Delsarte–Goethals Codes

The Delsarte-Goethals code DG(u, r) for even u ≥ 4 and 1 ≤ r ≤ u/2 is a binary, non-linear (2^u, 2^{(u−1)(u/2-r+1)+u+1}, 2^m−1 – 2^{m−1−d})-code [1]. For r = u/2 − 1 it has strength 7 and is therefore an orthogonal array OA(2^3u−1, 2^u, 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 ℤ₄.

Special Cases

• DG(u, 1) is the second-order Reed-Muller code RM(2, u).

• DG(u, u/2) is the Kerdock code K(u). DG(4, 2) is therefore the Nordstrom-Robinson code.

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

• [3, Chapter 15, Section 5] and [4, Section 5.11]

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 ℤ4-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: 2024-09-05. http://mint.sbg.ac.at/desc_CDelsarteGoethals.html