In this paper, we construct a family of new pooling designs based on the singular linear spaces and related counting theorems. By analyzing its disjunct properties, it turns out that the new design is superior to the existing designs under the same experimental efficiency. Additionally, we compare our pooling design with Liu et al. with regard to the parameters of construction. At last, the relevant proof of our pooling designs is given.

Coded caching technology can better alleviate network traffic congestion. Since many of the centralized coded caching schemes now in use have high subpacketization, which makes scheme implementation more challenging, coded caching schemes with low subpacketization offer a wider range of practical applications. It has been demonstrated that the coded caching scheme can be achieved by creating a combinatorial structure named placement delivery array (PDA). In this work, we employ vector space over a finite field to obtain a class of PDA, calculate its parameters, and consequently achieve a coded caching scheme with low subpacketization. Subsequently, we acquire a new MN scheme and compare it with the new scheme developed in this study. The subpacketization \(F\) of the new scheme has significant advantages. Lastly, the number of users \(K\), cache fraction \(\frac{M}{N}\), and subpacketization \(F\) have advantages to some extent at the expense of partial transmission rate \(R\) when compared to the coded caching scheme in other articles.

Sum-rank metric codes, as a generalization of Hamming codes and rank metric codes, have important applications in fields such as multi-shot linear network coding, space-time coding and distributed storage systems. The purpose of this study is to construct sum-rank metric codes based on orthogonal spaces over finite fields, and calculate the list sizes outputted by different decoding algorithms. The following achievements have been obtained. In this study, we construct a cyclic orthogonal group of order q^n-1 and an Abelian non-cyclic orthogonal group of order (q^n-1)^2 based on the companion matrices of primitive polynomials over finite fields. By selecting different subspace generating matrices, maximum rank distance (MRD) codes with parameters (n ×2n, q^2n, n)_q and (n ×4n, q^4n, n)_q are constructed respectively. Two methods for constructing sum-rank metric codes are proposed for the constructed MRD codes, and the list sizes outputted under the list decoding algorithm are calculated. Subsequently, the [n,k,d]_q^n/q-system is used to relate sum-rank metric codes to subspace designs. The list size of sum-rank metric codes under the list decoding algorithm is calculated based on subspace designs. This calculation method improves the decoding success rate compared with traditional methods.

Sidon space is an important tool for constructing cyclic subspace codes. In this letter, we construct some Sidon spaces by using primitive elements and the roots of some irreducible polynomials over finite fields. Let q be a prime power, k; m; n be three positive integers and rho = [m/2k] - 1, theta = [n/2m] -1. Based on these Sidon spaces and the union of some Sidon spaces, new cyclic subspace codes with size 3(q(n-)1)/q-1 and theta rho q(k) (q(n-)1)/q-1 are obtained. The size of these codes is lager compared to the known constructions from [14] and [10].

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.

In this paper, several inertial algorithms which combine the established viscosity iteration and cyclic processes for finding a solution of a multi-set split common fixed problem are investigated. The inertial parameters in our algorithms can be positive or negative, which may have fewer steps to meet the requirements than selecting all positive parameters. Under the appropriate conditions, our proposed algorithms are proven to converge strongly. A compressed sensing example is reported to demonstrate the computational implementation of our algorithms and the quality of image restoration.

The Erdos-Ko-Rado (EKR) theorem involves the nature of set-intersections, is the earliest result of studying finite set intersections, and is one of the classic conclusions in the combination and extreme value theory. Therefore, the EKR theorem has great research value and development prospects. In the paper, the geometric space under the symplectic group is used as a theoretical tool, utilizing the concept and its counting theorem of non-isotropic subspace in symplectic space, combining with the research method of Erdos-Ko-Rado theorem, by studying the monotonicity of functions and the method of adding vectors, the upper bounds of r-intersecting of type (2m, m) non-isotropic subspaces in symplectic spaces are determined, and the EKR theorem of non-isotropic subspaces in symplectic spaces on finite spaces is studied.

