Set Of Minimal Vectors Research Articles

Spherical designs and lattices

In this article we prove that integral lattices with minimum \(\le 7\) (or \(\le 9\)) whose set of minimal vectors form spherical \(9\)-designs (or \(11\)-designs respectively) are extremal, even and unimodular. We furthermore show that there does not exist an integral lattice with minimum \(\le 11\) which yields a \(13\)-design.

The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice

We show that the Minkowski sum of the Voronoi polytope of the root lattice and the zonotope is a 7-dimensional parallelohedron if and only if the set consists of minimal vectors of the dual lattice up to scalar multiplication, and does not contain forbidden sets. The minimal vectors of are the vectors of the classical root system . If the -norm of the roots is set equal to 2, then the scalar products of minimal vectors from the dual lattice only take the values . A set of minimal vectors is referred to as forbidden if it consists of six vectors, and the directions of some of these vectors can be changed so as to obtain a set of six vectors with all the pairwise scalar products equal to .Bibliography: 11 titles.

ON WELL-ROUNDED IDEAL LATTICES

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.

Minimal vectors in linear codes

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.

ITA: An algorithm for I DDQ testability analysis

This paper presents the IDDQ Testability Analysis (ITA) algorithm for the estimation of a circuit design's leakage fault testability. The algorithm is based on the calculation of the probability of applying each of a set of “essential vectors” to each gate in the circuit. The essential vectors for each gate represent the minimal vector set that provides maximal leakage fault coverage.

Structure functions with finite minimal vector sets

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

Structure functions with finite minimal vector sets

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.