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Weight Functions and Generalized Hamming Weights of Linear Codes.

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Názov: Weight Functions and Generalized Hamming Weights of Linear Codes.
Autori: D. Yu. Nogin
Zdroj: Problems of Information Transmission; Apr2005, Vol. 41 Issue 2, p91-104, 14p
Abstrakt: Abstract We prove that the weight function wt: $$\mathbb{F}_q^k \to \mathbb{Z}$$ on a set of messages uniquely determines a linear code of dimension k up to equivalence. We propose a natural way to extend the rth generalized Hamming weight, that is, a function on r-subspaces of a code C, to a function on $$\mathbb{F}_q^{\left( {_r^k } \right)} \cong \Lambda ^r C$$ . Using this, we show that, for each linear code C and any integer r ≰ k = dim C, a linear code exists whose weight distribution corresponds to a part of the generalized weight spectrum of C, from the rth weights to the kth. In particular, the minimum distance of this code is proportional to the rth generalized weight of C. [ABSTRACT FROM AUTHOR]
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  Data: Weight Functions and Generalized Hamming Weights of Linear Codes.
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  Data: Problems of Information Transmission; Apr2005, Vol. 41 Issue 2, p91-104, 14p
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  Data: Abstract We prove that the weight function wt: <img src="/fulltext-image.asp?format=htmlnonpaginated&src=T674768PQ0804441_html\11122_2005_Article_14_TeX2GIFIE1.gif" border="0" alt=" $$\mathbb{F}_q^k \to \mathbb{Z}$$ " /> on a set of messages uniquely determines a linear code of dimension k up to equivalence. We propose a natural way to extend the rth generalized Hamming weight, that is, a function on r-subspaces of a code C, to a function on <img src="/fulltext-image.asp?format=htmlnonpaginated&src=T674768PQ0804441_html\11122_2005_Article_14_TeX2GIFIE2.gif" border="0" alt=" $$\mathbb{F}_q^{\left( {_r^k } \right)} \cong \Lambda ^r C$$ " />. Using this, we show that, for each linear code C and any integer r ≰ k = dim C, a linear code exists whose weight distribution corresponds to a part of the generalized weight spectrum of C, from the rth weights to the kth. In particular, the minimum distance of this code is proportional to the rth generalized weight of C. [ABSTRACT FROM AUTHOR]
– Name: Abstract
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  Group: Ab
  Data: <i>Copyright of Problems of Information Transmission is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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