In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.
We classify all the cocyclic Butson Hadamard matrices BH(n, p) of order n over the pth roots of unity for an odd prime p and np ≤ 100 . That is, we compile a list of matrices such that any cocyclic BH(n, p) for these n, p is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.
We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders by Reingold et al. (Ann Math 155(1):157–187, 2002). We say that a function $$f:\{-1,1\}^d\rightarrow \{-1,1\}^D$$ f : { - 1 , 1 } d → { - 1 , 1 } D is a correlation amplifier with threshold $$0\le \tau \le 1$$ 0 ≤ τ ≤ 1 , error $$\gamma \ge 1$$ γ ≥ 1 , and strength p an even positive integer if for all pairs of vectors $$x,y\in \{-1,1\}^d$$ x , y ∈ { - 1 , 1 } d it holds that (i) $$|\langle x,y\rangle |<\tau d$$ | ⟨ x , y ⟩ | < τ d implies $$|\langle f(x),f(y)\rangle |\le (\tau \gamma )^pD$$ | ⟨ f ( x ) , f ( y ) ⟩ | ≤ ( τ γ ) p D ; and (ii) $$|\langle x,y\rangle |\ge \tau d$$ | ⟨ x , y ⟩ | ≥ τ d implies $$\left (\frac{\langle x,y\rangle }{\gamma d}\right )^pD \le \langle f(x),f(y)\rangle \le \left (\frac{\gamma \langle x,y\rangle }{d}\right )^pD$$ ⟨ x , y ⟩ γ d p D ≤ ⟨ f ( x ) , f ( y ) ⟩ ≤ γ ⟨ x , y ⟩ d p D .
MSC: 05B05 42C15 94A08 Keywords: Compressed sensing Pairwise balanced designsFor any rational number h and all sufficiently large n we give a deterministic construction for an n × hn com-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 produces good compressed sensing matrices from any appropriately chosen pairwise balanced design. The ( 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.
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