On the symmetry of Welch- and Golomb-constructed Costas arrays

AbstractWe prove that Welch Costas arrays are in general not symmetric and that there exist two special families of symmetric Golomb Costas arrays: one is the well-known Lempel family, while the other, although less well known, leads actually to the construction of dense Golomb rulers

A review of Costas arrays

Costas arrays are not only useful in radar engineering, but they also present many interesting, and still open, mathematical problems. This work collects in it all important knowledge about them available today: some history of the subjects, density results, construction methods, construction algorithms with full proofs, and open questions. At the same time all the necessary mathematical background is offered in the simplest possible format and terms, so that this work can play the role of a reference for mathematicians and mathematically inclined engineers interested in the field

On topics related to sum systems

For m ∈ N, we say that the m integer sets A1, . . . , Am ⊂ N0, form an m-part sum system if their sumset is the target set Xm j=1 Aj = n a1 + · · · + am : aj ∈ Aj , j ∈ {1, . . . , m} o = { 0, 1, 2, . . . ,Ym j=1 |Aj | − 1 } . That is to say, the sum over each element of the sets A1, . . . , Am uniquely generates the consecutive integers from 0 to Qm j=1 |Aj | − 1 with each integer appearing exactly once. Huxley, Lettington and Schmidt, in 2018, established a bijection between sum systems and sum-and-distance systems, utilising joint ordered factorisations, a specific form of ordered multi-factorisations, historically considered by MacMahon. They proved that for each m-part sum system there exists a corresponding m-part sum-and-distance system which generates the centro-symmetric set of consecutive (half) integers symmetric around the origin { − 1/2 (Ym j=1 |Aj | − 1 ), . . . , 1/2 (Ym j=1 |Aj | − 1 ) . In this thesis, we extend the results of Huxley, Lettington and Schmidt to obtain a unifying theory underpinning sum-and-distance systems, expressing their structures in terms of joint ordered-factorisations, thus enabling explicit construction formulae to be established via these factorisations. This unifying theory occurs when one allows consecutive half integers in the target set, when at least one component sum-and-distance set has even cardinality, leading to an invariance in the sum over weighted averages of the sum of squares across the sum-and-distance system component sets to be deduced. Further results include the application of associated divisor functions and Stirling numbers of the second kind, to enumerate all m-part joint ordered factorisations Nm(N) for a given positive integer N = n1 × n2 × . . . nm. We go on to show that the counting function Nm(N) satisfies an implicit three term recurrence relation proving an important relation in additive combinatorics. Additionally, sum systems (mod N + z), are considered, as well as orbit structures arising from very simple joint ordered factorisations. The latter leads to connections with cyclotomy