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.

Construction
Let m be an odd number, and n = 2m-1. We first describe the extended Preparata code of length 2n+2 = 2m+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 Peparata code is obtained by deleting the position in X corresponding to 0 in GF(2m).

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

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