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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]

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]

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]

Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of n≥2 qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A theorem by Arkhipov (arXiv:1209.3819) states that n-qubit magic sets in which each observable is in exactly two subsets of compatible observables can be reduced either to the two-qubit magic square or the three-qubit magic pentagram [N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)PRLTAO0031-900710.1103/PhysRevLett.65.3373]. An open question is whether there are magic sets that cannot be reduced to the square or the pentagram. If they exist, a second key question is whether they require n>3 qubits, since, if this is the case, these magic sets would capture minimal state-independent quantum advantage that is specific for n-qubit systems with specific values of n. Here, we answer both questions affirmatively. We identify magic sets that cannot be reduced to the square or the pentagram and require n=3, 4, 5, or 6 qubits. In addition, we prove a generalized version of Arkhipov's theorem providing an efficient algorithm for, given a hypergraph, deciding whether or not it can accommodate a magic set, and solve another open problem, namely, given a magic set, obtaining the tight bound of its associated noncontextuality inequality.