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 & J. J. SEIDEL, "A skew Hadamard matrix of order 36,"*J. Austral. Math. Soc.*, v. 11, 1970, pp. 343-344. MR**0269527 (42:4422)****[2]**M. J. E. GOLAY, "Complementary series,"*IRE Trans. Information Theory*, v. IT-7, 1961, pp. 82-87. MR**0125799 (23:A3096)****[3]**M. J. E. GOLAY, "Note on complementary series,"*Proc. IRE*, v. 50, 1962, p. 84. MR**0125799 (23:A3096)****[4]**E. SPENCE, "Hadamard matrices of order and ,"*Notices Amer. Math. Soc.*, v. 23, 1976, p. A-353.**[5]**E. SPENCE, "Hadamard matrices of the Goethals-Seidel type,"*Canad. J. Math.*, v. 27, 1975, pp. 555-560. MR**0384572 (52:5446)****[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*, v. 16, 1974, pp. 313-333. MR**0345847 (49:10577)****[8]**R. J. TURYN, "An infinite class of Williamson matrices,"*J. Combinatorial Theory Ser. A*, v. 12, 1972, pp. 319-321. MR**0299503 (45:8551)****[9]**R. J. TURYN, "Computation of certain Hadamard matrices,"*Notices Amer. Math. Soc.*, v. 20, 1973, p. A-1.**[10]**Y. TAKI et al., "Even-shift orthogonal sequences,"*IEEE Trans. Information Theory*, V. IT-IS, 1969, pp. 295-300. MR**0255290 (40:8495)****[11]**J. S. WALLIS, "On Hadamard matrices,"*J. Combinatorial Theory Ser. A*, v. 18, 1975, pp. 149-164. MR**0379239 (52:145)****[12]**A. L. WHITEMAN, "Skew Hadamard matrices of Goethals-Seidel type,"*Discrete Math.*, v. 2, 1972, pp. 397-405. MR**0304207 (46:3342)****[13]**A. L. WHITEMAN, "Williamson type matrices of order ,"*Notices Amer. Math. Soc.*, v. 21, 1974, p. A-623.**[14]**A. L. WHITEMAN, "An infinite family of Hadamard matrices of Williamson type,"*J. Combinatorial Theory Ser. A*, v. 14, 1973, pp. 334-340. MR**0316274 (47:4822)****[15]**J. WILLIAMSON, "Hadamard's determinant theorem and sum of four squares,"*Duke Math. J.*, v. 11, 1944, pp. 65-81. MR**0009590 (5:169g)****[16]**C. H. YANG, "On Hadamard matrices constructible by circulant submatrices,"*Math. Comp.*, v. 25, 1971, pp. 181-186. MR**0288037 (44:5235)****[17]**C. H. YANG, "Maximal binary matrices and sum of two squares,"*Math. Comp.*, v. 30, 1976, pp. 148-153. MR**0409235 (53:12995)**

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