18,599 research outputs found
Optimal Partitioned Cyclic Difference Packings for Frequency Hopping and Code Synchronization
Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise
to optimal frequency-hopping sequences and optimal comma-free codes. New
constructions for PCDPs, based on almost difference sets and cyclic difference
matrices, are given. These produce new infinite families of optimal PCDPs (and
hence optimal frequency-hopping sequences and optimal comma-free codes). The
existence problem for optimal PCDPs in , with base blocks
of size three, is also solved for all .Comment: to appear in IEEE Transactions on Information Theor
Constructing Numerical Semigroups of a Given Genus
Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros
conjectured that n_g possesses certain Fibonacci-like properties. Almost all
previous attempts at proving this conjecture were based on analyzing the
semigroup tree. We offer a new, simpler approach to counting numerical
semigroups of a given genus. Our method gives direct constructions of families
of numerical semigroups, without referring to the generators or the semigroup
tree. In particular, we give an improved asymptotic lower bound for n_g.Comment: 11 pages, 3 figures, 2 tables; accepted by Semigroup Foru
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