Improved Bounds on Sidon Sets via Lattice Packings of Simplices

Abstract

Abstract.A B h set (or Sidon set of order h) in an Abelian group G is any subset {b0, b1, . . . , bn} of G with the property that all the sums bi 1 + · · · + bi h are different up to the order of the summands. Let φ(h, n) denote the order of the smallest Abelian group containing a B h set of cardinality n + 1. It is shown thatwhere δl(△ n ) is the lattice packing density of an n-simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known (n ≤ 3) and gives improved bounds on φ(h, n) in the remaining cases. The corresponding geometric characterization of bases of order h in finite Abelian groups in terms of lattice coverings by simplices is also given.

Codes in the Space of Multisets—Coding for Permutation Channels With Impairments

“…The idea is to set l large enough so that the first summand on the right-hand side of (30) is minimized, but small enough so that the second summand is still negligible compared to the first one. Observe the relation (29) and the second summand on the right-hand side of (30). From (29) we see that, as n → ∞ and q ∼qn, the sum |△ q−1 n | = (see (34)), and since it has linearly many summands, there must exist λ ∈ (0, 1) such that q λn n−1 λn grows exponentially in n with the same exponent.…”

“…Observe the relation (29) and the second summand on the right-hand side of (30). From (29) we see that, as n → ∞ and q ∼qn, the sum |△ q−1 n | = (see (34)), and since it has linearly many summands, there must exist λ ∈ (0, 1) such that q λn n−1 λn grows exponentially in n with the same exponent. By using Stirling's approximation, one can find the exponent of q λn n−1 λn as a function of λ, and check by differentiation that it is maximized for unique λ = λ * =q 1+q .…”

“…Let also φ(h, q) denote the order of the smallest Abelian group containing a B h set of cardinality q. The lower bounds that follow will be expressed in terms of this quantity; more explicit lower bounds stated in terms of the parameters h, q can be obtained from the known upper bounds on φ(h, q) [44], [7], [25], [29].…”

“…By using Stirling's approximation, one can find the exponent of q λn n−1 λn as a function of λ, and check by differentiation that it is maximized for unique λ = λ * =q 1+q . Informally speaking, the term q λ * n n−1 λ * n in the sum (29), and the terms "immediately around it", account for most of the space △ q−1 n , and the remaining terms are negligible. More precisely, there exists a sublinear function f (n) = o(n) such that [46].…”

Runlength-Limited Sequences and Shift-Correcting Codes: Asymptotic Analysis

“…. , bm} ⊆ G with the property that all its t-sums (b i 1 + · · · + b it ) are distinct, up to the order of the summands; see[24],[34].…”