Bounds on generalized Frobenius numbers

Fukshansky, Lenny; Schürmann, Achill

doi:10.1016/j.ejc.2010.11.001

Cited by 20 publications

(25 citation statements)

References 26 publications

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“…Bounds with almost the same dependencies on s were recently obtained by Fukshansky and Schürmann [11]. Their lower bound, however, is only valid for sufficiently large s. Aliev and Gruber [1] applied the results of Schinzel [21] to obtain a sharp lower bound for the Frobenius number in terms of the covering radius of a simplex.…”

Section: #G(t )mentioning

confidence: 56%

Generalized Frobenius numbers: bounds and average behavior

2012

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Abstract. Let n ≥ 2 and s ≥ 1 be integers and a = (a1, . . . , an) be a relatively prime integer n-tuple. The s-Frobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as n i=1 aixi in at least s different ways, where x1, ..., xn are non-negative integers. This natural generalization of the classical Frobenius number, F1(a), has been studied recently by a number of authors. We produce new upper and lower bounds for the s-Frobenius number by relating it to the so called s-covering radius of a certain convex body with respect to a certain lattice; this generalizes a well-known theorem of R. Kannan for the classical Frobenius number. Using these bounds, we obtain results on the average behavior of the s-Frobenius number, extending analogous recent investigations for the classical Frobenius number by a variety of authors. We also derive bounds on the s-covering radius, an interesting geometric quantity in its own right.

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Section: #G(t )mentioning

confidence: 56%

Generalized Frobenius numbers: bounds and average behavior

2012

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“…Accurately counting lattice points in more general domains is a topic of the utmost interest in lattice theory. In [18], an upper bound on the quantity | \Lambda \cap P | , where \Lambda \subset \BbbR n is a full lattice and P \subset \BbbR n an arbitrary polytope of dimension n \prime \leq n, is given. Further, [15] gives an upper bound on | \Lambda \cap S| , where S \subset \BbbR n is a bounded domain, of general narrow class s \geq 1.…”

Section: Empirical Study and Discussionmentioning

confidence: 99%

“…The reason for this small set of lattices with known closed form theta series is that efficient counting of lattice points in domains in arbitrary dimensions is still an open problem. While many results have been obtained over the last two decades, such as the results in [15,16,18], the settings are so general that the upper bounds on the number of lattice points in bounded domains are far from being tight, even for very simple lattices and domains. Thus, being able to efficiently compute even an approximated version of the theta series of an arbitrary lattice is a problem which is interesting in its own right.…”

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confidence: 99%

An Approximation of Theta Functions with Applications to Communications

Barreal¹,

Damir²,

Freij-Hollanti³

et al. 2020

SIAM J. Appl. Algebra Geometry

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Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However, the theta series is in general not known in closed form, excluding a small set of very special lattices. In this article, motivated by the practical applications as well as the mathematical problem itself, a simple approximation of the theta series of a lattice is derived. A rigorous analysis of its accuracy is provided. In relation to this, maximum-likelihood decoding in the context of compute-and-forward relaying is studied. Following previous work, it is shown that the related metric can exhibit a flat behavior, which can be characterized by the flatness factor of the decoding function. Contrary to common belief, we note that the decoding metric can be rewritten as a sum over a random lattice only when at most two sources are considered. Using a particular matrix decomposition, a link between the random lattice and the code lattice employed at the transmitter is established, which leads to an explicit criterion for code design, in contrast to implicit criteria derived previously. Finally, candidate lattices are examined with respect to the proposed criterion using the derived theta series approximation.

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“…Now one is interested on finding the largest number b that cannot be represented in more than k ways. Beck and Robins gave formulas for n = 2 of the k-Frobenius number, but for general n and k only bounds on the k-Frobenius number F k (a) are available (see [4], [7] and [48] for prior work). Since then, many number theorists have contributed to the topic (e.g., see [27,71] and the references there).…”

Section: Motivation Prior and Related Workmentioning

confidence: 99%

Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

Aliev

Loera

Louveaux

2016

Recent Trends in Combinatorics

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Given an integral d × n matrix A, the well-studied affine semigroup Sg(A) = {b : Ax = b, x ∈ Z n , x ≥ 0} can be stratified by the number of lattice points inside the parametric polyhedra P A (b) = {x : Ax = b, x ≥ 0}. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset Sg ≥k (A) of all vectors b ∈ Sg(A) such that P A (b) ∩ Z n has at least k solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-handside vectors b for which P A (b) ∩ Z n has exactly k solutions or fewer than k solutions. (2) A computational complexity theory. We show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of Sg ≥k (A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least k solutions.(3) Applications and computation for the k-Frobenius numbers. Using Generating functions we prove that for fixed n, k the k-Frobenius number can be computed in polynomial time. This generalizes a well-known result for k = 1 by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of k-Frobenius numbers and their relatives.

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Bounds on generalized Frobenius numbers

Cited by 20 publications

References 26 publications

Generalized Frobenius numbers: bounds and average behavior

Generalized Frobenius numbers: bounds and average behavior

An Approximation of Theta Functions with Applications to Communications

Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

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