TY - JOUR

T1 - Circulant partial hadamard matrices

T2 - Construction via general difference sets and its application to fMRI experiments

AU - Lin, Yuan Lung

AU - Phoa, Frederick Kin Hing

AU - Kao, Ming-Hung

N1 - Funding Information: The authors deeply appreciate the careful corrections and constructive suggestions of an associate editor and two reviewers. This work was supported by Career Development Award of Academia Sinica (Taiwan) grant number 103-CDA-M04, and Ministry of Science and Technology (Taiwan) grant numbers

PY - 2017/10

Y1 - 2017/10

N2 - 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 AAT = 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.

AB - 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 AAT = 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.

KW - Circulant partial hadamard matrices

KW - Functional magnetic resonance imaging (fMRI)

KW - General difference sets

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U2 - 10.5705/ss.202016.0254

DO - 10.5705/ss.202016.0254

M3 - Article

SN - 1017-0405

VL - 27

SP - 1715

EP - 1724

JO - Statistica Sinica

JF - Statistica Sinica

IS - 4

ER -