Construction of random pooling designs based on singular linear space over finite fields

<abstract><p>Faced with a large number of samples to be tested, if there are requiring to be tested one by one and complete in a short time, it is difficult to save time and save costs at the same time. The random pooling designs can deal with it to some degree. In this paper, a family of random pooling designs based on the singular linear spaces and related counting theorems are constructed. Furtherly, based on it we construct an $ \alpha $-$ almost\ d^e $-disjunct matrix and an $ \alpha $-$ almost\ (d, r, z] $-disjunct matrix, and all the parameters and properties of these random pooling designs are given. At last, by comparing to Li's construction, we find that our design is better under certain condition.</p></abstract>

Class dimension of association schemes in singular linear spaces

A resolving set for an association scheme [Formula: see text] is a set of points [Formula: see text] such that, for all [Formula: see text], the ordered list of relations [Formula: see text] uniquely determines [Formula: see text], where [Formula: see text] denotes the relation [Formula: see text] containing the pair [Formula: see text] in [Formula: see text]. In this paper, we determine upper bounds on class dimension for a family of association schemes in singular linear spaces, and construct their resolving sets for a special case.

t-Singular Linear Spaces

As a generalization of singular linear spaces, we introduce the concept of t-singular linear spaces, make some anzahl formulas of subspaces, and determine the suborbits of t-singular linear groups.

Quasi-regular semilattices in singular linear spaces

Let [Formula: see text] denote the [Formula: see text]-dimensional singular linear space over a finite field [Formula: see text]. For a fixed integer [Formula: see text], denote by [Formula: see text] the set of all subspaces of type [Formula: see text], where [Formula: see text]. Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute its all parameters.