Hadamard matrices, finite sequences, and polynomials defined on the unit circle
HTML articles powered by AMS MathViewer
- by C. H. Yang PDF
- Math. Comp. 33 (1979), 688-693 Request permission
Abstract:
If a $(\ast )$-type Hadamard matrix of order 2n (i.e. a pair (A, B) of $n \times n$ circulant (1,-1) matrices satisfying $AA\prime + BB\prime = 2nI$) exists and a pair of Golay complementary sequences (or equivalently, two-symbol $\delta$-code) of length m exists, then a $(\ast )$-type Hadamard matrix of order 2mn also exists. If a Williamson matrix of order 4n (i.e. a quadruple (W, X, Y, Z) of $n \times n$ symmetric circulant (1,-1) matrices satisfying ${W^2} + {X^2} + {Y^2} + {Z^2} = 4nI$) exists and a four-symbol $\delta$-code of length m exists, then a Goethals-Seidel matrix of order 4mn (i.e. a quadruple (A, B, C, D) of $mn \times mn$ circulant (1, -1) matrices satisfying $AA\prime + BB\prime + CC\prime + DD\prime = 4mnI$) also exists. Other related topics are also discussed.References
- J.-M. Goethals and J. J. Seidel, A skew Hadamard matrix of order $36$, J. Austral. Math. Soc. 11 (1970), 343–344. MR 0269527
- Marcel J. E. Golay, Complementary series, IRE Trans. IT-7 (1961), 82–87. MR 0125799, DOI 10.1109/tit.1961.1057620
- Marcel J. E. Golay, Complementary series, IRE Trans. IT-7 (1961), 82–87. MR 0125799, DOI 10.1109/tit.1961.1057620 E. SPENCE, "Hadamard matrices of order $2{q^r}(q + 1)$ and ${q^r}(q + 1)$," Notices Amer. Math. Soc., v. 23, 1976, p. A-353.
- Edward Spence, Skew-Hadamard matrices of the Goethals-Seidel type, Canadian J. Math. 27 (1975), no. 3, 555–560. MR 384572, DOI 10.4153/CJM-1975-066-9 E. SPENCE, "Skew-Hadamard matrices of order $2(q + 1)$," Notices Amer. Math. Soc., v. 22, 1975, p. A-303.
- 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, DOI 10.1016/0097-3165(74)90056-9
- Richard J. Turyn, An infinite class of Williamson matrices, J. Combinatorial Theory Ser. A 12 (1972), 319–321. MR 299503, DOI 10.1016/0097-3165(72)90095-7 R. J. TURYN, "Computation of certain Hadamard matrices," Notices Amer. Math. Soc., v. 20, 1973, p. A-1.
- Yasuo Taki, Hiroshi Miyakawa, Mitsutoshi Hatori, and Seiichi Namba, Even-shift orthogonal sequences, IEEE Trans. Inform. Theory IT-15 (1969), 295–300. MR 255290, DOI 10.1109/tit.1969.1054284
- Jennifer Seberry Wallis, On Hadamard matrices, J. Combinatorial Theory Ser. A 18 (1975), 149–164. MR 379239, DOI 10.1016/0097-3165(75)90003-5
- Albert Leon Whiteman, Skew Hadamard matrices of Goethals—Seidel type, Discrete Math. 2 (1972), no. 4, 397–405. MR 304207, DOI 10.1016/0012-365X(72)90017-9 A. L. WHITEMAN, "Williamson type matrices of order $2q(q + 1)$," Notices Amer. Math. Soc., v. 21, 1974, p. A-623.
- Albert Leon Whiteman, An infinite family of Hadamard matrices of Williamson type, J. Combinatorial Theory Ser. A 14 (1973), 334–340. MR 316274, DOI 10.1016/0097-3165(73)90010-1
- John Williamson, Hadamard’s determinant theorem and the sum of four squares, Duke Math. J. 11 (1944), 65–81. MR 9590
- C. H. Yang, On Hadamard matrices constructible by circulant submatrices, Math. Comp. 25 (1971), 181–186. MR 288037, DOI 10.1090/S0025-5718-1971-0288037-7
- C. H. Yang, Maximal binary matrices and sum of two squares, Math. Comput. 30 (1976), no. 133, 148–153. MR 0409235, DOI 10.1090/S0025-5718-1976-0409235-X
Additional Information
- © Copyright 1979 American Mathematical Society
- 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