We study the problem of finding the largest numberT(n,m) of ternary vectors of lengthnsuch that for any three distinct vectors there are at leastmcoordinates where they pairwise differ. This problem is a special case of the perfectk-hashing problem in theoretical computer science, corresponding to thek= 3 case. Form= 1, we get the classical trifference problem which is wide open. We prove upper and lower bounds onT(n,m) for various ranges of the parametermand determine the phase transition threshold onm=m(n) whereT(n,m) jumps from constant to exponential in n. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters. Moreover, we relate the trifference problem to the sunflower conjecture.

Abstract In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $${{\,\textrm{PG}\,}}(n, q)$$ PG ( n , q ) such that every plane is incident with at least one line of the set. This is the first main open problem concerning the minimum size of (s, t)-blocking sets in $${{\,\textrm{PG}\,}}(n,q)$$ PG ( n , q ) , where we set $$s=2$$ s = 2 and $$t=1$$ t = 1 . In $${{\,\textrm{PG}\,}}(n,q)$$ PG ( n , q ) , an (s, t)-blocking set refers to a set of t-spaces such that each s-space is incident with at least one chosen t-space. This is a notoriously difficult problem, as it is equivalent to determining the size of certain q-Turán designs and q-covering designs. We present an improvement on the upper bounds of Etzion and of Metsch via a refined scheme for a recursive construction, which in fact enables improvement in the general case as well.

In this paper we introduce some example algorithms that can work with the Client-based Mobility Frame System presented recently by us. In this solution all the logic and structure is transferred from the network to the Mobile Node and thus each individuum can use the resources in a personalized and efficient way. In the present work modeling and detailed discussion on the parameters of the algorithms is given and comparison to existing mobility approaches and protocols is done. We prepared a simulation to test our protocol and to back up the proposals we provide the reader with simulation results. We stress that one the most important benefit of our findings is that all the MNs can run different management strategies and can optimize mobility for themselves.