BSC4024105 Project Details
| Supervisor | Paul M. E. Shutler |
| Project Code | BSC4024105 |
| Title of Project | Families of Cyclic Difference Sets |
| Description | Cyclic difference sets are best thought of as a loop of tape with holes cut into it according to a certain pattern, so that when two copies of the tape are cyclically displaced relative to each other, the number of holes which coincide happens always to be a constant irrespective of the displacement. In technical language, this means that the cyclic auto-correlation of a cyclic difference set has flat side lobes. This defining property of cyclic difference sets (or CDS for short) means that they are enormously useful for a wide variety of image and signal processing applications, in particular something called coded aperture imaging (or CAI). Although this project will only touch briefly on these practical applications, they are important as they will guide certain aspects of the mathematics i.e. some of the mathematical choices will be made so as to favour outcomes which are useful to these practical applications. The basic aims of this projects are: a) to generate some small CDS and use them for simple signal/image processing, b) to understand how the main families of cyclic difference sets are generated, c) to prove that these main families of CDS have the required CDS properties, In particular, the project will focus on three principle sources of CDS: 1) small cyclic difference sets obtained by exhaustive search algorithms, 2) quadratic residue sets and their bi-quadratic and octic generalisations, 3) maximum length polynomial sequences and their Singer generalisations. |
| Pre-requisites | • Year 1-2 level undergraduate algebra is sufficient for the mathematics. • Familiarity with computer programming not required but advantageous. The selected candidate will be given a crash course in Yabasic at the start of the project if requried. The latest version of Yabasic is available from the official website www.yabasic.deand the Crash Course in Yabasic is available at http://math.nie.edu.sg/shutler/crashcourse.zip |
| References | L.D. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics 182, Springer-Verlag, Berlin, 1971. This is available as an e-book at NTU library also http://math.nie.edu.sg/shutler/baumert.zip P.M.E. Shutler, S.V. Springham, A. Talebitaher, Nucl. Instr. Methods Phys. Res. A 709 (2013) 129-142. This is available at http://math.nie.edu.sg/shutler/nima.zip |