Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Author

Levenshtein, Vladimir I.

Author_Institution

M.V. Keldysh Inst. of Appl. Math., Acad. of Sci., Moscow, Russia

Volume

41

Issue

5

fYear

1995

fDate

9/1/1995 12:00:00 AM

Firstpage

1303

Lastpage

1321

Abstract

Universal bounds for the cardinality of codes in the Hamming space F_rⁿ with a given minimum distance d and/or dual distance d´ are stated. A self-contained proof of optimality of these bounds in the framework of the linear programming method is given. The necessary and sufficient conditions for attainability of the bounds are found. The parameters of codes satisfying these conditions are presented in a table. A new upper bound for the minimum distance of self-dual codes and a new lower bound for the crosscorrelation of half-linear codes are obtained

Keywords

correlation methods; dual codes; linear codes; linear programming; polynomials; Hamming spaces; Krawtchouk polynomials; cardinality; crosscorrelation; dual distance; half-linear codes; linear programming method; lower bound; minimum distance; necessary conditions; optimality; self-dual codes; sufficient conditions; universal bounds; upper bound; Associate members; Bismuth; Helium; Information theory; Linear programming; Mathematics; Polynomials; Sufficient conditions; Upper bound; Vectors;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/18.412678

Filename

412678