86 research outputs found
Almost separating and almost secure frameproof codes over q-ary alphabets
The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-015-0060-zIn this paper we discuss some variations of the notion of separating code for alphabets of arbitrary size. We show how the original definition can be relaxed in two different ways, namely almost separating and almost secure frameproof codes, yielding two different concepts. The new definitions enable us to obtain codes of higher rate, at the expense of satisfying the separating property partially. These new definitions become useful when complete separation is only required with high probability, rather than unconditionally. We also show how the codes proposed can be used to improve the rate of existing constructions of families of fingerprinting codes.Peer ReviewedPostprint (author's final draft
Constructions of almost secure frameproof codes with applications to fingerprinting schemes
The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-017-0359-zThis paper presents explicit constructions of fingerprinting codes. The proposed constructions use a class of codes called almost secure frameproof codes. An almost secure frameproof code is a relaxed version of a secure frameproof code, which in turn is the same as a separating code. This relaxed version is the object of our interest because it gives rise to fingerprinting codes of higher rate than fingerprinting codes derived from separating codes. The construction of almost secure frameproof codes discussed here is based on weakly biased arrays, a class of combinatorial objects tightly related to weakly dependent random variables.Peer ReviewedPostprint (author's final draft
A construction of traceability set systems with polynomial tracing algorithm
© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.A family F of w-subsets of a finite set X is called a set system with the identifiable parent property if for any w-subset contained in the union of some t sets, called traitors, of F at least one of these sets can be uniquely determined, i.e. traced. A set system with traceability property (TSS, for short) allows to trace at least one traitor by minimal distance decoding of the corresponding binary code, and hence the complexity of tracing procedure is of order O(M), where M is the number of users or the code's cardinality. We propose a new construction of TSS which is based on the old Kautz-Singleton concatenated construction with algebraic-geometry codes as the outer code and Guruswami-Sudan decoding algorithm. The resulting codes (set systems) have exponentially many users (codevectors) M and polylog(M) complexity of code construction and decoding, i.e. tracing traitors. This is the first construction of traceability set systems with such properties.Peer ReviewedPostprint (author's final draft
Spherical codes, maximal local packing density, and the golden ratio
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd
involves the placement of N nonoverlapping spheres of unit diameter near an
additional fixed unit-diameter sphere such that the greatest distance from the
center of the fixed sphere to the centers of any of the N surrounding spheres
is minimized. Solutions to the DLP problem are relevant to the realizability of
pair correlation functions for packings of nonoverlapping spheres and might
prove useful in improving upon the best known upper bounds on the maximum
packing fraction of sphere packings in dimensions greater than three. The
optimal spherical code problem in Rd involves the placement of the centers of N
nonoverlapping spheres of unit diameter onto the surface of a sphere of radius
R such that R is minimized. It is proved that in any dimension, all solutions
between unity and the golden ratio to the optimal spherical code problem for N
spheres are also solutions to the corresponding DLP problem. It follows that
for any packing of nonoverlapping spheres of unit diameter, a spherical region
of radius less than or equal to the golden ratio centered on an arbitrary
sphere center cannot enclose a number of sphere centers greater than one more
than the number that can be placed on the region's surface.Comment: 12 pages, 1 figure. Accepted for publication in the Journal of
Mathematical Physic
Estimates of the optimal density and kissing number of sphere packings in high dimensions
The problem of finding the asymptotic behavior of the maximal density of
sphere packings in high Euclidean dimensions is one of the most fascinating and
challenging problems in discrete geometry. One century ago, Minkowski obtained
a rigorous lower bound that is controlled asymptotically by , where
is the Euclidean space dimension. An indication of the difficulty of the
problem can be garnered from the fact that exponential improvement of
Minkowski's bound has proved to be elusive, even though existing upper bounds
suggest that such improvement should be possible. Using a
statistical-mechanical procedure to optimize the density associated with a
"test" pair correlation function and a conjecture concerning the existence of
disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental
Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found
with an asymptotic behavior controlled by . Using the same
methods, we investigate whether this exponential improvement can be further
improved by exploring other test pair correlation functions correponding to
disordered packings. We demonstrate that there are simpler test functions that
lead to the same asymptotic result. More importantly, we show that there is a
wide class of test functions that lead to precisely the same exponential
improvement and therefore the asymptotic form is much
more general than previously surmised.Comment: 23 pages, 4 figures, submitted to Phys. Rev.
Robust parent-identifying codes and combinatorial arrays
An -word over a finite alphabet of cardinality is called a descendant of a set of words if for all A code \cC=\{x^1,\dots,x^M\} is said to have the -IPP property if for any -word that is a descendant of at most parents belonging to the code it is possible to identify at least one of them. From earlier works it is known that -IPP codes of positive rate exist if and only if .
We introduce a robust version of IPP codes which allows {unconditional} identification of parents even if some of the coordinates in can break away from the descent rule, i.e., can take arbitrary values from the alphabet, or become completely unreadable. We show existence of robust -IPP codes
for all and some positive proportion of such coordinates.
The proofs involve relations between IPP codes and combinatorial arrays with separating properties such as perfect hash functions and hash codes, partially hashing families and separating codes.
For we find the exact proportion of mutant coordinates (for several error scenarios) that permits unconditional identification
of parents
Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps
We apply the semidefinite programming approach developed in
arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps.
We compute new upper bounds for the one-sided kissing number in several
dimensions where we in particular get a new tight bound in dimension 8.
Furthermore we show how to use the SDP framework to get analytic bounds.Comment: 15 pages, (v2) referee comments and suggestions incorporate
Finding passwords by random walks: How long does it take?
We compare an efficiency of a deterministic "lawnmower" and random search
strategies for finding a prescribed sequence of letters (a password) of length
M in which all letters are taken from the same Q-ary alphabet. We show that at
best a random search takes two times longer than a "lawnmower" search.Comment: To appear in J. Phys. A, special issue on "Random Search Problem:
Trends and Perspectives", eds.: MEG da Luz, E Raposo, GM Viswanathan and A
Grosber
Application of Edwards' statistical mechanics to high dimensional jammed sphere packings
The isostatic jamming limit of frictionless spherical particles from Edwards'
statistical mechanics [Song \emph{et al.}, Nature (London) {\bf 453}, 629
(2008)] is generalized to arbitrary dimension using a liquid-state
description. The asymptotic high-dimensional behavior of the self-consistent
relation is obtained by saddle-point evaluation and checked numerically. The
resulting random close packing density scaling is
consistent with that of other approaches, such as replica theory and density
functional theory. The validity of various structural approximations is
assessed by comparing with three- to six-dimensional isostatic packings
obtained from simulations. These numerical results support a growing accuracy
of the theoretical approach with dimension. The approach could thus serve as a
starting point to obtain a geometrical understanding of the higher-order
correlations present in jammed packings.Comment: 13 pages, 7 figure
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