35 research outputs found
Codes, Graphs and Schemes from Nonlinear Functions
AMS classifications: 05E30; 05B20; 94B0
Codes, graphs and schemes from nonlinear functions
AbstractWe consider functions on binary vector spaces which are far from linear functions in different senses. We compare three existing notions: almost perfect nonlinear functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and differential attacks on certain cryptosystems. We give a new combinatorial characterization of AB functions in terms of the number of solutions to a certain system of equations, and a characterization of CR functions in terms of the Fourier transform. We also show how these functions can be used to construct several combinatorial structures; such as semi-biplanes, difference sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes
Permutation complexity of the fixed points of some uniform binary morphisms
An infinite permutation is a linear order on the set N. We study the
properties of infinite permutations generated by fixed points of some uniform
binary morphisms, and find the formula for their complexity.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Infinite permutations vs. infinite words
I am going to compare well-known properties of infinite words with those of
infinite permutations, a new object studied since middle 2000s. Basically, it
was Sergey Avgustinovich who invented this notion, although in an early study
by Davis et al. permutations appear in a very similar framework as early as in
1977. I am going to tell about periodicity of permutations, their complexity
according to several definitions and their automatic properties, that is, about
usual parameters of words, now extended to permutations and behaving sometimes
similarly to those for words, sometimes not. Another series of results concerns
permutations generated by infinite words and their properties. Although this
direction of research is young, many people, including two other speakers of
this meeting, have participated in it, and I believe that several more topics
for further study are really promising.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Local Equivalence of Transversals in Matroids
Given any system of n subsets in a matroid M , a transversal of this system is an n-tuple of elements of M , one from each set, which is independent. Two transversals differing by exactly one element are adjacent, and two transversals connected by a sequence of adjacencies are locally equivalent, the distance between them being the minimum number of adjacencies in such a sequence. We give two sufficient conditions for all transversals of a set system to be locally equivalent. Also we propose a conjecture that the distance between any two locally equivalent transversals can be bounded by a function of n only, and provide an example showing that such function, if it exists, must grow at least exponentially. Let M be a matroid, and V = (V 1 ; : : : ; V n ) a collection of subsets of M . By a transversal of V we mean a sequence (v 1 ; : : : ; v n ) of elements of M such that v i 2 V i for i = 1; : : : ; n, and v 1 ; : : : ; v n are independent. By the well-known Rado's Theorem, transver..