Hexacode

The Hexacode is a linear, cyclic, Hermitian self-dual, projective [6, 3, 4]-code over F4 with weight distribution A0 = 1, A4 = 45, and A6 = 18. A possible generator matrix is

$\displaystyle \left(\vphantom{\begin{array}{cccccc} 1 & 0 & 0 & 1 & \omega & \o… … 0 & \omega & 1 & \omega\\ 0 & 0 & 1 & \omega & \omega & 1\end{array}}\right.$$\displaystyle \begin{array}{cccccc} 1 & 0 & 0 & 1 & \omega & \omega\\ 0 & 1 & 0 & \omega & 1 & \omega\\ 0 & 0 & 1 & \omega & \omega & 1\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cccccc} 1 & 0 & 0 & 1 & \omega & \o… … 0 & \omega & 1 & \omega\\ 0 & 0 & 1 & \omega & \omega & 1\end{array}}\right)$

with ω denoting a primitive element of F4.

Optimality

The Hexacode meets the Singleton bound with equality and is therefore an MDS-codes. The dual orthogonal array is an OA with index unity.

See also

References

[1]Jürgen Bierbrauer.
Introduction to Coding Theory.
Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004.
MR2079734 (2005f:94001)

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Hexacode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CHexa.html

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