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Problemy Peredachi Informatsii, 2005, Volume 41, Issue 2, Pages 26–41
(Mi ppi93)
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Coding Theory
Weight Functions and Generalized Weights of Linear Codes
D. Yu. Nogin Institute for Information Transmission Problems, Russian Academy of Sciences
Abstract:
We prove that the weight function
$\mathrm{wt}\colon\mathbb F_q^k\to\mathbb Z$
on a set of messages uniquely determines a linear code of dimension $k$ up to equivalence. We propose a natural way to extend the $r$th generalized Hamming weight, that is, a function on $r$-subspaces of a code $C$, to a function on
$\mathbb F_q^{\binom kr}\cong\Lambda^rC$.
Using this, we show that, for each linear code $C$ and any integer
$r\le k=\dim C$, a linear code exists whose weight distribution corresponds to a part of the
generalized weight spectrum of $C$, from the $r$th weights to the $k$th. In particular, the minimum
distance of this code is proportional to the $r$th generalized weight of $C$.
Received: 02.12.2003 Revised: 16.11.2004
Citation:
D. Yu. Nogin, “Weight Functions and Generalized Weights of Linear Codes”, Probl. Peredachi Inf., 41:2 (2005), 26–41; Problems Inform. Transmission, 41:2 (2005), 91–104
Linking options:
https://www.mathnet.ru/eng/ppi93 https://www.mathnet.ru/eng/ppi/v41/i2/p26
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Abstract page: | 524 | Full-text PDF : | 397 | References: | 54 | First page: | 2 |
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