Algebraic and SAT models for SCA generation

Abstract

In this paper, we compute sequence covering arrays (SCAs), which are arrays, consisting of sequences, such that all subsequences with pairwise different entries of some length are covered, via a novel approach based on commutative algebra and symbolic computation. Hereby, we provide various algebraic models being capable to characterize possibly small sets of permutations collectively containing particular shorter subsequences. These models take the form of multivariate polynomial systems of equations and are then processed via supercomputing by a Gröbner Basis solver in order to compute solutions from them. If the variety is not empty, i.e. the Gröbner basis is non-trivial, then each point in the computed variety can be transformed to a SCA. In our experiments, we observed varying computational performance depending on the chosen model, while all of them exhibited scalability issues. Additionally and for comparison, we give new SAT descriptions modelling SCAs. By employing a SAT solver on our provided SAT models, we are able to provide upper bounds, one of which is best among literature results. Lastly, we adapt our SAT approach to answer a question posed by Yuster (Des Codes Cryptogr 88(3):585–593, 2020). As a result, we find a characterization of the dimensions of all perfect SCAs with coverage multiplicity two of strength three.