main result ofthis paper is the construction of the first optimal linear quaternary [28, 20, 6] code which contains itsHermitian dual code and yields the first optimal quantum [[28, 12, 6]] code

Keywords

Quantum code, Optimal code, quaternary

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

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 \cite{CRSS}. 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.

Keywords

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References

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• A. E. Brouwer, Tables of linear codes, http://www.win.tue.nl/ aeb/.
• M. Grassl, http://www.codetables.de.
• F. J. MacWilliams and N. J. A. Sloane,
• The Theory of Error-Correcting Codes, North-Holland, Amsterdam 1977.
• G. Nebe, E. M. Rains, N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• V. D. Tonchev, Quantum codes from caps, Discrete Math., 308, 6368-6372, 2008.