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In Neri et al. (2022) [20] , the authors extended the family of 2-dimensional F q 2 t -linear MRD codes recently found in Longobardi et al. (2023) [13]. Also, for t ≥ 5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t ≥ 5. Moreover, we prove that in the case when t = 4 , the linear sets of the projective line PG (1 , q 8) ensuing from codes in the relevant family, are not equivalent to any one known so far. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field F p. We apply symmetric (row and column) elementary transformations to the adjacency matrix over F p in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over F p for any odd prime p. Our key tool is quadratic residues of F p. [ABSTRACT FROM AUTHOR]

We showed that isomorphism classes of idempotent evolution algebras are in bijection with the orbits of the semidirect product group of the symmetric group and the torus, considered the combinatoric problem of enumeration of isomorphism classes for these algebras over arbitrary finite fields, derived a general counting formula, and obtained explicit formulas for the numbers of isomorphism classes in dimensions 2, 3, and 4 over any finite field. [ABSTRACT FROM AUTHOR]

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure. [ABSTRACT FROM AUTHOR]

Let V be a finite-dimensional vector space over the finite field F q and suppose W and W ˜ are subspaces of V. Two linear transformations T : W → V and T ˜ : W ˜ → V are said to be similar if there exists a linear isomorphism S : V → V with S W = W ˜ such that S ∘ T = T ˜ ∘ S W. Given a linear map T defined on a subspace W of V , we give an explicit formula for the number of linear maps that are similar to T. Our results extend a theorem of Philip Hall that settles the case W = V where the above problem is equivalent to counting the number of square matrices over F q in a conjugacy class. [ABSTRACT FROM AUTHOR]

Nonsingular plane curves over a finite field F q of degree q + 2 passing through all the F q -points of the plane admit a representation by 3 × 3 matrices over F q. We classify their degenerations by means of the matrix representation. We also discuss the similar problem for the affine-plane filling projective curves of degree q + 1. [ABSTRACT FROM AUTHOR]

Abstract Let q be a prime power. For u = (u 1 , ... , u n) , v = (v 1 , ... , v n) ∈ F q 2 n let 〈 u , v 〉 : = ∑ i = 1 n u i q v i be the Hermitian form of F q 2 n. For any n × n matrix M over F q 2 set Num k (M) : = { 〈 u , M u 〉 | u ∈ F q 2 n , 〈 u , u 〉 = k }. Num 1 (M) is the numerical range of q. In this paper we use Num 1 to give a characterization of hermitian matrices, give some examples of 3 × 3 matrices M with Num k (M) = F q 2 for all k ∈ F q and study F q and F q 2 linear subspaces V ⊂ M n , n (F q 2 ) such that either ♯ (Num 1 (M)) ≤ q for all M ∈ V or ♯ (Num 1 (M)) > q for all M ∈ V ∖ { 0 }. [ABSTRACT FROM AUTHOR]

In this paper we classify the isomorphism classes of four dimensional nilpotent associative algebras over a field F , studying regular subgroups of the affine group AGL 4 ( F ) . In particular we provide explicit representatives for such classes when F is a finite field, the real field R or an algebraically closed field. [ABSTRACT FROM AUTHOR]

The columns of an m × n ACI-matrix over a field F are independent affine subspaces of F m . An ACI-matrix has constant rank ρ if all its completions have rank ρ . Huang and Zhan (2011) [4] characterized the m × n ACI-matrices of constant rank when | F | ≥ min ⁡ { m , n + 1 } . We complete their result characterizing the m × n ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) [8] proved that every partial matrix of constant rank ρ has a ρ × ρ submatrix of constant rank ρ if and only | F | ≥ ρ . We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility. [ABSTRACT FROM AUTHOR]

In an entry pattern matrix A , all entries are indeterminates but the same indeterminate can appear in numerous positions. For a field F , an F -completion of A results from assigning a value from F to each indeterminate entry. We define the generic F -rank of an entry pattern matrix to be its rank when considered over the function field generated over F by its indeterminate entries. We investigate the situation where the generic F -rank of A is not attained by any F -completion of A , which can occur only if the generic F -rank exceeds the field order. [ABSTRACT FROM AUTHOR]

Let q be a prime power. Following a paper by Coons, Jenkins, Knowles, Luke and Rault (case q a prime p ≡ 3 ( mod 4 ) ) we define the numerical range Num ( M ) ⊆ F q 2 of an n × n -matrix M with coefficients in F q 2 in terms of the usual Hermitian form. We prove that ♯ ( Num ( M ) ) > q (case q ≠ 2 ), unless M is unitarily equivalent to a diagonal matrix with eigenvalues contained in an affine F q -line. We study in details Num ( M ) when n = 2 . [ABSTRACT FROM AUTHOR]

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps A ↦ P A P ⁎ , A ↦ A σ , and A ↦ A − 1 , where P is an invertible matrix, P ⁎ is its conjugate transpose, and σ is an automorphism of the underlying field. Bijectivity of maps is not an assumption but a conclusion. Moreover, adjacency is assumed to be preserved in one direction only. [ABSTRACT FROM AUTHOR]

Fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction. In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and finite geometry. [ABSTRACT FROM AUTHOR]

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over F q [ x ] . We also propose a new conjecture on the density of unimodular matrix polynomials. [ABSTRACT FROM AUTHOR]

Fix a field K , a subset P ⊆ { 1 , … , k } × { 1 , … , m } and an integer δ ≤ min ⁡ { k , m } . Let C ( m , k , P , K ) be the vector space of all k × m matrices with entries a i , j = 0 if ( i , j ) ∉ P . Let α ( δ , K ) be the maximal dimension of a linear subspace V ⊆ C ( m , k , P , K ) such that all A ∈ V ∖ { 0 } have rank ≥ δ . We show that known lower bounds for α ( δ , K ) , for K one (resp. several, resp. almost all) finite field give the same lower bounds for some (resp. many, resp. all) number fields. [ABSTRACT FROM AUTHOR]

In this paper, we provide a construction of association scheme based on the subspaces of type ( 2 , 1 , 0 ) in singular symplectic space over finite fields. Moreover, all parameters of the association scheme are computed. [ABSTRACT FROM AUTHOR]

Given an n × n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0 , 1 , … , n , a 1 in the k th position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported. [ABSTRACT FROM AUTHOR]