Preparata code

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 − 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.

References

• http://www.encyclopediaofmath.org/index.php/Preparata_code
• http://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes