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.
Although non-linear over GF the Preparata codes are linear over Z4 with the Lee distance.

Construction

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 of 2m-tuples, each corresponding to subsets of the finite field GF in some fixed way.
The extended code contains the words satisfying three conditions
  1. X, Y each have even weight;
The Preparata code is obtained by deleting the position in X corresponding to 0 in GF.

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.