Finding nonnormal bent functions

The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.     

On the spectrum of n quasigroups with given conjugate invariant subgroup

Let tⁿ be a vector of n positive integers that sum to 2n ? 1. Let u denote a vector of n or more positive integers that sum to n², and call u, n-universal if for every possible choice of t¹, t²,…, tⁿ, the components of the tⁱ can be arranged in the successive rows of an n-row matrix (with 0 in each unused cell) so that u is the vector of column sums.It is shown that (n,…, n)_{(n times)} is n-universal for every n. More generally, for odd n, any choice of t¹, t³,…, tⁿ can be placed in rows so that the column sums are (n, n?1,…, 2, 1); for even n, any choice of t², t⁴,…, tⁿ can be placed in rows so that the column sums are (n, n ?1,…, 2, 1). Hence, any u that can be obtained from the sum of two rows whose nonzero components are, respectively, n, n ?1,…, 2, 1 and n ?1, n ?2,…, 2, 1 (in any order, with 0's elsewhere) is n-universal.The problem examined is closely related to the graph conjecture that trees on 2, 3,…, n + 1 vertices can be superposed to yield the complete graph on n + 1 vertices.     

Two new triangles of q-integers via q-Eulerian polynomials of type A and B

The classical Eulerian polynomials can be expanded in the basis t ^k?1(1+t) ^n+1?2k (1≤k≤?(n+1)/2?) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a q-analogue of this expansion for Carlitz’s q-Eulerian polynomials as well as a similar formula for Chow–Gessel’s q-Eulerian polynomials of type B. We shall give some applications of these two formulas, which involve two new sequences of polynomials in the variable q with positive integral coefficients. It is an open problem to give a combinatorial interpretation for these polynomials.     

On digraphs without antidirected cycles

Let t(n) denote the greatest number of arcs in a diagraph of orders n which does not contain any antidrected cycles. We show that [16/5(n ? 1)] ≤ t(n) ≤ 1/2 (n ? 1) for n ≥ 5. Let t_r (n) denote the corresponding quantity for r-colorable digraphs. We show that [16/5(n ? 1)] ≤ t₅(n) ≤ t₆(n) ≤ 10/3(n ? 1) for n ≥ 5 and that t₄(n) = 3(n ? 1) for n ≥ 3.     

Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1,k) and ordern exist for everyk < n whenn = 2 ^t+2?3 andn = 2 ^t+2?5 (wheret is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that theweighing matrix conjecture ‘For every ordern ≡ 0 (mod 4) there is, for eachk ?n, a square {0, 1, ? 1} matrixW = W(n, k) satisfyingWW ^t =kIn’ is resolved in the affirmative for all ordersn = 2^t+1?3,n = 2^t+1?5 (t a positive integer). The fact that the matrices we find are skew-symmetric for allk < n whenn ≡ 0 (mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders ? 64.     

The 2-intersection number of paths and bounded-degree trees

We represent a graph by assigning each vertex a finite set such that vertices are adjacent if and only if the corresponding sets have at least two common elements. The 2-intersection number θ₂(G) of a graph G is the minimum size of the union of sets in such a representation. We prove that the maximum order of a path that can be represented in this way using t elements is between (t² - 19t + 4)/4 and (t² - t + 6)/4, making θ₂(P_n) asymptotic to 2√n. We also show the existence of a constant c depending on ? such that, for any tree T with maximum degree at most d, θ₂(T) ≤ c(√n)^1+?. When the maximum degree is not bounded, there is an n-vertex tree T with θ₂(T) > .945n^2/3. © 1995 John Wiley & Sons, Inc.     

On a cardinal group invariant related to decompositions of Abelian groups

For each Abelian groupG, a cardinal invariant χ(G) is introduced and its properties are studied. In the special caseG = ? ⁿ , the cardinalχ(? ⁿ ) is equal to the minimal cardinality of an essential subset of ? ⁿ , i.e., a of a subsetA ? ? ⁿ such that, for any coloring of the group ? ⁿ inn colors, there exists an infinite one-color subset that is symmetric with respect to some pointα ofA. The estimaten( ^{n +} l)/2 ≤χ(? ⁿ ) < 2n is proved for alln and the relationχ(? ⁿ ) =n(n + 1)/2 forn ≤ 3. The structure of essential subsets of cardinalityχ(? ⁿ ) in ? ⁿ is completely described forn ≤ 3.     

A class of binary cyclic codes with five weights

In this paper, the dual code of the binary cyclic code of length 2 n-1 with three zeros α, α t 1 and α t 2 is proven to have five nonzero Hamming weights in the case that n 4 is even and t1 = 2 n/2 + 1, t2 = 2 n-1-2 n/2+1 + 1 or 2 n/2 + 3, where α is a primitive element of the finite field F 2 n . The dual code is a divisible code of level n/2+1, and its weight distribution is also completely determined. When n = 4, the dual code satisfies Ward's bound.     

Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions

We survey the properties of two parameters introduced by C. Ding and the author for quantifying the balancedness of vectorial functions and of their derivatives. We give new results on the distribution of the values of the first parameter when applied to F + L, where F is a fixed function and L ranges over the set of linear functions: we show an upper bound on the nonlinearity of F by means of these values, we determine then the mean of these values and we show that their maximum is a nonlinearity parameter as well, we prove that the variance of these values is directly related to the second parameter. We briefly recall the known constructions of bent vectorial functions and introduce two new classes obtained with Gregor Leander. We show that bent functions can be used to build APN functions by concatenating the outputs of a bent (n, n/2)-function and of some other (n, n/2)-function. We obtain this way a general infinite class of quadratic APN functions. We show that this class contains the APN trinomials and hexanomials introduced in 2008 by L. Budaghyan and the author, and a class of APN functions introduced, in 2008 also, by Bracken et al.; this gives an explanation of the APNness of these functions and allows generalizing them. We also obtain this way the recently found Edel?CPott cubic function. We exhibit a large number of other sub-classes of APN functions. We eventually design with this same method classes of quadratic and non-quadratic differentially 4-uniform functions.     

