A Lower Bound for Generalized Hamming Weights and a Condition for <I>t</I>-th Rank MDS

Tomoharu SHIBUYA; Ryo HASEGAWA; Kohichi SAKANIWA

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

Tomoharu SHIBUYA, Ryo HASEGAWA, Kohichi SAKANIWA

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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 C_ab, and determine g(C) for each code.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E82-A No.6 pp.1090-1101

Publication Date: 1999/06/25

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Type of Manuscript: PAPER

Category: Information Theory and Coding Theory

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Tomoharu SHIBUYA, Ryo HASEGAWA, Kohichi SAKANIWA, "A Lower Bound for Generalized Hamming Weights and a Condition for t-th Rank MDS" in IEICE TRANSACTIONS on Fundamentals, vol. E82-A, no. 6, pp. 1090-1101, June 1999, doi: .
Abstract: 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 C_ab, and determine g(C) for each code.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e82-a_6_1090/_p

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@ARTICLE{e82-a_6_1090,
author={Tomoharu SHIBUYA, Ryo HASEGAWA, Kohichi SAKANIWA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Lower Bound for Generalized Hamming Weights and a Condition for t-th Rank MDS},
year={1999},
volume={E82-A},
number={6},
pages={1090-1101},
abstract={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 C_ab, and determine g(C) for each code.},
keywords={},
doi={},
ISSN={},
month={June},}

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TY - JOUR
TI - A Lower Bound for Generalized Hamming Weights and a Condition for t-th Rank MDS
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1090
EP - 1101
AU - Tomoharu SHIBUYA
AU - Ryo HASEGAWA
AU - Kohichi SAKANIWA
PY - 1999
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E82-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 1999
AB - 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 C_ab, and determine g(C) for each code.
ER -