Volume 33, Issue 4

There exist well-known distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph. An example is given by the Johnson graph J (8, 3) with the intersection array {15, 8, 3;1, 4, 9}. The paper is concerned with the problem of the existence of distance-regular graphs Γ with the intersection arrays {78, 50, 9;1, 15, 60} and {174, 110, 18;1, 30, 132} for which Γ 3 is a triangle-free graph.

We consider a critical Galton – Watson branching process starting with N particles; the number of offsprings is supposed to have the distribution p k =( k  + 1) − τ −( k  + 2) − τ , k =0, 1, 2, … Limit distributions of the maximal tree size are obtained for the corresponding Galton – Watson forest with N trees and n non-root vertices as N , n  → ∞, n / N τ ⩾ C  > 0.

We prove that any Boolean function in n variables can be modeled by a testable Boolean circuit with two additional inputs in the basis “conjunction, oblique conjunction, disjunction, negation” so that the circuit admits a complete diagnostic test of the length at most 2 n + 3 with respect to stuck-at faults of the type 1 at gate outputs.

The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity.

For sequences of independent random variables having a Bernoulli distribution the joint distribution and the limit joint distribution of statistics of three tests of the NIST statistical package («Monobit Test», «Frequency Test within a Block», and «Cumulative Sums Test») are obtained. In the case when two blocks are used in «Frequency Test within a Block», pairwise covariance of these statistics is given.

We consider predicates on a finite set that are invariant with respect to an affine operation f G , where G is some Abelian group. Such predicates are said to be multiaffine for the group G . Special attention is paid to predicates that are affine for a group G q of addition modulo q = p s , where p is a prime number and s =1. We establish the predicate multiaffinity criterion for a group G q . Then we introduce disjunctive normal forms (DNF) for predicates on a finite set and obtain properties of DNFs of predicates that are multiaffine for a group G q . Finally we show how these properties can be used to design a polynomial algorithm that decides satisfiability of a system of predicates which are multiaffine for a group G q , if predicates are specified by DNF.

Earlier, the author introduced the concept of a universal function and proved the existence of universal functions for classes of linear k -valued functions of two variables for k ≥ 5 . In this paper, we show that the product modulo k is a universal function for the class of linear k -valued functions of two variables if and only if k = 6l ± 1 .