A difference triangle set (D$\Delta$S) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct D$\Delta$Ss, we present a new type of combinatorial design, monotonic directed $(v,k,\lambda)$-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (D$\Delta$Ss). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) D$\Delta$Ss if the existence of a single regular D$\Delta$S is known.

Using cyclotomic numbers, a simple construction is presented for frequency-hopping (FH) sequences having optimal autocorrelation with respect to the well-known Lempel-Greenberger bound. Some optimal families of FH sequences are constructed. The simplicity of this technique makes it attractive for practical use.

Summary form only given. Topology transparent scheduling for medium access control is an attractive technique for mobile ad hoc networks (MANETs) and sensor networks. The transmission schedule for each node is fixed and guarantees a bounded delay independent of which nodes are its neighbours, as long as the network is not too dense. Constructions of and performance criteria for topology transparent schedules have been extensively studied however, to date, frame synchronization is assumed. Synchronization is a difficult problem for MANETs and sensor networks. We study the relationships among topology transparent schedules, synchronization, and maximum delay. Frame synchronization, slot synchronization, and asynchronous transmission are the three synchronization models for this study. For each synchronization model, the first question is: (1) Do topology transparent schedules exist? If the answer to this question is yes, then two further questions are natural: (2) How to construct topology transparent schedules? (3) What is the least maximum delay? For frame and slot synchronization these three questions are answered in earlier work. We give answers for these three basic questions for asynchronous networks.

We formulate a combinatorial model of impulse radio sequences (IRSs) to study sequence or signal design for ultra-wideband (UWB) radio with unmodulated time hopping. Using this combinatorial model for IRSs, we develop necessary and sufficient conditions for the existence of IRSs. Several novel constructions for IRSs with optimal correlation properties are given. The constructions involve the Welch construction for Costas arrays, a quadratic polynomial construction over finite fields, recursive techniques for optical orthogonal codes, and combinatorial design techniques using perfect Mendelsohn designs (PMDs).

Zhang (see IEEE Trans. Commun., vol.47, p.967-973, July 1999) proposed a special family of optical address codes, called strict optical orthogonal codes (S-OOCs), was proposed for fiber-optic code-division multiple-access (FO-CDMA) networks. Such codes can strictly guarantee both cross-correlation and autocorrelation functions constrained to have the value one in fully asynchronous data communications and ultra fast switching. In Zhang's work the theory and designs of S-OOC, plus several examples, comparison tables, and performance analyses were presented. In this article, we set up the equivalence between S-OOC and so-called difference triangle sets (DTS), which have been extensively studied previously. Thus, all the known constructions, bounds, and analyses for DTS can be directly applied to S-OOC.

We present a new recursive construction for (n,/spl omega/,/spl lambda//sub a/,/spl lambda//sub c/) optical orthogonal codes. For the case of /spl lambda//sub a/ = /spl lambda//sub c/ = /spl lambda/, this recursive construction enlarges the original family with /spl lambda/ unchanged, and produces a new family of asymptotically optimal codes, if the original family is asymptotically optimal. We call a code asymptotically optimal, following the definition of O. Moreno et al. (see ibid., vol.41, p.448-55, 1995), if, as n, the length of code, goes to infinity, the ratio of the number of codewords to the corresponding Johnson bound approaches unity.