A Lower Bound for Generalized Hamming Weights and a Condition for t-th Rank MDS

Tomoharu SHIBUYA  Ryo HASEGAWA  Kohichi SAKANIWA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E82-A   No.6   pp.1090-1101
Publication Date: 1999/06/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Information Theory and Coding Theory
Keyword: 
generalized Hamming weights,  t-th rank MDS,  Reed-Solomon codes,  Reed-Muller codes,  codes on affine algebraic variety,  AG codes on Cab,  

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Summary: 
In this paper, we introduce a lower bound for the generalized Hamming weights, which is applicable to arbitrary linear code, in terms of the notion of well-behaving. We also show that any [n,k] linear code C over a finite field F is the t-th rank MDS for t such that g(C)+1 t k where g(C) is easily calculated from the basis of Fn so chosen that whose first n-k elements generate C. Finally, we apply our result to Reed-Solomon, Reed-Muller and algebraic geometry codes on Cab, and determine g(C) for each code.