Tetracode

The Tetracode C is a linear, cyclic, self-dual, [4, 2, 3]-code over 3. It is equivalent to the Simplex code S(2, 3) and to the Hamming code H(2, 3). Its weight distribution is A0 = 1 and A3 = 8, thus all non-zero code words have weight 3 and C ∖ { 0} is a constant-weight (4, 8, 3)-code. A possible generator matrix is

$\displaystyle \left(\vphantom{\begin{array}{cccc} 1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2\end{array}}\right.$$\displaystyle \begin{array}{cccc} 1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cccc} 1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2\end{array}}\right)$.

Optimality

The Tetracode meets the sphere-packing bound and the Singleton bound with equality and is therefore a perfect MDS-code. The dual orthogonal array is a tight OA with index unity.

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. “Tetracode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CTetra.html

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