# Preparata code

2010 Mathematics Subject Classification: *Primary:* 94B [MSN][ZBL]

A class of non-linear binary double-error-correcting codes. They are named after Franco P. Preparata who first described them in 1968. Although non-linear over the finite field $\mathrm{GF}(2)$, it is known that the Kerdock and Preparata codes are linear over $\mathbb{Z}/4$.

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 $\mathrm{GF}(2^m)$) 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 Peparata code is obtained by deleting the position in $X$ corresponding to 0 in $\mathrm{GF}(2^m)$.

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, "A class of optimum nonlinear double-error-correcting codes",
*Information and Control***13**(1968) 378-400 DOI 0.1016/S0019-9958(68)90874-7 - J.H. van Lint,
*Introduction to Coding Theory*(2nd ed), Springer-Verlag (1992) ISBN 3-540-54894-7. pp.111-113

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Preparata code.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Preparata_code&oldid=35015