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Let p be a prime, p a power of p and 𝔽q the finite field with q elements. Any function φ: 𝔽q → 𝔽q can be unqiuely represented by a polynomial, 𝔽φ of degree < q. If the map x ↦ Fφ(x) induces a permutation on the underlying field we say Fφ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F(x) induce a permutation on 𝔽q. In the second chapter we are concerned the permutation behavior of the polynomial gn,q, a q-ary version of the reversed Dickson polynomial, when the integer n is of the form n = qa - qb - 1. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial. In the fourth chapter we are concerned with permutation polynomials of 𝔽q, where q is even, that can be represented as the sum of a power function and a linearized polynomial. These types of permutation polynomials have applications in cryptography. Lastly, chapter five is concerned with a conjecture on monomial graphs that can be formulated in terms of polynomials over finite fields.

Let p be a prime and q = pk. The polynomial gn,q isin Fp[x] defined by the functional equation Sigmaa isin Fq (x+a)n = gn,q(xq- x) gives rise to many permutation polynomials over finite fields. We are interested in triples (n,e;q) for which gn,q is a permutation polynomial of Fqe. In Chapters 2, 3, and 4 of this dissertation, we present many new families of permutation polynomials in the form of gn,q. The permutation behavior of gn,q is becoming increasingly more interesting and challenging. As we further explore the permutation behavior of gn,q, there is a clear indication that gn,q is a plenteous source of permutation polynomials. We also describe a piecewise construction of permutation polynomials over a finite field Fq which uses a subgroup of Fq*, a “selection” function, and several “case” functions. Chapter 5 of this dissertation is devoted to this piecewise construction which generalizes several recently discovered families of permutation polynomials.

Modern wireless communication systems have seen an increased usage of various channel coding techniques to facilitate improved throughput and latency. Interleavers form an integral part of these coding techniques and play a critical role by making the communication more robust and resilient to noise and other interference. The ever increasing need for higher throughputs and lower latencies has made designers to pursue a more parallel design approach giving rise to parallel adaptations of these encoding/decoding techniques. A bulk of the modern telecommunication occurs over Wireless Wide Area Network (WWAN), commonly referred to as cellular networks. The 3rd Generation Partnership Project (3GPP), Long Term Evolution (LTE) develops and specifies the standards that are used in cellular communication. Their current most widely used "4G" standard employs Turbo coding techniques and a Quadratic Permutation Polynomial (QPP) interleaver. Silicon Large Grain Object or SiLago is a Coarse Grain Reconfigurable Fabric facilitating a modular approach towards electronics hardware development. The concept is similar to LEGO bricks, that is to have a library of hardened blocks (similar to Lego bricks) out of which systems of various types and functionalities can be built. This thesis investigates the state-of-the-art parallel interleavers and parallel interleaving techniques available for the 3GPP LTE QPP interleavers, and implements two interleaver designs, one for Radix 2 and another for Radix 4 decoding techniques. A physical synthesis is carried out in 28nm technology and the results in terms of power and area are reported. / Moderna trådlösa kommunikationssystem har sett ökad användning av olika kanaler kodningstekniker för att underlätta förbättrad genomströmning och latens. Interleavers utgör en integrerad del av dessa kodningstekniker och spelar en viktig roll genom att göra kommunikation mer robust och fjädrande för brus och andra störningar. Det ökande behovet av högre genomströmningar och lägre latenser har gjort konstruktörer att driva en mer parallell design tillvägagångssätt som ger upphov till parallella anpassningar av dessa kodningstekniker. En stor del av modern telekommunikation är via Wireless Wide Area Network (WWAN), vanligen kallad mobilnät. Det Third Generation Partnership Project (3GPP), Long Term Evolution (LTE) utvecklar och specificerar de standarder som används i mobil kommunikation. Deras nuvarande mest använda "4G" standard använder Turbo-kodning tekniker och en Quadratic Permutation Polynomial (QPP) interleaver. Silicon Large Grain Object eller SiLago är ett grovt kornkonfigurerbart tygstöd ett modulärt tillvägagångssätt för elektronikutveckling. Konceptet är liknande LEGO-tegelstenar, det är med ett library av härdade block (liknande Lego-tegelstenar), varav system av olika typer och funktioner kan byggas. Denna avhandling undersöker de toppmoderna parallella interleaversna och parallellinterfolieringen tekniker som är tillgängliga för 3GPP LTE QPP interleavers, och implementerar tvåinterleavers mönster, en för Radix 2 och en annan för Radix 4avkodningstekniker. En fysisk syntes utförs i 28nm-teknik och resultaten i kraft och area rapporteras.