Time-discrete one- and two- dimensional sequences and arrays find applications in signal processing and as aperture functions for electromagnetic and acoustic imaging. Applications of two dimensional perfect binary arrays are found in 2-D synchronization and time-frequency coding. These are also used in channel coding and cryptographic coding. This research involves the study of perfect arrays and their related mathematical objects. Wide-band digital communications, optical signal processing, radar and audio coding are other disciplines that have applications of these objects.
The conventional binary case { +1, -1} as the alphabet. The investigator extends the alphabet to the ternary and quaternary cases: the corresponding alphabets being {0, +1, -1} and {i,-i,+1,-1}. Here i denotes the complex fourth roots of unity. The problems studied pertain to perfect binary arrays, perfect ternary arrays, perfect quaternary arrays, almost perfect arrays and periodic complementary sequences. The investigator studies these via their algebraic formulations viz. difference sets, supplementary difference sets and divisible difference sets. New structural theorems will be proved using character sums of group ring elements over suitable abelian groups. These results would enable to produce new classes of sequences and arrays with desirable correlation properties.
