Estimates of lengths of shortest nonzero vectors in some lattices. I
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Alexander S. Rybakov
Abstract
In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here we give a better estimate for the cardinality of the set of exceptional lattices for which the above estimates are not valid. In the case of dimension 4 we improve the upper estimate for an arbitrary chosen parameter that controls the accuracy of these lower estimates and for the number of exceptions. In this (first) part of the paper, we also prove some auxiliary results, which will be used in the second (main) part of the paper, in which an analogue of A. Friese et al. result will be given for dimension 5.
Originally published in Diskretnaya Matematika (2021) 33, №1, 31–46 (in Russian).
References
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Articles in the same Issue
- Frontmatter
- Diagnostic tests for discrete functions defined on rings
- New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk
- On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions
- On some limit properties for the power series distribution
- Estimates of lengths of shortest nonzero vectors in some lattices. I
Articles in the same Issue
- Frontmatter
- Diagnostic tests for discrete functions defined on rings
- New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk
- On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions
- On some limit properties for the power series distribution
- Estimates of lengths of shortest nonzero vectors in some lattices. I
