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We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length p and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form g 2 X ℓ + g Y ℓ + 1 = 0 over the finite field F p for some primitive root g modulo p. We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field F q of F p. We show that for q greater than a lower bound of the order of magnitude O (ℓ 2) there exists a generator g of F q × such that the equation in question is solvable over F q. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3. [ABSTRACT FROM AUTHOR]

The correlation measure of order k is an important measure of pseudorandomness for binary sequences. This measure tries to look for dependence between several shifted versions of a sequence. We study the relation between the correlation measure of order k and two other pseudorandom measures: the N th linear complexity and the N th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it. [ABSTRACT FROM AUTHOR]

Sequences with perfect linear complexity profile were defined more than thirty years ago in the study of measures of randomness for binary sequences. More recently apwenian sequences , first with values ±1, then with values in { 0 , 1 } , were introduced in the study of Hankel determinants of automatic sequences. We explain that these two families of sequences are the same up to indexing, and give consequences and questions that this implies. We hope that this will help gathering two distinct communities of researchers. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notion of functional graph. Specifically, given an equation E (X , Y) = 0 with variables X and Y over a finite field F q of odd characteristic, we define a digraph by choosing the elements in F q as vertices and drawing an edge from x to y if and only if E (x , y) = 0. We call this graph as equational graph. In this paper, we study the equational graph when choosing E (X , Y) = (Y 2 − f (X)) (λ Y 2 − f (X)) with f (X) a polynomial over F q and λ a non-square element in F q. We show that if f is a permutation polynomial over F q , then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected. [ABSTRACT FROM AUTHOR]