33 research outputs found
Computing coset leaders and leader codewords of binary codes
In this paper we use the Gr\"obner representation of a binary linear code
to give efficient algorithms for computing the whole set of coset
leaders, denoted by and the set of leader codewords,
denoted by . The first algorithm could be adapted to
provide not only the Newton and the covering radius of but also to
determine the coset leader weight distribution. Moreover, providing the set of
leader codewords we have a test-set for decoding by a gradient-like decoding
algorithm. Another contribution of this article is the relation stablished
between zero neighbours and leader codewords
Geodesic rewriting systems and pregroups
In this paper we study rewriting systems for groups and monoids, focusing on
situations where finite convergent systems may be difficult to find or do not
exist. We consider systems which have no length increasing rules and are
confluent and then systems in which the length reducing rules lead to
geodesics. Combining these properties we arrive at our main object of study
which we call geodesically perfect rewriting systems. We show that these are
well-behaved and convenient to use, and give several examples of classes of
groups for which they can be constructed from natural presentations. We
describe a Knuth-Bendix completion process to construct such systems, show how
they may be found with the help of Stallings' pregroups and conversely may be
used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory,
Dortmund and Carleton Conferences". Series: Trends in Mathematics.
Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009,
Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause
Complete Presentations Of Coxeter Groups â
A complete group presentation is a useful tool for performing computations with the specified group. Here we give complete presentations for the irreducible finite Coxeter groups Dn, and prove some of its properties. With this result, it becomes possible to construct the complete presentation of any finite Coxeter group. 1 Introduction. A complete presentation of a group or monoid M is a presentation of M that is complete when regarded as a string rewriting system. Such a presentation provides a straightforward solution to the word problem in M in a âsyntactical â fashion, since any word on the generators of M can be rewritten in a unique way as a canonical o
Computing Gröbner Bases by FGLM Techniques in a Noncommutative Setting
A generalization of the FGLM technique is given to compute Gröbner bases for two-sided ideals of free finitely generated algebras. Specializations of this algorithm are presented for the cases in which the ideal is determined by either functionals or monoid (group) presentations. Generalizations are discussed in order to compute Gröbner bases on (twisted) semigroup rings. c â 2000 Academic Press 1
Complete presentations of direct products of groups
Complete presentations provide a natural solution to the word problem in monoids and groups. Here we give a simple way to construct complete presentations for the direct product of groups, when such presentations are available for the factors. Actually, the construction we are referring to is just th