Abstract.
In this paper, we study a special class of recently introduced quasigroups called multivariate quadratic quasigroups (MQQs) and solve several open research problems about them. Our main contributions are threefold. The first is to provide a standard form of the MQQ generating function. Secondly, we show how to explicitly construct MQQs of higher orders and give lower bounds on the number of MQQs, which so far were open problems. Last, but not least, we refine the definition of MQQs of different types by introducing the notion of “MQQs of strict type”. The new concept has the advantage of being invariant under invertible affine transformations over the set of the rows, columns and symbols of the multiplication table of an MQQ. It is therefore better suited to characterize the complexity of the underneath multivariate quadratic system.
© 2013 by Walter de Gruyter Berlin Boston
This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.