Nordstrom–Robinson OA

The Nordstrom-Robinson code [1] is a non-linear (16, 28, 6)-code over ℤ2. The strength of the Nordstrom-Robinson code is 5, therefore it is an orthogonal array OA(28, 16, ℤ2, 5). A linear code or linear orthogonal array with these parameters cannot exist.

A simple construction for the Nordstrom-Robinson code is given in [2]: Consider the ℤ4-linear, self-dual Octacode generated by the matrix

$\displaystyle \left(\vphantom{\begin{array}{cccccccc} 1 & 3 & 1 & 2 & 1 & 0 & 0… … 0 & 0 & 3 & 1 & 2 & 1 & 0\\ 1 & 0 & 0 & 0 & 3 & 1 & 2 & 1\end{array}}\right.$$\displaystyle \begin{array}{cccccccc} 1 & 3 & 1 & 2 & 1 & 0 & 0 & 0\\ 1 & 0 &… …\\ 1 & 0 & 0 & 3 & 1 & 2 & 1 & 0\\ 1 & 0 & 0 & 0 & 3 & 1 & 2 & 1\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cccccccc} 1 & 3 & 1 & 2 & 1 & 0 & 0… … 0 & 0 & 3 & 1 & 2 & 1 & 0\\ 1 & 0 & 0 & 0 & 3 & 1 & 2 & 1\end{array}}\right)$.

After mapping 0 $ \mapsto$ 00, 1 $ \mapsto$ 01, 2 $ \mapsto$ 11, and 3 $ \mapsto$ 10 the Nordstrom-Robinson code is obtained.

See Also

References

[1]Alan W. Nordstrom and John P. Robinson.
An optimum nonlinear code.
Information and Control, 11(5–6):613–616, 1967.
doi:10.1016/S0019-9958(67)90835-2
[2]G. D. Forney, Jr., Neil J. A. Sloane, and M. D. Trott.
The Nordstrom-Robinson code is the binary image of the octacode.
In A. R. Calderbank, G. D. Forney, Jr., and N. Moayeri, editors, Proceedings DIMACS/IEEE Workshop on Coding and Quantization, pages 19–26. American Mathematical Society, 1993.
[3]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. “Nordstrom–Robinson OA.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CNordstromRobinson.html

Show usage of this method