Preparata code: Difference between revisions

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==Construction==

Let ''m'' be an odd number, and ''n'' = 2^''m''- − 1. We first describe the '''extended Preparata code''' of length 2''n'' + 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

# ''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 GF(2^''m'').

==Properties==

The Preparata code is of length 2^''m''+1-&nbsp;&minus;&nbsp;1, size 2^''k'' where ''k'' = 2^{''m''&nbsp;+&nbsp;1} - &nbsp;&minus;&nbsp;2''m'' - &nbsp;&minus;&nbsp;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 | pages=378–400 | doi=10.1016/S0019-9958(68)90874-7 }}

* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd ed | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 | pages=111-113}}

[[Category:Error detection and correction]]