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

Abstract

It is shown that key instruments for composition of four-symbol δ \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 t and a set of normal (or near normal) sequences for length n n exist then four-symbol δ \delta -codes of length ( 2 n + 1 ) t ( or  n t ) \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 4 u w 4uw can be constructed, where w w is the order of Williamson matrices and u = ( 2 n + 1 ) t ( or  n t ) u = \left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right ) . Other related topics are also discussed.