Delsarte–Goethals Codes

The Delsarte-Goethals code DG(u, r) for even u ≥ 4 and 1 ≤ ru/2 is a binary, non-linear (2u, 2(u−1)(u/2-r+1)+u+1, 2m−1 – 2m−1−d)-code [1]. For r = u/2 − 1 it has strength 7 and is therefore an orthogonal array OA(23u−1, 2u, ℤ2, 7).

In [2] it is shown that these codes can be constructed by applying the Gray-code mapping 0→00, 1→01, 2→11, and 3→10 to linear, cyclic codes over 4.

Special Cases

Partition in Translates

DG(u, r) is the union of disjoint translates of DG(u, r + 1). In particular, DG(u, u/2 − 1) is the union of disjoint translates of Kerdock codes K(u) = DG(u, u/2), which are again the disjoint union of translates of first-order Reed-Muller codes RM(1, u). Construction X, Construction X4, and Construction XX can be applied to these codes.

See Also

References

[1]Philippe Delsarte and Jean-Marie Goethals.
Alternating bilinear forms over GFq.
Journal of Combinatorial Theory, Series A, 19(1):26–50, July 1975.
doi:10.1016/0097-3165(75)90090-4
[2]A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, Neil J. A. Sloane, and P. Solé.
The 4-linearity of Kerdock, Preparata, Goethals and related codes.
IEEE Transactions on Information Theory, 40(2):301–319, March 1994.
doi:10.1109/18.312154
[3]F. Jessie MacWilliams and Neil J. A. Sloane.
The Theory of Error-Correcting Codes.
North-Holland, Amsterdam, 1977.
[4]A. S. Hedayat, Neil J. A. Sloane, and John Stufken.
Orthogonal Arrays.
Springer Series in Statistics. Springer-Verlag, 1999.

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

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