On composition of four-symbol 𝛿-codes and Hadamard matrices

Yang, C. H.

doi:10.1090/S0002-9939-1989-0979054-5

On composition of four-symbol $\delta$-codes and Hadamard matrices
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Proc. Amer. Math. Soc. 107 (1989), 763-776 Request permission

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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On Golay sequences and near normal sequences