Let m be an odd number, and . We first describe the extended Preparata code of length : 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
- X, Y each have even weight;
The Preparata code is obtained by deleting the position in X corresponding to 0 in GF(2m).
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.