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 '''Z'''<sub>4</sub> 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<sup>''m''</sup>-tuples, each corresponding to subsets of the finite field GF(2<sup>''m''</sup>) 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<sup>''m''</sup>).
==Properties== The Preparata code is of length 2<sup>''m''+1</sup> − 1, size 2<sup>''k''</sup> where ''k'' = 2<sup>''m'' + 1</sup> − 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=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