On the non-minimal codewords in binary Reed--Muller codes
Discrete Applied Mathematics - Special issue: International workshop on coding and cryptography (WCC 2001)
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Communications and Information Theory
Secret Sharing Schemes with Nice Access Structures
Fundamenta Informaticae - SPECIAL ISSUE ON TRAJECTORIES OF LANGUAGE THEORY Dedicated to the memory of Alexandru Mateescu
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Information Processing Letters
On the covering structures of two classes of linear codes from perfect nonlinear functions
IEEE Transactions on Information Theory
Minimal codewords in Reed---Muller codes
Designs, Codes and Cryptography
Covering and secret sharing with linear codes
DMTCS'03 Proceedings of the 4th international conference on Discrete mathematics and theoretical computer science
On correctable errors of binary linear codes
IEEE Transactions on Information Theory
On the De Boer-Pellikaan method for computing minimum distance
Journal of Symbolic Computation
Secret sharing schemes from binary linear codes
Information Sciences: an International Journal
Smaller decoding exponents: ball-collision decoding
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Proceedings of the 4th international conference on Security of information and networks
European Journal of Combinatorics
Secret Sharing Schemes with Nice Access Structures
Fundamenta Informaticae - SPECIAL ISSUE ON TRAJECTORIES OF LANGUAGE THEORY Dedicated to the memory of Alexandru Mateescu
Note: The maximum number of minimal codewords in long codes
Discrete Applied Mathematics
Secret sharing schemes based on graphical codes
Cryptography and Communications
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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