On Hadamard matrices constructible by circulant submatrices

Author:
C. H. Yang

Journal:
Math. Comp. **25** (1971), 181-186

MSC:
Primary 05.25

DOI:
https://doi.org/10.1090/S0025-5718-1971-0288037-7

Corrigendum:
Math. Comp. **28** (1974), 1183-1184.

Corrigendum:
Math. Comp. **28** (1974), 1183-1184.

MathSciNet review:
0288037

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Abstract | References | Similar Articles | Additional Information

Abstract: Let ${V_{2n}}$ be an *H*-matrix of order 2*n* constructible by using circulant $n \times n$ submatrices. A recursive method has been found to construct ${V_{4n}}$ by using circulant $2n \times 2n$ submatrices which are derived from $n \times n$ submatrices of a given ${V_{2n}}$. A similar method can be applied to a given ${W_{4n}}$, an *H*-matrix of Williamson type with odd *n*, to construct ${W_{8n}}$. All ${V_{2n}}$ constructible by the standard type, for $1 \leqq n \leqq 16$, and some ${V_{2n}}$, for $n \geqq 20$, are listed and classified by this method.

- L. D. Baumert and Marshall Hall Jr.,
*Hadamard matrices of the Williamson type*, Math. Comp.**19**(1965), 442โ447. MR**179093**, DOI https://doi.org/10.1090/S0025-5718-1965-0179093-2 - Marshall Hall Jr.,
*Combinatorial theory*, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1967. MR**0224481** - Herbert John Ryser,
*Combinatorial mathematics*, The Carus Mathematical Monographs, No. 14, Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1963. MR**0150048** - John Williamson,
*Hadamardโs determinant theorem and the sum of four squares*, Duke Math. J.**11**(1944), 65โ81. MR**9590** - C. H. Yang,
*On designs of maximal $(+1,\,-1)$-matrices of order $n\equiv 2({\rm mod}\ 4)$*, Math. Comp.**22**(1968), 174โ180. MR**225476**, DOI https://doi.org/10.1090/S0025-5718-1968-0225476-4 - C. H. Yang,
*On designs of maximal $(+1,\,-1)$-matrices of order $n\equiv 2({\rm mod}\ 4)$. II*, Math. Comp.**23**(1969), 201โ205. MR**239748**, DOI https://doi.org/10.1090/S0025-5718-1969-0239748-1 - L. D. Baumert,
*Hadamard matrices of orders $116$ and $232$*, Bull. Amer. Math. Soc.**72**(1966), 237. MR**186567**, DOI https://doi.org/10.1090/S0002-9904-1966-11481-7

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Additional Information

Keywords:
Construction of Hadamard matrices,
circulant matrices,
standard type <I>H</I>-matrices,
Williamson type <I>H</I>-matrices,
recursive method for <I>H</I>-matrices,
table for some <I>H</I>-matrices

Article copyright:
© Copyright 1971
American Mathematical Society