Delsarte-Goethals code
A code belonging to a family of non-linear binary error-correcting codes (cf. also Error-correcting code). Delsarte–Goethals codes were first presented in a joint paper [a2] by Ph. Delsarte and J.-M. Goethals.
Let be an even integer. Let
be an integer satisfying
. For each
and
there is a Delsarte–Goethals code, denoted
. This code has length
, and is sandwiched between the Kerdock code
and the second-order Reed–Muller code
of the same length (cf. also Kerdock and Preparata codes; Error-correcting code):
![]() |
The number of codewords in is
and the minimum distance is
. As
increases, the number of codewords increases and the minimum distance decreases. When
, the Delsarte–Goethals code coincides with the Kerdock code
, and when
the Delsarte–Goethals code coincides with
.
The construction of involves taking the union of certain cosets of
in
. These cosets are determined by certain bilinear forms. The rank of these forms, and the rank of the sum of any two of them, is at least
, and this property determines the minimum distance. The fact that it is possible to find
such forms is proved in [a2] (see also [a5]).
The Delsarte–Goethals codes have been shown to have another construction. It was shown in [a3] that they are the Gray image of a -linear code. A direct proof of the minimum distance from the
construction was given in [a1].
There exist non-linear binary codes whose distance distribution is the MacWilliams transform of the distribution of the Delsarte–Goethals codes, see [a4]. These codes act like dual codes, and the construction gives an explanation for their existence, see [a3].
References
[a1] | A.R. Calderbank, G. McGuire, "![]() |
[a2] | P. Delsarte, J.M. Goethals, "Alternating bilinear forms over ![]() |
[a3] | A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, P. Sole, "The ![]() |
[a4] | F.B. Hergert, "On the Delsarte–Goethals codes and their formal duals" Discr. Math. , 83 (1990) pp. 249–263 |
[a5] | F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes" , North-Holland (1977) |
Delsarte-Goethals code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Delsarte-Goethals_code&oldid=22339