Minimal vectors in linear codes
Ashikhmin, A.
Barg, A.
Los Alamos Nat. Lab., NM;
This paper appears in: Information Theory, IEEE Transactions on
Publication Date: Sep 1998
Volume: 44,
Issue: 5
On page(s): 2010-2017
ISSN: 0018-9448
References Cited: 21
CODEN: IETTAW
INSPEC Accession Number: 6013610
Digital Object Identifier: 10.1109/18.705584
Current Version Published: 2002-08-06
Abstract
Minimal vectors in linear codes arise in numerous applications,
particularly, in constructing decoding algorithms and studying linear
secret sharing schemes. However, properties and structure of minimal
vectors have been largely unknown. We prove basic properties of minimal
vectors in general linear codes. Then we characterize minimal vectors of
a given weight and compute their number in several classes of codes,
including the Hamming codes and second-order Reed-Muller codes. Further,
we extend the concept of minimal vectors to codes over rings and compute
them for several examples. Turning to applications, we introduce a
general gradient-like decoding algorithm of which minimal-vectors
decoding is an example. The complexity of minimal-vectors decoding for
long codes is determined by the size of the set of minimal vectors.
Therefore, we compute this size for long randomly chosen codes. Another
example of algorithms in this class is given by zero-neighbors decoding.
We discuss relations between the two decoding methods. In particular, we
show that for even codes the set of zero neighbors is strictly optimal
in this class of algorithms. This also implies that general asymptotic
improvements of the zero-neighbors algorithm in the frame of
gradient-like approach are impossible. We also discuss a link to
secret-sharing schemes
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