| In this work two applications for sequence designs are explored. The first is for ultra-wideband systems. These systems have the ability to operate in an electromagnetic spectrum that is already occupied by other narrowband users with little degradation to the performance of these existing users. However, the FCC issued a ruling containing regulations allowing transmission in the range of 3.1 GHz--10.6 GHz with some variations depending on the application. In order to satisfy this ruling, spectral shaping with code design is proposed. For this, constructions of families of polynomials over finite fields are presented, some of them based on permutation polynomials and the use of the trace function. Bounds on the code energy spectrum are presented and it is shown that with these coding techniques the FCC regulation can be satisfied.;The second application is for turbo codes. In 1993 this class of near channel-capacity-achieving codes was introduced by Berrou et al. The prototype turbo code consists of a pair of convolutional codes concatenated in parallel with a data interleaver (scrambler) between the inputs to the convolutional encoders. The interleaver plays a fundamental role in the performance of this type of code. Normally, interleavers are built by resorting to computer searches to identify them. In this work, we study algebraically constructed deterministic interleavers. Specifically, interleavers corresponding to families of permutation polynomials are considered. It is shown that this kind of interleaver performs as well as the random interleaver and that in some specific cases it outperforms them. |
