Preparata code

Construction

Let m be an odd number, and n = 2 m − 1 . We first describe the extended Preparata code of length 2 n + 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 2^m-tuples, each corresponding to subsets of the finite field GF(2^m) in some fixed way.

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

1. X, Y each have even weight;
2. ∑ x ∈ X x = ∑ y ∈ Y y ;
3. ∑ x ∈ X x 3 + ( ∑ x ∈ X x ) 3 = ∑ y ∈ Y y 3 .

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

Properties

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

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