Statistica Sinica 27 (2017), 1715-1724

CIRCULANT PARTIAL HADAMARD MATRICES:

CONSTRUCTION VIA GENERAL DIFFERENCE SETS

AND ITS APPLICATION TO fMRI EXPERIMENTS

Yuan-Lung Lin^{1},Frederick Kin Hing Phoa^{1} and Ming-Hung Kao^{2}

Abstract: An is circulant if 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 **А** such that . 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.

Key words and phrases: Circulant partial hadamard matrices, functional magnetic resonance imaging (fMRI), general difference sets.