In this paper we study finite dimensional algebras, in particular finite semifields, through their correspondence with nonsingular threefold tensors. We study a particular embedding of the tensor product space into a projective space. This cyclic model allows us to understand tensors and their contractions in a new geometric way, relating the contraction of a tensor to a natural subspace of a subgeometry. This leads us to new results on invariants and classifications of tensors and algebras and on nonsingular fourfold tensors. A detailed study of the geometry of this setup for the case of the threefold tensor power of a vector space of dimension two over a finite field surprisingly leads to a new construction of quasi-Hermitian varieties in PG(3, q2). (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

In this paper we completely classify semifields of order 2^8=256 containing a nucleus of order 2^4=16. We introduce new invariants for semifields, and apply new computational techniques for calculating old invariants. Together these make the computational classification significantly quicker.

We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.

Let PG(n-1,q) denote the (n-1)-dimensional projective space over 𝔽_q. We investigate the intersection of two Desarguesian (h-1)-spreads of PG(kh-1,q) and show that it is determined by a subgeometry over a suitable extension field. Our approach combines a characterization of subsets of points of PG(k-1,q^h) closed under q-order subgeometries with a matrix model for Desarguesian spreads based on Moore matrices. This leads naturally to the notion of generalized Segre varieties 𝒮^r_kr-1,h-1(q) and a geometric description of their maximal subspaces. As a main application, we prove that if two distinct Desarguesian (h-1)-spreads of PG(kh-1,q) contain a common pseudo-arc of size k+1, then their intersection is precisely the system ℛ^r_h,q of (h-1)-dimensional subspaces of 𝒮^r_kr-1,h-1(q), for some proper divisor r of h.

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance. In this paper, we establish the tensor rank of some semifields and finite fields of characteristics 2 and 3. We also propose new upper and lower bounds on their additive complexity, and give new associated algorithms improving on the state-of-the-art in terms of overall complexity. We achieve this by considering short straight line programs for encoding linear codes with given parameters.

ABSTRACT In this paper, we construct new optimal subspace designs and, consequently, new optimal codes in the sum‐rank metric. We construct new 1‐designs by finding sets of disjoint maximum scattered linear sets, and use these constructions to also find new ‐designs for . As a means of achieving this, we establish a correspondence between the metric properties of sum‐rank metric codes and the geometric properties of subspace designs. Specifically, we determine the geometric counterpart of the coding‐theoretic notion of generalised weights for the sum‐rank metric in terms of subspace designs and determine a geometric characterisation of MSRD codes. This enables us to characterise subspace designs via their intersections with hyperplanes and via duality operations.

In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results.

We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a non-constructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank-metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank-metric codes.

In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order $64$. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces of maximum dimension with respect to the associated spread.

Two-weight linear codes are linear codes in which any nonzero codeword can have only two possible distinct weights. Those in the Hamming metric have proven to be very interesting for their connections with authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we characterize two-weight codes in the rank metric, answering a recent question posed by Pratihar and Randrianarisoa.

In this paper we study geometric aspects of codes in the sum-rank metric. We establish the geometric description of generalised weights, and analyse the Delsarte and geometric dual operations. We establish a correspondence between maximum sum-rank distance codes and h-designs, extending the well-known correspondence between MDS codes and arcs in projective spaces and between MRD codes and h-scatttered subspaces. We use the geometric setting to construct new h-designs and new MSRD codes via new families of pairwise disjoint maximum scattered linear sets.

A subspace of matrices in ${\mathbb F}_{q^{e}}^{m\times n}$ can be naturally embedded as a subspace of matrices in ${\mathbb F}_{q}^{em\times en}$ with the property that the rank of any of its matrix is a multiple of $e$ . It is quite natural to ask whether or not all subspaces of matrices with such a property arise from a subspace of matrices over a larger field. In this paper we explore this question, which corresponds to studying divisible codes in the rank metric. We determine some cases for which this question holds true, and describe counterexamples by constructing subspaces with this property which do not arise from a subspace of matrices over a larger field.

We classify symplectic 4-dimensional semifields over F-q, for q <= 9, thereby extending (and confirming) the previously obtained classifications for q <= 7. The classification is obtained by classifying all symplectic semifield subspaces in PG(9, q) for q < 9 up to K-equivalence, where K <= PGL(10, q) is the lift of PGL(4, q) under the Veronese embedding of PG(3, q) in PG(9, q) of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for q even, q <= 8. For q odd, and q <= 9, our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over F(q )is contained in the Knuth orbit of a Dickson commutative semifield.

We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.

We classify symplectic 4-dimensional semifields over 𝔽_q , for q≤ 9 , thereby extending (and confirming) the previously obtained classifications for q≤ 7 . The classification is obtained by classifying all symplectic semifield subspaces in PG(9,q) for q≤ 9 up to K-equivalence, where K≤PGL(10,q) is the lift of PGL(4,q) under the Veronese embedding of PG(3,q) in PG(9,q) of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for q even, q≤ 8 . For q odd, and q≤ 9 , our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over 𝔽_q is contained in the Knuth orbit of a Dickson commutative semifield.

We classify symplectic 4-dimensional semifields over F q , for q ≤ 9 , thereby extending (and confirming) the previously obtained classifications for q ≤ 7 . The classification is obtained by classifying all symplectic semifield subspaces in PG ( 9 , q ) for q ≤ 9 up to K -equivalence, where K ≤ PGL ( 10 , q ) is the lift of PGL ( 4 , q ) under the Veronese embedding of PG ( 3 , q ) in PG ( 9 , q ) of degree two. Our results imply the non-existence of non-associative symplectic 4-dimensional semifields for q even, q ≤ 8 . For q odd, and q ≤ 9 , our results imply that the isotopism class of a symplectic non-associative 4-dimensional semifield over F q is contained in the Knuth orbit of a Dickson commutative semifield.

In this paper we study finite dimensional algebras, in particular finite semifields, through their correspondence with nonsingular threefold tensors. We introduce a alternative embedding of the tensor product space into a projective space. This model allows us to understand tensors and their contractions in a new geometric way, relating the contraction of a tensor with a natural subspace of a subgeometry. This leads us to new results on invariants and classifications of tensors and algebras and on nonsingular fourfold tensors. A detailed study of the geometry of this setup for the case of the threefold tensor power of a vector space of dimension two over a finite field surprisingly leads to a new construction of quasi-hermitian varieties in $\mathrm{PG}(3,q^2)$.

We determine the tensor rank of all semifields of order 16 over F2 and of all semifields of order 81 over F3. Our results imply that some semifields of order 81 have lower multiplicative complexity than the finite field F81 over F3. We prove new results on the correspondence between linear codes and tensor rank, including a generalisation of a theorem of Brockett and Dobkin to arbitrary tensors, which makes the problem computationally feasible.