Hadamard matrices, finite sequences, and polynomials defined on the unit circle

Author:
C. H. Yang

Journal:
Math. Comp. **33** (1979), 688-693

MSC:
Primary 05B20; Secondary 15A57

DOI:
https://doi.org/10.1090/S0025-5718-1979-0525685-8

MathSciNet review:
525685

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If a -type Hadamard matrix of order 2*n* (i.e. a pair (*A, B*) of circulant (1,-1) matrices satisfying ) exists and a pair of Golay complementary sequences (or equivalently, two-symbol -code) of length *m* exists, then a -type Hadamard matrix of order 2*mn* also exists. If a Williamson matrix of order 4*n* (i.e. a quadruple (*W, X, Y, Z*) of symmetric circulant (1,-1) matrices satisfying ) exists and a four-symbol -code of length *m* exists, then a Goethals-Seidel matrix of order 4*mn* (i.e. a quadruple (*A, B, C, D*) of circulant (1, -1) matrices satisfying ) also exists. Other related topics are also discussed.

**[1]**J.-M. Goethals and J. J. Seidel,*A skew Hadamard matrix of order 36*, J. Austral. Math. Soc.**11**(1970), 343–344. MR**0269527****[2]**Marcel J. E. Golay,*Complementary series*, IRE Trans.**IT-7**(1961), 82–87. MR**0125799**, https://doi.org/10.1109/tit.1961.1057620**[3]**Marcel J. E. Golay,*Complementary series*, IRE Trans.**IT-7**(1961), 82–87. MR**0125799**, https://doi.org/10.1109/tit.1961.1057620**[4]**E. SPENCE, "Hadamard matrices of order and ,"*Notices Amer. Math. Soc.*, v. 23, 1976, p. A-353.**[5]**Edward Spence,*Skew-Hadamard matrices of the Goethals-Seidel type*, Canadian J. Math.**27**(1975), no. 3, 555–560. MR**384572**, https://doi.org/10.4153/CJM-1975-066-9**[6]**E. SPENCE, "Skew-Hadamard matrices of order ,"*Notices Amer. Math. Soc.*, v. 22, 1975, p. A-303.**[7]**R. J. Turyn,*Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, and surface wave encodings*, J. Combinatorial Theory Ser. A**16**(1974), 313–333. MR**345847**, https://doi.org/10.1016/0097-3165(74)90056-9**[8]**Richard J. Turyn,*An infinite class of Williamson matrices*, J. Combinatorial Theory Ser. A**12**(1972), 319–321. MR**299503**, https://doi.org/10.1016/0097-3165(72)90095-7**[9]**R. J. TURYN, "Computation of certain Hadamard matrices,"*Notices Amer. Math. Soc.*, v. 20, 1973, p. A-1.**[10]**Yasuo Taki, Hiroshi Miyakawa, Mitsutoshi Hatori, and Seiichi Namba,*Even-shift orthogonal sequences*, IEEE Trans. Inform. Theory**IT-15**(1969), 295–300. MR**255290**, https://doi.org/10.1109/tit.1969.1054284**[11]**Jennifer Seberry Wallis,*On Hadamard matrices*, J. Combinatorial Theory Ser. A**18**(1975), 149–164. MR**379239**, https://doi.org/10.1016/0097-3165(75)90003-5**[12]**Albert Leon Whiteman,*Skew Hadamard matrices of Goethals—Seidel type*, Discrete Math.**2**(1972), no. 4, 397–405. MR**304207**, https://doi.org/10.1016/0012-365X(72)90017-9**[13]**A. L. WHITEMAN, "Williamson type matrices of order ,"*Notices Amer. Math. Soc.*, v. 21, 1974, p. A-623.**[14]**Albert Leon Whiteman,*An infinite family of Hadamard matrices of Williamson type*, J. Combinatorial Theory Ser. A**14**(1973), 334–340. MR**316274**, https://doi.org/10.1016/0097-3165(73)90010-1**[15]**John Williamson,*Hadamard’s determinant theorem and the sum of four squares*, Duke Math. J.**11**(1944), 65–81. MR**9590****[16]**C. H. Yang,*On Hadamard matrices constructible by circulant submatrices*, Math. Comp.**25**(1971), 181–186. MR**288037**, https://doi.org/10.1090/S0025-5718-1971-0288037-7**[17]**C. H. Yang,*Maximal binary matrices and sum of two squares*, Math. Comput.**30**(1976), no. 133, 148–153. MR**0409235**, https://doi.org/10.1090/S0025-5718-1976-0409235-X

Retrieve articles in *Mathematics of Computation*
with MSC:
05B20,
15A57

Retrieve articles in all journals with MSC: 05B20, 15A57

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0525685-8

Article copyright:
© Copyright 1979
American Mathematical Society