Constructing New Turyn Type Sequences, T-Sequences and Hadamard Matrices

This study investigates computational methods for the construction and enumeration of Turyn Type sequences, which can be used to construct T-Sequences and Hadamard matrices. Extending and optimizing an approach originally introduced by Hadi Kharaghani, we have performed the most comprehensive search for Turyn Type sequences to date. We have checked the enumeration of Turyn Type sequences and found it to be correct for lengths up to 30 and identified 66 additional sequences for length 32. We then estimated the total number of Turyn Type sequences of lengths 34 and 36. In order to identify longer sequences, the search algorithm was further optimized. This approach allowed the identification of 119 new Turyn Type sequences of lengths 34 and 101 new Turyn Type Sequences of length 36. We verified that each sequence leads to a unique Hadamard matrix. This calculation has allowed the identification of 101 new Hadamard matrices of order 428; only one had been identified prior to this analysis. We have also found 10 Turyn Type sequences of length 38. This directly leads to 10 of the 11 known T-sequences of length 113. Finally, we discovered the first three Turyn Type sequences of length 40 which are the longest known Turyn Type sequences.