Fano Kaleidoscopes and their generalizations

In this work we introduce Fano Kaleidoscopes, Hesse Kaleidoscopes and their generalizations. These are a particular kind of colored designs for which we will discuss general theory, present some constructions and prove existence results. In particular, using difference methods we show the existence of both a Fano and a Hesse Kaleidoscope on $v$ points when $v$ is a prime or prime power congruent to 1$\pmod{6}$, $v\ne13$. In the Fano case this, together with known results on pairwise balanced designs, allows us to prove the existence of Kaleidoscopes of order $v$ for many other values of $v$; we discuss what the situation is, on the other hand, in the Hesse and general case.Comment: 19 page

Periodic Golay pairs and pairwise balanced designs

In this paper we exploit a relationship between certain pairwise balanced designs with v points and periodic Golay pairs of length v, to classify periodic Golay pairs of length less than 40. In particular, we construct all pairwise balanced designs with v points under specific block conditions having an assumed cyclic automorphism group, and using isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs, we complete a classification up to equivalence. This is done using the theory of orbit matrices and some compression techniques which apply to complementary sequences. We use similar tools to construct new periodic Golay pairs of lengths greater than 40 where classifications remain incomplete and demonstrate that under some extra conditions on its automorphism group, a periodic Golay pair of length 90 will not exist. Length 90 remains the smallest length for which existence of a periodic Golay pair is undecided. Some quasi-cyclic self-orthogonal codes are constructed as an added application

Resistant And Susceptible BIB Designs

Statistic

An asymptotic existence result on compressed sensing matrices

For any rational number $h$ and all sufficiently large $n$ we give a deterministic construction for an $n\times \lfloor hn\rfloor$ compressed sensing matrix with $(\ell_1,t)$-recoverability where $t=O(\sqrt{n})$. Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of $\epsilon$-equiangular frames, which we introduce as a generalisation of equiangular tight frames. The method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design. The $(\ell_1,t)$-recoverability performance is specified as a simple function of the parameters of the design. To obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201