Preparata code

In coding theory, the Preparata codes form a class of non-linear double-error-correcting codes. They are named after Franco P. Preparata who first described them in 1968.

Although non-linear over GF(2) the Preparata codes are linear over Z4 with the Lee distance.

Construction
Let m be an odd number, and $$n = 2^m-1$$. We first describe the extended Preparata code of length $$2n+2 = 2^{m+1}$$: the Preparata code is then derived by deleting one position. The words of the extended code are regarded as pairs (X, Y) of 2m-tuples, each corresponding to subsets of the finite field GF(2m) in some fixed way.

The extended code contains the words (X, Y) satisfying three conditions


 * 1) X, Y each have even weight;
 * 2) $$\sum_{x \in X} x = \sum_{y \in Y} y;$$
 * 3) $$\sum_{x \in X} x^3 + \left(\sum_{x \in X} x\right)^3 = \sum_{y \in Y} y^3.$$

The Preparata code is obtained by deleting the position in X corresponding to 0 in GF(2m).

Properties
The Preparata code is of length 2m+1 &minus; 1, size 2k where k = 2m + 1 &minus; 2m &minus; 2, and minimum distance 5.

When m = 3, the Preparata code of length 15 is also called the Nordstrom–Robinson code.