View source for Preparata code

{{TEX|done}}{{MSC|94B}}

A class of non-linear binary double-[[error-correcting code]]s.  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;
# <math>\sum_{x \in X} x = \sum_{y \in Y} y</math>;
# <math>\sum_{x \in x} x^3 + \left(\sum_{x \in X} x\right)^3 = \sum_{y \in Y} y^3</math>.

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