IJPAM: Volume 65, No. 1 (2010)

FROM IDEALS IN POLYNOMIAL RINGS TO
LINEAR CODES USING GROEBNER BASES

Mehwish Saleemi$^1$, Karl-Heinz Zimmermann$^2$
$^{1,2}$Institute of Computer Technology (E-13)
Hamburg University of Technology
Schwarzenbergstr. 95E, Hamburg, 21073, GERMANY
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]


Abstract.In this paper, we investigate linear codes as ideals in the group algebra over an elementary abelian $p$-group. We provide a description of these codes in terms of Groebner bases and supply corresponding encoding and decoding procedures. In particular, we study generalizations of primitive Reed-Muller codes, construct their Groebner bases, and give their code parameters. Finally, we show that the class of codes studied contains an interesting family of linear codes. These codes have a designed Hamming distance and turn out to be superior to the primitive Reed-Muller codes in the non-binary case.

Received: May 12, 2010

AMS Subject Classification: 13P10, 94B05

Key Words and Phrases: commutative polynomial rings, ideals, Groebner bases, Reed-Muller codes, decoding

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 65
Issue: 1