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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 {\displaystyle n=2^{m}-1} . We first describe the extended Preparata code of length 2 n + 2 = 2 m + 1 {\displaystyle 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-tuples, each corresponding to subsets of the finite field GF(2) in some fixed way.

The extended code contains the words (X, Y) satisfying three conditions

  1. X, Y each have even weight;
  2. ∑ x ∈ X x = ∑ y ∈ Y y ; {\displaystyle \sum _{x\in X}x=\sum _{y\in Y}y;}
  3. ∑ x ∈ X x 3 + ( ∑ x ∈ X x ) 3 = ∑ y ∈ Y y 3 . {\displaystyle \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).

Properties

The Preparata code is of length 2 − 1, size 2 where k = 2 − 2m − 2, and minimum distance 5.

When m = 3, the Preparata code of length 15 is also called the Nordstrom–Robinson code.

References

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