Functions with derivatives given by polynomials in the function itself or a related function

We construct the set of holomorphic functions S ₁ = {f: Ω_f ? ? → ?} whose members have n-th order derivatives which are given by a polynomial of degree n+1 in the function itself. We also construct the set of holomorphic functions S ₂ = {g: Ω_g ? ? → ?} whose members have n-th order derivatives which are given by the product of the function itself with a polynomial of degree n in an element of S ₁. The union S ₁ ∪ S ₂ contains all the hyperbolic and trigonometric functions. We study the properties of the polynomials involved and derive explicit expressions for them. As particular results, we obtain explicit and elegant formulas for the n-th order derivatives of the hyperbolic functions tanh, sech, coth and csch and the trigonometric functions tan, sec, cot and csc.     

Reconstruction theorems for centered functions and perfect codes

The article addresses the centered functions and perfect codes in the space of all binary n-tuples. We prove that all values of a centered function in a ball of radius k ≤ (n + 1)/2 are uniquely defined from its radial sums with respect to the vertices of the corresponding sphere. We present some theorems of full and partial reconstruction of a centered function from part of its values and derive a new property of the symmetry groups of centered functions.     

Cycle Covers of Planar 2-Edge-Connected Graphs

A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n ? 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n ? 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.     

Diameter vulnerability of graphs

Let f(t, D) denote the maximum possible diameter of a graph obtained from a (t+1)-edge-connected graph of diameter D by deleting t edges. F.R.K. Chung and M.R. Garey have shown that for D≥4,(t+1)(D?2)≤ f(t, D)≤(t+1)D+t. Here we consider the cases D=2 and D=3 and show that f(t,2)=4 and $\text{3}\text{2t}\text{?3≤f(t,3)≤3}\text{2t}\text{+4}$ if t is large enough. We solve also the problem for the directed case (answering a question of F.R.K. Chung and M.R. Garey) by showing that if D ≥ 3 the diameter of a diagraph obtained from a (t + 1)-arc-connected digraph of order n by deleting t arcs is at most n?2t+1. In the case D=.....2, the maximum possible diameter of the resulting digraph is (like in the undirected case) 4. We also consider the same problem for vertices.     

On the summability of spectral expansions of piecewise smooth functions

For piecewise smooth functions of n variables, we prove the uniform Riesz summability of order s > (n ? 3)/2 of their spectral expansions associated with an arbitrary elliptic operator with constant coefficients. For s = (n ? 3)/2, the corresponding Riesz means are bounded.     

Rotation symmetric Boolean functions—Count and cryptographic properties

Rotation symmetric (RotS) Boolean functions have been used as components of different cryptosystems. This class of Boolean functions are invariant under circular translation of indices. Using Burnside's lemma it can be seen that the number of n-variable rotation symmetric Boolean functions is 2^g_n, where g_n=(1/n)∑_t|nφ(t)2^n/t, and φ(.) is the Euler phi-function. In this paper, we find the number of short and long cycles of elements in having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We study RotS bent functions up to 10 variables and observe (experimentally) that there is no homogeneous rotation symmetric bent function having degree >2. Further, we studied the RotS functions on 5,6,7 variables by computer search for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier.     

A linear Vizing‐like relation relating the size and total domination number of a graph

We prove that m ≤ Δ (n ? γ_t) for every graph each component of which has order at least 3 of order n, size m, total domination number γ_t, and maximum degree Δ ≥ 3. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 285–290, 2005     

Subdivisions of graphs: A generalization of paths and cycles

One of the basic results in graph theory is Dirac's theorem, that every graph of order n?3 and minimum degree ?n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree ?n/2 contains a cycle C then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every connected component of H is non-trivial and contains at most one cycle. The degree bound can be improved to (n-t)/2 if H has t components that are trees.We attempt a similar generalization of the Corrádi-Hajnal theorem that every graph of order ?3k and minimum degree ?2k contains k disjoint cycles. Again, this may be restated as: every graph of order ?3k and minimum degree ?2k contains a subdivision of kK₃. We show that if H is any graph of order n with k components, each of which is a cycle or a non-trivial tree, then every graph of order ?n and minimum degree ?n-k contains a subdivision of H.     

Digraphs containing every possible pair of dicycles

Let D = (V, A) be a directed graph of order n ≥ 4. Suppose that the minimum degree of D is at least (3n − 3)/2. Then for any two integers s and t with s ≥ 2, t ≥ 2 and s + t ≤ n, D contains two vertex‐disjoint directed cycles of lengths s and t, respectively. Moreover, the condition on the minimum degree is sharp. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 154–162, 2000     

Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q ⁺ (2m + 1, q), and yields infinite classes of q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

A non-cyclic triple-error-correcting BCH-like code and some minimum distance results

In this paper, we give the first example of a non-cyclic triple-error-correcting code which is not equivalent to the primitive BCH code. It has parameters [63, 45, 7]. We also give better bounds on minimum distances of some [2 ⁿ ? 1, 2 ⁿ - 3n - 1] cyclic codes with three small zeroes. Finally, we reprove weight distribution results of Kasami for triple-error-correcting BCH-like codes using direct methods.