On the existence of Turyn sequences of length less than 43

Edmondson, Genet M.; Seberry, Jennifer; Anderson, Malcolm R.

doi:10.1090/S0025-5718-1994-1203733-8

On the existence of Turyn sequences of length less than $43$
HTML articles powered by AMS MathViewer

by Genet M. Edmondson, Jennifer Seberry and Malcolm R. Anderson PDF

Math. Comp. 62 (1994), 351-362 Request permission

Abstract:

Some theoretical results and computational algorithms are described which verify previous calculations, prove some theoretical nonexistence results, and totally enumerate all inequivalent Turyn sequences of length less than 43, that is, the longer sequence has length less than 43.

References

Some properties of complementary sequences

H. Davenport, The higher arithmetic: An introduction to the theory of numbers, Harper Torchbooks/The Science Library, Harper & Brothers, New York, 1960. MR 0109802

More non-existent Turyn sequences

Shalom Eliahou, Michel Kervaire, and Bahman Saffari, A new restriction on the lengths of Golay complementary sequences, J. Combin. Theory Ser. A 55 (1990), no. 1, 49–59. MR 1070014, DOI 10.1016/0097-3165(90)90046-Y
Anthony V. Geramita and Jennifer Seberry, Orthogonal designs, Lecture Notes in Pure and Applied Mathematics, vol. 45, Marcel Dekker, Inc., New York, 1979. Quadratic forms and Hadamard matrices. MR 534614
Marcel J. E. Golay, Complementary series, IRE Trans. IT-7 (1961), 82–87. MR 0125799, DOI 10.1109/tit.1961.1057620

Note on complementary series

50

Malcolm Griffin, There are no Golay complementary sequences of length $2\cdot 9^{t}$, Aequationes Math. 15 (1977), no. 1, 73–77. MR 490419, DOI 10.1007/BF01837875

Golay sequences

Christos Koukouvinos, Stratis Kounias, and Jennifer Seberry, Further results on base sequences, disjoint complementary sequences, $\textrm {OD}(4t;t,t,t,t)$ and the excess of Hadamard matrices, Ars Combin. 30 (1990), 241–255. MR 1088294
S. Kounias, C. Koukouvinos, and K. Sotirakoglou, On Golay sequences, Discrete Math. 92 (1991), no. 1-3, 177–185. MR 1140585, DOI 10.1016/0012-365X(91)90279-B
C. Koukouvinos, S. Kounias, and K. Sotirakoglou, On base and Turyn sequences, Math. Comp. 55 (1990), no. 192, 825–837. MR 1023764, DOI 10.1090/S0025-5718-1990-1023764-7
Christos Koukouvinos and Jennifer Seberry, Addendum to: “Further results on base sequences, disjoint complementary sequences, $\textrm {OD}(4t;t,t,t,t)$ and the excess of Hadamard matrices” [Ars Combin. 30 (1990), 241–255; MR1088294 (91m:05037)] by Koukouvinos, S. Kounias and Seberry, Proceedings of the Twenty-second Southeastern Conference on Combinatorics, Graph Theory, and Computing (Baton Rouge, LA, 1991), 1991, pp. 97–103. MR 1152061
Marcel J. E. Golay, Complementary series, IRE Trans. IT-7 (1961), 82–87. MR 0125799, DOI 10.1109/tit.1961.1057620
Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences, and block designs, Contemporary design theory, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992, pp. 431–560. MR 1178508

Signal multiplexing in surface-wave delay lines using orthogonal pairs of Golay’s complementary sequences

18

C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inform. Theory IT-18 (1972), 644–652. MR 398660, DOI 10.1109/tit.1972.1054860
Richard J. Turyn, Four-phase Barker codes, IEEE Trans. Inform. Theory IT-20 (1974), 366–371. MR 376236, DOI 10.1109/tit.1974.1055224
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

Quaternary codes for pulsed radar

IT-6

Autocorrelation of

sequences

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
C. H. Yang, Hadamard matrices, finite sequences, and polynomials defined on the unit circle, Math. Comp. 33 (1979), no. 146, 688–693. MR 525685, DOI 10.1090/S0025-5718-1979-0525685-8
C. H. Yang, Hadamard matrices and $\delta$-codes of length $3n$, Proc. Amer. Math. Soc. 85 (1982), no. 3, 480–482. MR 656128, DOI 10.1090/S0002-9939-1982-0656128-3
C. H. Yang, A composition theorem for $\delta$-codes, Proc. Amer. Math. Soc. 89 (1983), no. 2, 375–378. MR 712655, DOI 10.1090/S0002-9939-1983-0712655-2
C. H. Yang, Lagrange identity for polynomials and $\delta$-codes of lengths $7t$ and $13t$, Proc. Amer. Math. Soc. 88 (1983), no. 4, 746–750. MR 702312, DOI 10.1090/S0002-9939-1983-0702312-0
C. H. Yang, On composition of four-symbol $\delta$-codes and Hadamard matrices, Proc. Amer. Math. Soc. 107 (1989), no. 3, 763–776. MR 979054, DOI 10.1090/S0002-9939-1989-0979054-5