Low-Density Parity-Check (LDPC) codes have low linear decoding complexity, which is a kind of good codes with excellent performance. Therefore, LDPC codes have great research value. Two kinds of LDPC codes are constructed based on vector space over finite field. The code length, code rate and minimum distance are given. Moreover, the two kinds of codes are compared with the existing codes, and the constructed codes are better than some existing ones in terms of code rate or minimum distance.

Compressed Sensing (CS) is a new signal processing theory under the condition that the signal is sparse or compressible. One of the central problems in compressed sensing is the construction of sensing matrices. In this paper, we provide a new deterministic construction via vector spaces over finite fields, which is superior to Devore's construction using polynomials over finite fields under some conditions. Moreover, we use the algorithm to perform numerical simulation experiments on sensing matrices. Simulation results also demonstrate that signal recovery performance performs better using the constructed matrices as compared with several state-of-the-art sensing matrices, such as DeVore's matrix and random Gaussian matrix.

Let ASG(2 ν + l , ν ;F q ) be the (2 ν + l )-dimensional affine-singular symplectic space over the finite field F q and ASp 2 ν + l , ν (F q ) be the affine-singular symplectic group of degree 2 ν + l over F q . Let O be any orbit of flats under ASp 2 ν + l , ν (F q ). Denote by L J the set of all flats which are joins of flats in O such that O ⊆ L J and assume the join of the empty set of flats in ASG (2 ν + l , ν ;F q ) is ∅. Ordering L J by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice L J , when the lattices form geometric lattice, lastly gives the characteristic polynomial of L J .

As a generalization of attenuated space, the concept of singular linear spaces was firstly introduced in [1]. In this paper, we construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular linear space over finite fields, and exhibit their disjunct properties. Moreover, we show that the new construction gives better ratio of efficiency than the former ones under conditions. At last, the paper gives the brief introduction about the relationship between the columns (rows) of the matrix and the related parameters.

Pooling designs have many applications in molecular biology. In this paper, we firstly construct a family of error-correcting pooling designs using the containment relationship of subspaces of vector spaces. Then by comparing with the test efficiencies, the new design is better than that in D’yachkov et al. (2005). At last, we analyze how the related parameters influence the test efficiency ts.

In the paper titled "Lattices generated by two orbits of subspaces under finite classical group" by Wang and Guo. The subspaces in the lattices are characterized and the geometricity is classified. In this paper, the result above is generalized to singular symplectic space. This paper characterizes the subspaces in these lattices, classifies their geometricity, and computes their characteristic polynomials.

Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular linear space over finite fields, and exhibit their disjunct properties.

Let ASG(2v,F(q)) be the 2v-dimensional affine-symplectic space over the finite field F(q) and let ASp(2v)(F(q)) be the affine-symplectic group of degree 2v over F(q). For any two orbits M' and M '' of flats under ASp(2v)(F(q)), let L'(resp. L '') be the set of all flats which are joins (resp. intersections) of flats in M' (resp. M '') such that M '' subset of L' (resp. M' subset of L '') and assume the join (resp. intersection) of the empty set of flats in ASG(2v,F(q)) is theta (resp. F(q)((2v))). Let L = L' boolean AND L ''. By ordering L', L '', L by ordinary or reverse inclusion, six lattices are obtained. This article discusses the relations between different lattices, and computes their characteristic polynomial.

Let ASG(2v + 1, v; F-q) be the (2v+l)-dimensional affine-singular symplectic space over the finite field IF and let ASp(2v+l,v)(F-q) be the affine-singular symplectic group of degree 2v + l over F-q. For any orbit O of flats under ASp(2v+l,v)(F-q), let be the set of all flats which are intersections of flats in O such that O subset of L and assume the intersection of the empty set of flats in ASG(2v + 1, v;Fq) is F-q((2v+1)). By ordering L by ordinary or reverse inclusion, two lattices are obtained. This article discusses the relations between different lattices,classify their geometricity and computes their characteristic polynomial.

For 1 <= d <= v - 1. Let V denote the 2v-dimensional symplectic space over a finite field F-q, and fix a (v - d)-dimensional totally isotropic subspace W of V. Let L(d, 2v) = P boolean OR {V}, where P = {A|A is a subspace of V, A boolean AND W = {0} and A subset of W-perpendicular to}. Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.