For any rational number h and all sufficiently large n we give a deterministic construction for an n× bhnc compressed sensing matrix with (`1, t)-recoverability where t = O( √ n). Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of -equiangular frames, which we introduce as a generalisation of equiangular tight frames. The method is general and produce...

We study binary linear codes constructed from fifty-four Hadamard 2-(71, 35, 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss the errorcorrecting capability of the obtained codes by majority l...

A pseudo (v, k, λ)-design is a pair (X,B) where X is a v-set and B = {B1, . . . , Bv−1} is a collection of k-subsets (blocks) of X such that each two distinct Bi, Bj intersect in λ elements; and 0 < λ < k < v − 1. We use the notion of pseudo designs to characterize graphs of order n whose spectrum contains either ±1 or ± √ 2 with multiplicity (n − 2)/2 or (n− 3)/2. It turns out that the subdivi...

Let H be a subgraph of G. An H-design (V, C) of order v and index μ is embedded into a G-design (X,B) of order v + w and index λ if μ ≤ λ, V ⊆ X and there is an injective mapping f : C → B such that B is subgraph of f (B) for every B ∈ C. For every pair of positive integers v, λ, (except when λ = 3 and v = 30, 34, 42, 46, 54, 58, 66 or λ = 5 and v = 19) we determine the minimum value of w such ...

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...

This work presents a recursive construction for simple t-designs using resolutions of the ingredient designs. The result extends a construction of t-designs in our recent paper [39]. Essentially, the method in [39] describes the blocks of a constructed design as a collection of block unions from a number of appropriate pairs of disjoint ingredient designs. Now, if some pairs of these ingredient...

A dodecagon quadrangle is the graph consisting of two cycles: a 12cycle (x1, x2, ..., x12) and a 4-cycle (x1, x4, x7, x10). A dodecagon quadrangle system [DQS] of order v and index λ [DQSλ] is a pair Σ = (X,B), where X is a finite set of v vertices and B is a collection of edge disjoint dodecagon quadrangles (called blocks) which partitions the edge set of λKv, the complete multigraph with vert...

Let G be a family of graphs whose edges are colored with elements from a set R of r colors. We assume no two vertices ofG are joined by more than one edge of color i for any i ∈ R, for each G ∈ G. K n will denote the complete graph with r edges joining any pair of distinct vertices, one of each of the r colors. We describe necessary and asymptotically sufficient conditions on n for the existenc...

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means t...

Let D be the triangle with attached edge (i. e. D is the graph having vertices {a0, a1, a2, a3} and edges {a0, a1}, {a0, a2}, {a1, a2}, {a0, a3}). J.C. Bermond and J. Schönheim [1], proved that a D-design of order n exists if and only if n ≡ 0 or 1 (mod 8). Let (W, C) be a nontrivial D-design of order n, n ≥ 8, and let V ⊂ W , |V | = v < n. We say that a handcuffed design (V,P) of order v and b...