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.
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Construction[edit]
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 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;
The Preparata code is obtained by deleting the position in X corresponding to 0 in GF(2^{m}).
Properties[edit]
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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