# Weight Functions and Generalized Hamming Weights of Linear Codes

Weight Functions and Generalized Hamming Weights of Linear Codes 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Weight Functions and Generalized Hamming Weights of Linear Codes

, Volume 41 (2) – Jul 15, 2005
14 pages

/lp/springer_journal/weight-functions-and-generalized-hamming-weights-of-linear-codes-oUlJrr6u4s
Publisher
Nauka/Interperiodica
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1007/s11122-005-0014-6
Publisher site
See Article on Publisher Site

### 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.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 15, 2005

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