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A Lower Bound for Generalized Hamming Weights and a Condition for t-th Rank MDS
Tomoharu SHIBUYA Ryo HASEGAWA Kohichi SAKANIWA
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Publication Date: 1999/06/25
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Information Theory and Coding Theory
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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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.