On the covering radius of binary, linear codes meeting the Griesmer bound

Author

Busschbach, Peter B. ; Gerretzen, Michiel G L ; Van Tilborg, Henk C A

Volume

Issue

fYear

1985

fDate

7/1/1985 12:00:00 AM

Firstpage

465

Lastpage

468

Abstract

Let $g(k, d) = \\sum _{i=0}^{k-1} \\lceil d / 2^{i} \\rceil$ . By the Griesmer bound, $n \\geq g(k, d)$ for any binary, linear $[n, k, d]$ code. Let $s = \\lceil d / 2^{k-1} \\rceil$ . Then, $s$ can be interpreted as the maximum number of occurrences of a column in the generator matrix of any code with parameters $[g(k, d), k, d]$ . Let $\\rho$ be the covering radius of a [g(k, d), k, d] code. It will be shown that $\\rho \\leq d - \\lceil s / 2 \\rceil$ . Moreover, the existence of a $[g(k, d), k, d]$ code with $\\rho = d - \\lceil s / 2 \\rceil$ is equivalent to the existence of a $[g(k + 1, d), k + 1, d]$ code. For $s \\leq 2$ , all $[g(k,d),k,d]$ codes with $\\rho = d - \\lceil s / 2 \\rceil$ are described, while for $s > 2$ a sufficient condition for their existence is formulated.

Keywords

Linear coding; Linear code; Mathematics; Sufficient conditions;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1985.1057073

Filename

1057073

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=940528