Switching codes and designs

AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters

A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij ) of order n over a field K as the set of zeros of the binomial ideal xi xj − xlij : (i, j) is a non-empty cell inL ⊆ K[x1, . . . , xn]. Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic

Permutation group approach to association schemes

AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed

Orthogonal Resolutions and Latin Squares

Resolutions which are orthogonal to at least one other resolution (RORs) and sets of m mutually orthogonal resolutions (m-MORs) of 2-(v, k, λ) designs are considered. A dependence of the number of nonisomorphic RORs and m-MORs of multiple designs on the number of inequivalent sets of v/k − 1 mutually orthogonal latin squares (MOLS) of size m is obtained. ACM Computing Classification System (1998): G.2.1.∗ This work was partially supported by the Bulgarian National Science Fund under Contract No I01/0003

No occurrence obstructions in geometric complexity theory

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.Comment: Substantial revision. This version contains an overview of the proof of the main result. Added material on the model of power sums. Theorem 4.14 in the old version, which had a complicated proof, became the easy Theorem 5.4. To appear in the Journal of the AM