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A 2- (v , k , λ) design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group G in such a way that its block set is contained in (or coincides with) the set of all zero-sum k-subsets of its point set. Explicit results on the additivity or strong additivity of symmetric designs and subspace 2-designs are presented. In particular, the strong additivity of PG d (n , q) , which was known to be additive only for q = 2 or d = n - 1 , is always established. [ABSTRACT FROM AUTHOR]

Based on the packing density of cross-polytopes in R n , more than 50 years ago Golomb and Welch proved that the packing density of Lee spheres in Z n must be strictly smaller than 1 provided that the radius r of the Lee sphere is large enough compared with n, which implies that there is no perfect Lee code for the corresponding parameters r and n. In this paper, we investigate the lattice packing density of Lee spheres with fixed radius r for infinitely many n. First we present a method to verify the nonexistence of the second densest lattice packing of Lee spheres of radius 2. Second, we consider the constructions of lattice packings with density δ n → 2 r (2 r + 1) r ! as n → ∞ . When r = 2 , the packing density can be improved to δ n → 2 3 as n → ∞ . [ABSTRACT FROM AUTHOR]

For a countable discrete amenable group G, it turns out that for any subsets H of G and E of Z with positive densities, there exists k ∈ N which depends only on the densities of H and E such that G k ⊂ (H · H - 1) E - E . [ABSTRACT FROM AUTHOR]

Device-to-Device (D2D) communication is proposed in cellular network as a vital technology to enhance spectral efficiency and system capacity, while reducing the energy consumption and the latency. Enabling D2D communications in multi-cell OFDMA systems poses two majors challenges: First, D2D equipements are characterizd by limited capacity of battery. Second, D2D devices inevitably suffer from co-channel interference (CCI) at cell edge. This paper addresses the mobility control (MC) for D2D in downlink multi-cell OFDMA system with cooperative relaying and Fractional Frequency Reuse (FFR) scheme. FFR and cooperative relaying schemes are proposed to improve energy efficiency (EE) and avoid CCI. In the edge region, according to a difference set and using the sectorization technique, the Frequency Reuse Factor (FRF) of 7/3 and 7/4 have been applied. The deployment of FFR with Amplify and Forward fixed relays improve significantly the system performance of FRF of 1 and 3. As shown in simulations results, the proposed cooperative MC scheme with FRF of 7/3 and 7/4 provides an improvement in term of EE nears 1 , 2. 10 6 bits/s/J and 1 , 3. 10 6 bits/s/J respectively in comparison with non-cooperative scheme with FRF=3. [ABSTRACT FROM AUTHOR]

Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein, which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we describe four direct constructions of optimal DSSs from finite projective geometries and present a recursive construction of DSSs by extending the known construction. As a consequence, new infinite families of optimal DSSs can be obtained. [ABSTRACT FROM AUTHOR]

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group's structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields. [ABSTRACT FROM AUTHOR]

The difference set Δ s , t of two (nondeterministic, in general) transducers s , t is the set of all input words for which the output sets of the two transducers are not equal. When the two transducers realize homomorphisms, their difference set is the complement of the well known equality set of the two homomorphisms. However, we show that transducer difference sets result in Chomsky-like classes of languages that are different than the classes resulting from equality sets. We also consider the following word problem: given transducers s , t and input w , tell whether the output sets s (w) and t (w) are different. In general the problem is PSPACE -complete, but it becomes NP -complete when at least one of the given transducers has finite outputs. We also provide a PRAX (polynomial randomized approximation) algorithm for the word problem as well as for the NFA (in)equivalence problem. Our presentation of PRAX algorithms improves the original presentation. • The difference set of two transducers is the set of all input words for which the corresponding output sets are not equal. • We show that transducer difference sets result in language classes that are different than classes resulting from equality sets. • We also consider the following word problem: given transducers s , t and input w , tell whether s (w) and t (w) are different. • The problem is PSPACE-complete, but it becomes NP-complete when at least one of the given transducers has finite outputs. • We also provide polynomial randomized approximation (PRAX) algorithms for the word problem and for the NFA equivalence problem. • Our presentation of PRAX algorithms improves the original presentation. [ABSTRACT FROM AUTHOR]