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(2) the Preparata codes are linear over Z 4 with the Lee distance.

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 Z 4 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 (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 X, Y each have even weight; \sum_{x \in X} x = \sum_{y \in Y} y; \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(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.

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