Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On designs of maximal $ (+1,\,-1)$-matrices of order $ n\equiv 2({\rm mod}\ 4)$. II


Author: C. H. Yang
Journal: Math. Comp. 23 (1969), 201-205
MSC: Primary 65.35
DOI: https://doi.org/10.1090/S0025-5718-1969-0239748-1
Corrigendum: Math. Comp. 28 (1974), 1183.
Corrigendum: Math. Comp. 28 (1974), 1183-1184.
MathSciNet review: 0239748
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Finding maximal $ ( + 1, - 1)$-matrices $ {M_{2m}}$ of order $ 2m$ (with odd $ m$) constructible in the standard form

$\displaystyle \left( {\begin{array}{*{20}{c}} A & B \\ { - {B^T}} & {{A^T}} \\ \end{array} } \right)$

is reduced to the finding of two polynomials $ C(w)$, $ D(w)$(corresponding to the circulant submatrices $ A$, $ B$) satisfying

$\displaystyle \vert C(w)\vert^2 + \vert D(w){\vert^2} = \tfrac{1}{2}(m - 1),$ ($ *$)

, where $ w$ is any primitive $ m$th root of unity. Thus, all $ {M_{2m}}$ constructible by the standard form (see [4]) can be classified by the formula $ \left( * \right)$. Some new matrices $ {M_{2m}}$ for $ m = 25,27,31$, were found by this method.

References [Enhancements On Off] (What's this?)

  • [1] H. Ehlich, ``Determinantenabschätzungen für binäre Matrizen,'' Math. Z., v. 83, 1964, pp. 123-132. MR 0160792 (28:4003)
  • [2] C. H. Yang, ``Some designs for maximal $ ( + 1, - 1)$-determinant of order $ n \equiv 2(\bmod 4)$,'' Math. Comp., v. 20, 1966, pp. 147-148. MR 32 #5534. MR 0188093 (32:5534)
  • [3] C. H. Yang, ``A construction for maximal $ ( + 1, - 1)$-matrix of order 54,'' Bull. Amer. Math. Soc., v. 72, 1966, p. 293. MR 32 #5678. MR 0188239 (32:5678)
  • [4] C. H. Yang, ``On designs of maximal $ ( + 1, - 1)$-matrices of order $ n \equiv 2(\bmod 4)$,'' Math. Comp., v. 22, 1968, pp. 174-180. MR 0225476 (37:1069)
  • [5] J. Williamson, ``Hadamard's determinant theorem and the sum of four squares,'' Duke Math. J., v. 11, 1944, pp. 65-81. MR 5, 169. MR 0009590 (5:169g)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.35

Retrieve articles in all journals with MSC: 65.35


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1969-0239748-1
Article copyright: © Copyright 1969 American Mathematical Society

American Mathematical Society