# Preparata code

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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 (XY) of 2m-tuples, each corresponding to subsets of the finite field GF(2m) in some fixed way.

The extended code contains the words (XY) 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.