On the cardinality spectrum and the number of latin bitrades of order 3

By a (latin) unitrade, we call a set of vertices of the Hamming graph that is intersects with every maximal clique in $0$ or $2$ vertices. A bitrade is a bipartite unitrade, that is, a unitrade splittable into two independent sets. We study the cardinality spectrum of the bitrades in the Hamming graph $H(n,k)$ with $k=3$ (ternary hypercube) and the growth of the number of such bitrades as $n$ grows. In particular, we determine all possible (up to $2.5\cdot 2^n$) and large (from $14\cdot 3^{n-3}$) cardinatities of bitrades and prove that the cardinality of a bitrade is compartible to $0$ or $2^n$ modulo $3$ (this result has a treatment in terms of a ternary code of Reed--Muller type). A part of the results is valid for any $k$. We prove that the number of nonequivalent bitrades is not less than $2^{(2/3-o(1))n}$ and is not greater than $2^{\alpha^n}$, $\alpha<2$, as $n\to\infty$.Comment: 18 pp. In Russia

Exact Synthesis of ESOP Forms

We present an exact synthesis approach for computing Exclusive-or Sum-of-Products (ESOP) forms with a minimum number of product terms using Boolean satisfiability. Our approach finds one or more ESOP forms for a given Boolean function. The approach can deal with incompletely specified Boolean functions defined over many Boolean variables and is particularly fast if the Boolean function can be expressed with only a few product terms. We describe the formalization of the ESOP synthesis problem with a fixed number of terms as a decision problem and present search procedures for determining ESOP forms of minimum size. We further discuss how the search procedures can be relaxed to find ESOP forms of small sizes in reasonable time. We experimentally evaluate the performance of the SAT-based synthesis procedures on completely and incompletely specified Boolean functions