We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.

We present an orderly algorithm for classifying triple systems. Subsequently, we show that there exist exactly 7,038,699,746 nonisomorphic simple 2-(11, 3, 3) designs. The method is also used to confirm the previously accomplished classifications of 2-(8,3,6), 2-(12, 3, 2) and 2-(19, 3, 1) designs.

In this paper, we determine the spectrum (the set of all possible volumes) of simple T(2, 3, v) trades for any even foundation size v.