Preparata code: Difference between revisions

Revision as of 06:36, 31 July 2008

Latest revision as of 23:51, 17 August 2023

(14 intermediate revisions by 13 users not shown)

Line 1:

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<sup>''^m''-1</supmath>-1. We first describe the '''extended Preparata code''' of length 2''n''<math>2n+2 = 2<sup>''^{m''+1}</supmath>: the Preparata code is then derived by deleting one position. The words of the extended code are regarded as pairs (''X'',&nbsp;''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'',&nbsp;''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 xX} x^3 + \left(\sum_{x \in X} x\right)^3 = \sum_{y \in Y} y^3.</math>.

The PeparataPreparata 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 | issue=4 | pages=378378–400 | doi=10.1016/S0019-4009958(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 ed | 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/111-113 111–113] | 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

Line 25 ⟶ 30:

[[Category:Finite fields]]

[[Category:Coding theory]]

{{algebra-stub}}