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In [[coding theory]], the '''Preparata codes''' form a class of non-linear double-[[Error detection and correction|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]].

==Construction==
Let ''m'' be an odd number, and <math>n = 2^m-1</math>. We first describe the '''extended Preparata code''' of length <math>2n+2 = 2^{m+1}</math>: 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;
# <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 Preparata 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 − 2''m'' − 2, and minimum distance 5.

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

== References ==
* {{cite journal | author=F.P. Preparata | authorlink=Franco P. Preparata | title=A class of optimum nonlinear double-error-correcting codes | journal=Information and Control | volume=13 | year=1968 | issue=4 | pages=378–400 | doi=10.1016/S0019-9958(68)90874-7 | doi-access=free | hdl=2142/74662 | hdl-access=free }}
* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 | pages=[https://archive.org/details/introductiontoco0000lint/page/111 111–113] | url=https://archive.org/details/introductiontoco0000lint/page/111 }}
* http://www.encyclopediaofmath.org/index.php/Preparata_code
* http://www.encyclopediaofmath.org/index.php/Kerdock_and_Preparata_codes

[[Category:Error detection and correction]]
[[Category:Finite fields]]
[[Category:Coding theory]]

Preparata code - Revision history

imported>OAbot: Open access bot: hdl updated in citation with #oabot.