Circulant partial hadamard matrices: Construction via general difference sets and its application to fMRI experiments

Yuan Lung Lin, Frederick Kin Hing Phoa, Ming-Hung Kao

Research output: Contribution to journalArticle


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.

Original languageEnglish (US)
Pages (from-to)1715-1724
Number of pages10
JournalStatistica Sinica
Issue number4
Publication statusPublished - Oct 1 2017



  • Circulant partial hadamard matrices
  • Functional magnetic resonance imaging (fMRI)
  • General difference sets

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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