81 research outputs found
The Unimodular Lattices of Dimension up to 23 and the Minkowski-Siegel Mass Constants
In an earlier paper we enumerated the integral lattices of determinant one and dimension not exceeding 20. The present paper extends this enumeration to dimension 23, finding 40 lattices of dimension 21, 68 of dimension 22, and 117 of dimension 23. We also give explicit formulae for the Minkowski-Siegel mass constants for unimodular lattices (apparently not stated correctly elsewhere in the literature) and an exact table of the mass constants up to 32 dimensions, which provided a valuable check on our enumeration
Weight enumerators of self-orthogonal codes
AbstractCanonical forms are given for (i) the weight enumerator of an |n, 12(n−1)| self-orthogonal code, and (ii) the split weight enumerator (which classifies the codewords according to the weight of the left-and right-half words) of an |n, 12n| self-dual code
Complete Weight Enumerators of Generalized Doubly-Even Self-Dual Codes
For any q which is a power of 2 we describe a finite subgroup of the group of
invertible complex q by q matrices under which the complete weight enumerators
of generalized doubly-even self-dual codes over the field with q elements are
invariant.
An explicit description of the invariant ring and some applications to
extremality of such codes are obtained in the case q=4
Physical Vacuum Properties and Internal Space Dimension
The paper addresses matrix spaces, whose properties and dynamics are
determined by Dirac matrices in Riemannian spaces of different dimension and
signature. Among all Dirac matrix systems there are such ones, which nontrivial
scalar, vector or other tensors cannot be made up from. These Dirac matrix
systems are associated with the vacuum state of the matrix space. The simplest
vacuum system realization can be ensured using the orthonormal basis in the
internal matrix space. This vacuum system realization is not however unique.
The case of 7-dimensional Riemannian space of signature 7(-) is considered in
detail. In this case two basically different vacuum system realizations are
possible: (1) with using the orthonormal basis; (2) with using the
oblique-angled basis, whose base vectors coincide with the simple roots of
algebra E_{8}.
Considerations are presented, from which it follows that the least-dimension
space bearing on physics is the Riemannian 11-dimensional space of signature
1(-)& 10(+). The considerations consist in the condition of maximum vacuum
energy density and vacuum fluctuation energy density.Comment: 19 pages, 1figure. Submitted to General Relativity and Gravitatio
Picture-Hanging Puzzles
We show how to hang a picture by wrapping rope around n nails, making a
polynomial number of twists, such that the picture falls whenever any k out of
the n nails get removed, and the picture remains hanging when fewer than k
nails get removed. This construction makes for some fun mathematical magic
performances. More generally, we characterize the possible Boolean functions
characterizing when the picture falls in terms of which nails get removed as
all monotone Boolean functions. This construction requires an exponential
number of twists in the worst case, but exponential complexity is almost always
necessary for general functions.Comment: 18 pages, 8 figures, 11 puzzles. Journal version of FUN 2012 pape
Directed Graph Representation of Half-Rate Additive Codes over GF(4)
We show that (n,2^n) additive codes over GF(4) can be represented as directed
graphs. This generalizes earlier results on self-dual additive codes over
GF(4), which correspond to undirected graphs. Graph representation reduces the
complexity of code classification, and enables us to classify additive (n,2^n)
codes over GF(4) of length up to 7. From this we also derive classifications of
isodual and formally self-dual codes. We introduce new constructions of
circulant and bordered circulant directed graph codes, and show that these
codes will always be isodual. A computer search of all such codes of length up
to 26 reveals that these constructions produce many codes of high minimum
distance. In particular, we find new near-extremal formally self-dual codes of
length 11 and 13, and isodual codes of length 24, 25, and 26 with better
minimum distance than the best known self-dual codes.Comment: Presented at International Workshop on Coding and Cryptography (WCC
2009), 10-15 May 2009, Ullensvang, Norway. (14 pages, 2 figures
Fermat-linked relations for the Boubaker polynomial sequences via Riordan matrices analysis
The Boubaker polynomials are investigated in this paper. Using Riordan
matrices analysis, a sequence of relations outlining the relations with
Chebyshev and Fermat polynomials have been obtained. The obtained expressions
are a meaningful supply to recent applied physics studies using the Boubaker
polynomials expansion scheme (BPES).Comment: 12 pages, LaTe
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