Motivated by a connection, described here for the first time, between the hidden normal subgroup problem (HNSP) and abelian hypergroups (algebraic objects that model collisions of physical particles), we develop a stabilizer formalism using abelian hypergroups and an associated classical simulation theorem (a la Gottesman-Knill). Using these tools, we develop the first provably efficient quantu...

A discrete DJS-hypergroup is constructed in connection with the linearization formula for the product of two spherical elements for a quantum Gelfand pair of two compact quantum groups. A similar construction is discussed for the case of a generalized quantum Gelfand pair, where the role of the quantum subgroup is taken over by a two-sided coideal in the dual Hopf algebra. The paper starts with...

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classiication of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, i...

An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible if Q(x1, . . . , xn) = P ( R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ) where P and R are (n− k+1)-ary and kary quasigroups, σ is a permutation, and 1 < k < n. Anm-ary ...

We count all latin cubes of order n ≤ 6 and latin hypercubes of order n ≤ 5 and dimension d ≤ 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-a...

The growing amount of data available in modern-day datasets makes the need to efficiently search and retrieve information. To make large-scale search feasible, Distance Estimation and Subset Indexing are the main approaches. Although binary coding has been popular for implementing both techniques, n-ary coding (known as Product Quantization) is also very effective for Distance Estimation. Howev...

In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a p-ary NPS of period n and type γ and a cyclic (n, p, n, n−γ p + γ, 0, n−γ p )-DPDS for an arbitrary integer γ. Next, we present the necessary conditions for the existence of a p-ary NPS of type γ. We apply this result for excluding the existence ...

An algorithm for finding the unequal error protection (UEP) capability of a q-ary image of a low-rate qm-ary cyclic code is presented by combining its concatenated structure with the UEP capability of concatenated codes. The results are independent of a choice of a basis to be used for expanding an element over GF(qm) into GF(q). A table of the UEP capability of binary images of low-rate Reed-S...

the aim of this paper is to initiate and investigate new soft sets over semihypergroups, named special soft sets and transitive soft sets and denoted by $s_{h}$ and  $t_{h},$ respectively. it is shown that $t_{h}=s_{h}$ if and only if $beta=beta^{*}.$ we also introduce the derived semihypergroup from a special soft set and study some properties of this class of semihypergroups.

We present some exact expressions for the number of paths of a given length in a perfect $m$-ary tree. We first count the paths in perfect rooted $m$-ary trees and then use the results to determine the number of paths in perfect unrooted $m$-ary trees, extending a known result for binary trees.