Circulant partial Hadamard matrices

Craigen, R.; Faucher, G.; Low, Richard M.; Wares, T.

doi:10.1016/j.laa.2013.09.004

Cited by 18 publications

(19 citation statements)

References 9 publications

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“…H ∈ M M ×N (±1), the existence of the completion is known to be automatic at any M = N − K, with K ≤ 7. See [8], [16], and for more recent analysis of related completion problems [6].…”

Section: Completion Problemsmentioning

confidence: 99%

The quantum algebra of partial Hadamard matrices

Banica

Skalski

2015

Linear Algebra and its Applications

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Abstract. A partial Hadamard matrix is a matrix H ∈ M M×N (T) whose rows are pairwise orthogonal. We associate to each such H a certain quantum semigroup G of quantum partial permutations of {1, . . . , M } and study the correspondence H → G. We discuss as well the relation between the completion problems for a given partial Hadamard matrix and completion problems for the associated submagic matrix P ∈ M M (M N (C)), in both cases introducing certain criteria for the existence of the suitable completions.

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Section: Completion Problemsmentioning

confidence: 99%

The quantum algebra of partial Hadamard matrices

Banica

Skalski

2015

Linear Algebra and its Applications

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“…It can be easily proved that the equality holds only if r = 2 by counting the allocation of lambdas. Craigen et al (2013) has proven that m < n/2 as r = 0 by linear algebra, it can also be proven by a discussion of differences.…”

Section: Resultsmentioning

confidence: 89%

“…Following (Kao (2015)), the Paley difference sets can also be adopted to obtain some 0-H(m × n), although this was not pointed out in that paper. While this method gives an infinite number of 0-H(m × n), the value of m for each n may be relatively small compared to the maximum possible m. For example, the Paley difference set can achieve an 0-H(5 × 20), but the maximum m for this n = 20 is m = 7 (Craigen et al (2013); Low et al (2005)). Clearly, the existence of r-H(m × n) implies the existence of r-H((m − 1) × n).…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…As a major drawback, this difference method works only for limited situations, and fails to find designs for other cases that might be encountered in practice. Another useful combinatorial construction in design of experiments is the 'r-row-regular circulant partial Hadamard matrix' (Craigen et al (2013)). An application of these matrices in constructing good fMRI designs is also discussed in (Kao (2015)).…”

Section: Introductionmentioning

confidence: 99%

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Circulant Partial Hadamard Matrices: Construction via General Difference Sets and Its Application to fMRI Experiments

2018

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An m × n matrix A = (ai,j) is circulant if ai+1,j+1 = ai,j where the subscripts are reduced modulo n. A question arising in stream cypher cryptanalysis is reframed as follows: For given n, what is the maximum value of m for which there exists a circulant m × n (±1)-matrix A such that AA T = nIm. In 2013, Craigen et al. called such matrices circulant partial Hadamard matrices (CPHMs). They proved some important bounds and compiled a table of maximum values of m for small n via computer search. The matrices and algorithm are not in the literature. In this paper, we introduce general difference sets (GDSs), and derive a result that connects GDSs and CPHMs. We propose an algorithm, the difference variance algorithm (DVA), which helps us to search GDSs. In this work, the GDSs with respect to CPHMs listed by Craigen et al. when r = 0, 2 are found by DVA, and some new lower bounds are given for the first time.

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“…Universally optimal designs in D N for comparing HRFs of length K ≤ G + 1 can also be obtained by using a (G + 1)-by-N circulant partial Hadamard matrix with zero row sums. As described in Craigen et al (2013), for such a matrix, C = ((c g,n )) g=1,..., G+1,n=1,...,N , we have that the entries c g,n ∈ {−1, 1}, Cj N = 0, CC ′ = N I G+1 , and the gth row is obtained by cyclically shifting the (g − 1)st row one position to the right with c g,1 = c g−1,N and c g,n = c g−1,n−1 for g = 2, . .…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Universally optimal fMRI designs for comparing hemodynamic response functions

Kao¹

2015

STAT SINICA

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We consider an experimental design problem in functional magnetic resonance imaging (fMRI), a dominant technology for studying brain activity in response to mental stimuli presented to the experimental subject. In contrast to previous studies, we develop analytical results on optimal designs for comparing hemodynamic response functions, each describing the effect of the corresponding type of stimulus. In particular, for studies with two stimulus types, we derive a sufficient condition for an fMRI design to be universally optimal, and show that designs constructed via m-sequences or Paley difference sets satisfy this sufficient condition.

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Circulant partial Hadamard matrices

Cited by 18 publications

References 9 publications

The quantum algebra of partial Hadamard matrices

The quantum algebra of partial Hadamard matrices

Circulant Partial Hadamard Matrices: Construction via General Difference Sets and Its Application to fMRI Experiments

Universally optimal fMRI designs for comparing hemodynamic response functions

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