On the Number of Minimum Weight Codewords of Subcodes of Reed-Muller Codes

Hitoshi TOKUSHIGE  Toyoo TAKATA  Tadao KASAMI  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E81-A   No.10   pp.1990-1997
Publication Date: 1998/10/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Coding Theory
Keyword: 
minimum weight codeword,  Reed-Muller code,  subcode,  

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Summary: 
In this paper, we consider linear subcodes of RMr,m whose bases are formed from the monomial basis of RMr,m by deleting ΔK monomials of degree r where ΔK < . For such subcodes, a procedure for computing the number of minimum weight codewords is presented and it is shown how to delete ΔK monomials in order to obtain a subcode with the smallest number of codewords of the minimum weight. For ΔK 3, a formula for the number of codewords of the minimum weight is presented. A (64,40) subcode of RM3,6 is being considered as an inner code in a concatenated coding system for NASA's high-speed satellite communications. For (64,40) subcodes, there are three equivalent classes. For each class, the number of minimum weight codewords, that of the second smallest weight codewords and simulation results on error probabilities of soft-decision maximum likelihood decoding are presented.