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On composition of four-symbol $ \delta$-codes and Hadamard matrices


Author: C. H. Yang
Journal: Proc. Amer. Math. Soc. 107 (1989), 763-776
MSC: Primary 94B60; Secondary 05B20, 62K10
DOI: https://doi.org/10.1090/S0002-9939-1989-0979054-5
MathSciNet review: 979054
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Abstract: It is shown that key instruments for composition of four-symbol $ \delta $-codes are the Lagrange identity for polynomials, a certain type of quasisymmetric sequences (i.e., a set of normal or near normal sequences) and base sequences. The following is proved: If a set of base sequences for length $ t$ and a set of normal (or near normal) sequences for length $ n$ exist then four-symbol $ \delta $-codes of length $ \left( {2n + 1} \right)t\left( {{\text{or }}nt} \right)$ can be composed by application of the Lagrange identity. Consequently a new infinite family of Hadamard matrices of order $ 4uw$ can be constructed, where $ w$ is the order of Williamson matrices and $ u = \left( {2n + 1} \right)t\left( {{\text{or }}nt} \right)$. Other related topics are also discussed.


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  • [A] S. S. Agaian, Hadamard matrices and their applications, Springer-Verlag, Berlin, 1985. MR 818740 (87k:05038)
  • [GS] A. V. Geramita and J. Seberry, Orthogonal designs, Dekker, New York, 1979. MR 534614 (82a:05001)
  • [Go] J. M. Goethals and J. J. Seidel, A skew Hadamard matrix of order 36, J. Aust. Math. Soc. 11 (1970), 343-344. MR 0269527 (42:4422)
  • [G] M. J. E. Golay, Complementary series, IRE Trans. Information Theory, IT-7 (1961), 82-87. MR 0125799 (23:A3096)
  • [H.] M. Hall, Jr., Combinatorial theory, 2nd ed., Wiley and Sons, New York, 1986. MR 840216 (87j:05001)
  • [HS] M. Harwit and N. J. A. Sloane, Hadamard transform optics, Academic Press, New York, 1979.
  • [HW] A. Hedayat and W. D. Wallis, Hadamard matrices and their applications, Ann. Math. Stat. 6 (1978), 1184-1238. MR 523759 (80e:05037)
  • [K] C. Koukouvinos, S. Kounias and J. Seberry, Further results on base sequences, disjoint complementary sequences, $ OD\left( {4t;t,t,t} \right)$ and the excess of Hadamard matrices, (to appear).
  • [M] A. C. Mukhopadyay, Some infinite classes of Hadamard matrices, J. Combin. Theory (A) 25 (1978), 128-141. MR 509438 (80c:05046)
  • [S] E. Spence, An infinite family of Williamson matrices, J. Aust. Math. Soc. (A) 24 (1977), 252-256. MR 0505639 (58:21697)
  • [T1] R. J. Turyn, Hadamard matrices, Baumert-Hall units, four symbol sequences, pulse compression, and surface wave encodings, J. Combin. Theory (A) 16 (1974), 313-333. MR 0345847 (49:10577)
  • [T2] -, Personal communication, 1980.
  • [T3] -, An infinite class of Williamson matrices, J. Combin. Theory (A) 12 (1972), 319-321. MR 0299503 (45:8551)
  • [W] J. Seberry Wallis, Construction of Williamson type matrices, J. Linear and Multilinear Alg. 3 (1975), 197-207. MR 0396299 (53:167)
  • [Wi] J. Williamson, Hadamard's determinant theorem and the sum of four squares, Duke Math. J. 11 (1944), 65-81. MR 0009590 (5:169g)
  • [Y1] C. H. Yang, A composition theorem for $ \delta $-codes, Proc. Amer. Math. Soc. 89 (1983), 375-378. MR 712655 (85i:94025)
  • [Y2] -, Lagrange identity for polynomials and $ \delta $-codes of lengths $ 7t$ and $ 13t$, Proc. Amer. Math. Soc. 88 (1983), 746-750. MR 702312 (85f:05034)
  • [Y3] -, Hadamard matrices and $ \delta $-codes of length $ 3n$, Proc. Amer. Math. Soc. 85 (1982),
  • [Y4] -, Hadamard matrices, finite sequences, and polynomials defined on the unit circle, Math. Comp. 33 (1979), 688-693. MR 525685 (80i:05024)
  • [Y5] -, On Golay sequences and near normal sequences, (to appear).

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DOI: https://doi.org/10.1090/S0002-9939-1989-0979054-5
Article copyright: © Copyright 1989 American Mathematical Society

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