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.

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

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 Peparata 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.

References

• F.P. Preparata (1968). "A class of optimum nonlinear double-error-correcting codes". Information and Control 13: 378-400. DOI:10.1016/S0019-9958(68)90874-7. Research Blogging.

• J.H. van Lint (1992). Introduction to Coding Theory, 2nd ed. Springer-Verlag, 111-113. ISBN 3-540-54894-7.