817 research outputs found
Exhaustive Generation of Linear Orthogonal Cellular Automata
We consider the problem of exhaustively visiting all pairs of linear cellular
automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal
Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of
coprime polynomials over a finite field having the same degree and a nonzero
constant term. While previous research showed how to count all such pairs for a
given degree and order of the finite field, no practical enumeration algorithms
have been proposed so far. Here, we start closing this gap by addressing the
case of polynomials defined over the field \F_2, which corresponds to binary
CA. In particular, we exploit Benjamin and Bennett's bijection between coprime
and non-coprime pairs of polynomials, which enables us to organize our study
along three subproblems, namely the enumeration and count of: (1) sequences of
constant terms, (2) sequences of degrees, and (3) sequences of intermediate
terms. In the course of this investigation, we unveil interesting connections
with algebraic language theory and combinatorics, obtaining an enumeration
algorithm and an alternative derivation of the counting formula for this
problem.Comment: 9 pages, 1 figure. Submitted to the exploratory track of AUTOMATA
2023. arXiv admin note: text overlap with arXiv:2207.0040
Computing random -orthogonal Latin squares
Two Latin squares of order are -orthogonal if, when superimposed,
there are exactly distinct ordered pairs. The spectrum of all values of
for Latin squares of order is known. A Latin square of order is
-self-orthogonal if and its transpose are -orthogonal. The spectrum
of all values of is known for all orders . We develop randomized
algorithms for computing pairs of -orthogonal Latin squares of order and
algorithms for computing -self-orthogonal Latin squares of order
Heuristic search of (semi-)bent functions based on cellular automata
An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata (CA), focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter
Bent functions in the partial spread class generated by linear recurring sequences
We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials are either 1 or 2. We then count the resulting sets of polynomials and prove that, for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all PSâ and PS+ bent functions of n=8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d=2 are not EA-equivalent to any MaioranaâMcFarland or Desarguesian partial spread function
Local permutation polynomials and the action of e-Klenian groups
Permutation polynomials of finite fields have many applications in Coding Theory, Cryptography and Combinatorics.
In the first part of this paper we present a new family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups. In the second part we use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in particular, we provide a new family of MOLS on size a prime power
Amenability of groups and -sets
This text surveys classical and recent results in the field of amenability of
groups, from a combinatorial standpoint. It has served as the support of
courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure.
The goals of the text are (1) to be as self-contained as possible, so as to
serve as a good introduction for newcomers to the field; (2) to stress the use
of combinatorial tools, in collaboration with functional analysis, probability
etc., with discrete groups in focus; (3) to consider from the beginning the
more general notion of amenable actions; (4) to describe recent classes of
examples, and in particular groups acting on Cantor sets and topological full
groups
The Cellular Automaton Interpretation of Quantum Mechanics
When investigating theories at the tiniest conceivable scales in nature,
almost all researchers today revert to the quantum language, accepting the
verdict from the Copenhagen doctrine that the only way to describe what is
going on will always involve states in Hilbert space, controlled by operator
equations. Returning to classical, that is, non quantum mechanical,
descriptions will be forever impossible, unless one accepts some extremely
contrived theoretical constructions that may or may not reproduce the quantum
mechanical phenomena observed in experiments.
Dissatisfied, this author investigated how one can look at things
differently. This book is an overview of older material, but also contains many
new observations and calculations. Quantum mechanics is looked upon as a tool,
not as a theory. Examples are displayed of models that are classical in
essence, but can be analysed by the use of quantum techniques, and we argue
that even the Standard Model, together with gravitational interactions, might
be viewed as a quantum mechanical approach to analyse a system that could be
classical at its core. We explain how such thoughts can conceivably be
reconciled with Bell's theorem, and how the usual objections voiced against the
notion of `superdeterminism' can be overcome, at least in principle. Our
proposal would eradicate the collapse problem and the measurement problem. Even
the existence of an "arrow of time" can perhaps be explained in a more elegant
way than usual.
Discussions added in v3: the role of the gravitational force, a mathematical
physics definition of free will, and an unconventional view on the arrow of
time, amongst others.Comment: 259 pages, 21 figures. Thoroughly rewritten versio
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