Peter Dukes

Constructions for Permutation Codes in Powerline Communications

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.

On constant composition codes

A constant composition code over a k-ary alphabet has the property that the numbers of occurrences of the k symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. In this paper, direct and recursive methods are developed for the construction of constant composition codes.

Ternary Schedules for Energy-Limited Sensor Networks

Medium access control for multihop wireless sensor networks (WSNs) must be energy efficient because the battery-operated nodes are not practical to recharge. We give constructions for ternary schedules in which each node is in one of three states: transmitting, receiving, or asleep. For each hop (vi, vj), communication is effective only when vi is transmitting, vj is receiving, and no other node in proximity of vj is also transmitting. Since sensor nodes are prone to failure, the schedules should be independent of the detailed topology while supporting spatial reuse. We use arc-decompositions of the complete lambda-fold directed graph Koarrn into directed complete bipartite subgraphs Koarra,b as a model for ternary scheduling in WSNs. We associate the vertices of Koarrn with the nodes of the WSN, and occurrences of Koarra,bs (blocks) in the decomposition with time slots in the schedule. A block with out-vertices A and in-vertices B corresponds to a slot in which the a nodes in A are transmitting, the b in B are receiving, and all others are asleep. Such a decomposition of lambdaKoarrnguarantees that every ordered pair of nodes in the WSN can communicate in lambda time slots.

An Existence Theory for Incomplete Designs

An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a ‘hole’. If there are v points, a hole of size w, and all (other) block sizes equal k, this is denoted IPBD((v;w), k). In addition to congruence restrictions on v and w, there is also a necessary inequality: v > (k − 1)w. _is article establishes two main existence results for IPBD((v;w), k): one in which w is ûxed and v is large, and the other in the case v > (k − 1+ є)w when w is large (depending on є). Several possible generalizations of the problem are also discussed.

Topology-transparent schedules for energy limited ad hoc networks

In an ad hoc network, each node can be in one of three states: asleep (powered down), listening, or transmitting. Communication is effective only when the sender is transmitting, the destination is receiving, and no other nodes in proximity to the receiver are also transmitting. Our strategy makes no assumptions about knowledge of neighbours or of geographical position; it is topology-transparent. A general combinatorial model for topology-transparent scheduling that treats energy conservation is described. As in the two state (transmit and receive) case, the combinatorial requirements are met by a D cover-free family. Graph designs, where an arc from vertex x to y indicates an opportunity for x to transmit and y to receive, are proposed as a model for schedule construction. In order to achieve reasonable throughput while obtaining a dramatic reduction in energy consumption, we focus on Koarra,a designs, where the number of nodes transmitting and receiving per slot is equal to a. Patterned on constructions for resolvable designs, we examine a computational search method to meet the required combinatorial conditions

On the structure of uniform one-factorizations from starters in finite fields

It is known that the Horton starters can be used to construct uniform one-factorizations of the complete graph. Of primary interest is the cycle structure of such one-factorizations. In this paper we give some general conditions for the existence of k-cycles, then specialize this to the cases k=4,6, completely characterizing the four-cycle case. We also show that for each even k>4 and any positive integer N there exists a uniform one-factorization in some large enough complete graph containing at least N k-cycles.

A combinatorial error bound for t -point-based sampling

The method of two-point-based sampling using orthogonal arrays (Inform. Process. Lett. 60 (1996) 91) is extended to consider t-wise independent sampling using orthogonal arrays of higher strength t. Using combinatorial considerations, an error bound is calculated which agrees with the previously known result when t = 2, and has the advantage of exponentially decreasing in t. The result is shown to be strictly sharper than that arising from the generalized Chebyshev inequality. Finally, the behavior of the family of error bounds we obtain for increasing values of t is analyzed.

Matching divisible designs with block size four

We consider edge-decompositions of the graph join of several equal-sized one-factors into cliques of a prescribed size. These objects are variants of group divisible designs and have applications to packings, coverings, and embeddings. Assuming block (clique) size four, we show that the obvious divisibility and counting conditions are sufficient for the existence of such designs.

Pairwise Balanced Designs with Prescribed Minimum Dimension

The dimension of a linear space is the maximum positive integer d such that any d of its points generate a proper subspace. For a set K of integers at least two, recall that a pairwise balanced design

Generalized Cover-Free Families for Topology-Transparent Channel Assignment

\operatorname{PBD}(v,K)