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Generators and weights of polynomial codes

Citation

Cazaran, J and Kelarev, AV, Generators and weights of polynomial codes, Archiv der Mathematik, 69, (6) pp. 479-486. ISSN 0003-889X (1997) [Refereed Article]

DOI: doi:10.1007/s000130050149

Abstract

Berman and Charpin proved that all generalized Reed-Muller codes coincide with powers of the radical of a certain algebra. The ring-theoretic approach was developed by several authors including Landrock and Manz, and helped to improve parameters of the codes. It is important to know when the codes have a single generator. We consider a class of ideals in polynomial rings containing all generalized Reed-Muller codes, and give conditions necessary and sufficient for the ideal to have a single generator. The main result due to Glastad and Hopkins (Comment. Math. Univ. Carolin. 21, 371-377 (1980)) is an immediate corollary to our theorem. We also describe all finite quotient rings ℤ/mℤ[x1, . . . ,Xn]/I which are commutative principal ideal rings where I is an ideal generated by univariate polynomials and then give formulas for the minimum Hamming weight of the radical and its powers in the algebra double-struck F sign[x1, . . . ,xn]/(x1 a1(1-x1 b1), . . . ,xn an(1-xn bn)) where double-struck F sign is an arbitrary field.

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Pure Mathematics
Research Field:Algebra and Number Theory
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Mathematical Sciences
Author:Cazaran, J (Professor Jilyana Cazaran)
Author:Kelarev, AV (Dr Andrei Kelarev)
ID Code:11483
Year Published:1997
Web of Science® Times Cited:26
Deposited By:Mathematics
Deposited On:1997-08-01
Last Modified:2011-08-12
Downloads:0

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