Thesis (M. Sc.)--Carleton University, 2004. / Includes bibliographical references (p. 38-39). Also available in electronic format on the Internet.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 143-146).

In the thesis a new way of checking whether a function is CCZ-equivalent to a permutation is given. The results for known families of almost perfect nonlinear (APN) functions are presented for functions defined over GF(2n ), for even n ≤ 12. The ways how to reduce the number of polynomials from each family are studied. For functions of the form x3 + a-1 tr1(a3 x9 ) it is shown, that they cannot be CCZ-equivalent to a permutation on fields GF(24n ) for n ∈ ℕ .

Orientador: Andreia Cristina Ribeiro / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T11:28:46Z (GMT). No. of bitstreams: 1 Assis_LucianaMafaldaEliasde_M.pdf: 10126493 bytes, checksum: bb2628e76f90df6deb24a9011e535714 (MD5) Previous issue date: 2008 / Resumo: Neste trabalho são apresentados vários resultados importantes da Análise Combinatória com destaque para o Princípio da Inclusão e Exclusão. Relevantes aplicações deste princípio são abordadas / Abstract: In this work we present important results from enumerative combinatorics, with an emphasis on the Principle of Inclusion and Exclusion. Relevant applications of this principle are presented to illustrate its use / Mestrado / Mestre em Matemática

D.Ing. / In this thesis we combined two techniques, namely a spectral shaping technique and a distance-preserving mapping technique to design new codes with both special spectrum shaping and error correction capabilities, in order to overcome certain communication problems like those that occur in a power-line communication channel. A new distance-preserving mapping construction based on graph theory is firstly presented. The k-cube graph construction from binary sequences to permutation sequences reached the upper bound on the sum of the Hamming distances for certain lengths of the permutation sequences and achieves the same sum of the Hamming distances as the best previously published constructions for most of the rest of the lengths. The k-cube graph construction is considered to be a simple and easy construction to understand the concept of mappings and especially the concept of a distance-reducing mapping.

D.Ing. (Electrical And Electronic Engineering) / We address the issue of synchronization, using sync-words (or markers), for encoded data. We focus on data that is encoded using permutation codes or Reed-Solomon codes. For each type of code (permutation code and Reed-Solomon code) we give a synchronization procedure or algorithm such that synchronization is improved compared to when the procedure is not employed. The gure of merit for judging the performance is probability of synchronization (acquisition). The word acquisition is used to indicate that a sync-word is acquired or found in the right place in a frame. A new synchronization procedure for permutation codes is presented. This procedure is about nding sync-words that can be used speci cally with permutation codes, such that acceptable synchronization performance is possible even under channels with frequency selective fading/jamming, such as the power line communication channel. Our new procedure is tested with permutation codes known as distance-preserving mappings (DPMs). DPMs were chosen because they have de ned encoding and decoding procedures. Another new procedure for avoiding symbols in Reed-Solomon codes is presented. We call the procedure symbol avoidance. The symbol avoidance procedure is then used to improve the synchronization performance of Reed-Solomon codes, where known binary sync-words are used for synchronization. We give performance comparison results, in terms of probability of synchronization, where we compare Reed-Solomon with and without symbol avoidance applied.

<p>Un code de permutations G(n,d) un sous-ensemble C de Sym(n) tel que la distance de Hamming D entre deux éléments de C est supérieure ou égale à d. Dans cette thèse, le groupe des isométries de (Sym(n),D) est déterminé et il est prouvé que ces isométries sont des automorphismes du schéma d'association induit sur Sym(n) par ses classes de conjugaison. Ceci mène, par programmation linéaire, à de nouveaux majorants de la taille maximale des G(n,d) pour n et d fixés et n compris entre 11 et 13. Des algorithmes de génération avec rejet d'objets isomorphes sont développés. Pour classer les G(n,d) non isométriques, des invariants ont été construits et leur efficacité étudiée. Tous les G(4,3) et les G(5,4) ont été engendrés à une isométrie près, il y en a respectivement 61 et 9445 (dont 139 sont maximaux et décrits explicitement). D’autres classes de G(n,d) sont étudiées.<p><p><p><p> <p><p><p><p>A permutation code G(n,d) is a subset C of Sym(n) such that the Hamming distance D between two elements of C is larger than or equal to d. In this thesis, we characterize the isometry group of the metric space (Sym(n),D) and we prove that these isometries are automorphisms of the association scheme induced on Sym(n) by the conjugacy classes. This leads, by linear programming, to new upper bounds for the maximal size of G(n,d) codes for n and d fixed and n between 11 and 13. We develop generating algorithms with rejection of isomorphic objects. In order to classify the G(n,d) codes up to isometry, we construct invariants and study their efficiency. We generate all G(4,3) and G(4,5)codes up to isometry; there are respectively 61 and 9445 of them. Precisely 139 out of the latter codes are maximal and explicitly described. We also study other classes of G(n,d)codes.<p><p><p><p> / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished

In this thesis, we examine a family of vectorial boolean functions on F22m inspired by Kim function, in order to find new APN permutations on F22m for m > 2. The functions of this family are defined as F(X) = X3 + bX3q + cX2q+1 + dXq+2 , where parameters b, c and d are from F2m . Necessary and sufficient conditions for this functions to be APN or equivalent to a permutation are presented in this thesis. To find conditions for being APN, Trace-0/Trace-1 decomposition method is used. A method using exponential sums is used to deduce which functions of this family is CCZ-equivalent to a certain type of permutation. These results were then used to search for APN permutations on F26 and F210 . 1

PermuNim is an impartial combinatorial game played on a board of squares where each player takes turns playing in rows and columns of the board which have not been played in, avoiding zero or more permutations. The game comes to an end when neither player can move. The first player unable to move on his or her turn loses the game. Many researchers have investigated combinatorial game theory as well as the idea of permutation pattern avoidance. PermuNim combines both of these ideas. When (12) or (1) is the forbidden permutation in PermuNim, or when the forbidden permutation is 'loses' in size to that of the smallest of the two dimensions of the board, we can say a great deal about the value of the game. For other permutations, the values of the options seem much more chaotic. Even (123) is chaotic as evidenced by our data in the appendix. We investigate the trend for even height boards which are `wide enough' to have options with all odd values and vice versa but we don't believe that this to be true in general. If a PermuNim board is stretched by adding columns, sometimes the value of the position is affected. We find that when any permutation is avoided and t moves have been made, as long at 2m columns are available together, there is a place where any number of columns may be added to the board without affecting the value of the position. We suspect that the number of columns necessary may be much lower for some permutations.

In this paper, new variations of some well-known permutation statistics are introduced and studied. Firstly, a double-descent of a permutation π is defined as a position i where πi ≥ 2πi+1. By proofs by induction and direct proofs, recursive and explicit expressions for the number of n-permutations with k double-descents are presented. Also, an expression for the total number of double-descents in all n-permutations is presented. Secondly, a double-inversion of a permutation π is defined as a pair (πi,πj) where i&lt;j but πi ≥ 2πj. The total number of double-inversions in all n-permutations is presented.