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We prove new non-existence results for vectorial monomial Dillon type bent functions mapping the field of order 2 2 m to the field of order 2 m / 3. When m is odd and m > 3 we show that there are no such functions. When m is even we derive a condition for the bent coefficient. The latter result allows us to find examples of bent functions with m = 6 in a simple way. [ABSTRACT FROM AUTHOR]

We introduce a new class of complex Hadamard matrices which have not been studied previously. We use these matrices to construct a new infinite family of parity proofs of the Kochen-Specker theorem. We show that the recently discovered simple parity proof of the Kochen-Specker theorem is the initial member of this infinite family. [ABSTRACT FROM AUTHOR]

We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Finally, we prove that if gcd ⁡ (3 n , ϕ (3 n)) = 1 , where ϕ is the Euler's totient function, then all permutation equivalent constacyclic codes of length n over F 4 are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes. [ABSTRACT FROM AUTHOR]

Abstract: Steganography is concerned with communicating hidden messages in such a way that no one apart from the sender and the intended recipient can detect the very existence of the message. We study the syndrome coding method (sometimes also called the “matrix embedding method”), which uses a linear code as an ingredient. Among all codes of a fixed block length and fixed dimension (and thus of a fixed information rate), an optimal code is one that makes it most difficult for an eavesdropper to detect the presence of the hidden message. We show that the average distance to code is the appropriate concept that replaces the covering radius for this particular application. We completely classify the optimal codes in the cases when the linear code used in the syndrome coding method is a one- or two-dimensional code over . In the steganography application this translates to cases when the code carries a high payload (has a high information rate). [Copyright &y& Elsevier]

Abstract: Let denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov. [Copyright &y& Elsevier]

Abstract: A proper double blocking set in is a set B of points such that for each line l. The smallest known example of a proper double blocking set in for large primes p is the disjoint union of two projective triangles of side ; the size of this set is . For each prime such that we construct a proper double blocking set with points, and for each prime we construct a proper double blocking set with points. [Copyright &y& Elsevier]

Abstract: Given a pair of Latin squares, we may remove from both squares those cells that contain the same symbol in corresponding positions. The resulting pair of partial Latin squares is called a Latin bitrade. The number of filled cells in is called the size of T. There are at least two natural ways to define the genus of a Latin bitrade; the bitrades of genus 0 are called spherical. We construct a simple bijection between the isomorphism classes of planar Eulerian triangulations on v vertices and the main classes of spherical Latin bitrades of size . Since there exists a fast algorithm (due to Batagelj, Brinkmann and McKay) for generating planar Eulerian triangulations up to isomorphism, our result implies that also spherical Latin bitrades can be generated very efficiently. [Copyright &y& Elsevier]

Abstract: We say that the sequence is quasi-polynomial in n if there exist polynomials and an integer such that, for all , where . We present several families of combinatorial objects with the following properties: Each family of objects depends on two or more parameters, and the number of isomorphism types of objects is quasi-polynomial in one of the parameters whenever the values of the remaining parameters are fixed to arbitrary constants. For each family we are able to translate the problem of counting isomorphism types of objects into the problem of counting integer points in a union of parametrized rational polytopes. The families of objects to which this approach is applicable include combinatorial designs, linear and unrestricted codes, and dissections of regular polygons. [Copyright &y& Elsevier]

We say that two points x, y of a cap C form a free pair of points if any plane containing x and y intersects C in at most three points. For given N and q, we denote by m ( N, q) the maximum number of points in a cap of PG( N, q) that contains at least one free pair of points. It is straightforward to prove that m ( N, q) ≤ ( q N-1 + 2 q − 3)/( q − 1), and it is known that this bound is sharp for q = 2 and all N. We use geometric constructions to prove that this bound is sharp for all q when N ≤ 4. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design. [ABSTRACT FROM AUTHOR]

Abstract: We show that all spherical Eulerian triangulations can be inductively generated from the set of all even double wheels using just one of the two local transformations used in the algorithm by Brinkmann and McKay and originally proposed by Batagelj. [Copyright &y& Elsevier]

We say that (x , y , z) ∈ Q 3 is an associative triple in a quasigroup Q (⁎) if (x ⁎ y) ⁎ z = x ⁎ (y ⁎ z). It is easy to show that the number of associative triples in Q is at least | Q | , and it was conjectured that quasigroups with exactly | Q | associative triples do not exist when | Q | > 1. We refute this conjecture by proving the existence of quasigroups with exactly | Q | associative triples for a wide range of values | Q |. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums. [ABSTRACT FROM AUTHOR]