Linear Codes from Some 2-Designs

A classical method of constructing a linear code over \gf(q) with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over \gf(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented

On bent and hyper-bent functions

Bent functions are Boolean functions which have maximum possible nonlinearity i.e. maximal distance to the set of affine functions. They were introduced by Rothaus in 1976. In the last two decades, they have been studied widely due to their interesting combinatorial properties and their applications in cryptography. However the complete classification of bent functions has not been achieved yet. In 2001 Youssef and Gong introduced a subclass of bent functions which they called hyper-bent functions. The construction of hyper-bent functions is generally more difficult than the construction of bent functions. In this thesis we give a survey of recent constructions of infinite classes of bent and hyper-bent functions where the classification is obtained through the use of Kloosterman and cubic sums and Dickson polynomials

On the dual of (non)-weakly regular bent functions and self-dual bent functions

For weakly regular bent functions in odd characteristic the dual function is also bent. We analyse a recently introduced construction of nonweakly regular bent functions and show conditions under which their dual is bent as well. This leads to the denition of the class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass. We analyse self-duality for bent functions in odd characteristic, and characterize quadratic self-dual bent functions. We construct non-weakly regular bent functions with and without a bent dual, and bent functions with a dual bent function of a dierent algebraic degree

Value Distributions of Perfect Nonlinear Functions

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.Comment: 28 pages. minor revisions of the previous version. The paper is now identical to the published version, outside of formattin

On complete mappings and value sets of polynomials over finite fields

In this thesis we study several aspects of permutation polynomials over nite elds with odd characteristic. We present methods of construction of families of complete mapping polynomials; an important subclass of permutations. Our work on value sets of non-permutation polynomials focus on the structure of the spectrum of a particular class of polynomials. Our main tool is a recent classi cation of permutation polynomials of Fq, based on their Carlitz rank. After introducing the notation and terminology we use, we give basic properties of permutation polynomials, complete mappings and value sets of polynomials in Chapter 1. We present our results on complete mappings in Fq[x] in Chapter 2. Our main result in Section 2.2 shows that when q > 2n + 1, there is no complete mapping polynomial of Carlitz rank n, whose poles are all in Fq. We note the similarity of this result to the well-known Chowla-Zassenhaus conjecture (1968), proven by Cohen (1990), which is on the non-existence of complete mappings in Fp[x] of degree d, when p is a prime and is su ciently large with respect to d. In Section 2.3 we give a su cient condition for the construction of a family of complete mappings of Carlitz rank at most n. Moreover, for n = 4, 5, 6 we obtain an explicit construction of complete mappings. Chapter 3 is on the spectrum of the class Fq,n of polynomials of the form F(x) = f(x)+x, where f is a permutation polynomial of Carlitz rank at most n. Upper bounds for the cardinality of value sets of non-permutation polynomials of the xed degree d or xed index l were obtained previously, which depend on d or l respectively. We show, for instance, that the upper bound in the case of a subclass of Fq,n is q -2, i.e., is independent of n. We end this work by giving examples of complete mappings, obtained by our methods