John Sheekey
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View article: Quotients of skew polynomial rings: new constructions of division algebras and MRD codes
Quotients of skew polynomial rings: new constructions of division algebras and MRD codes Open
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 method…
View article: Generalised evasive subspaces
Generalised evasive subspaces Open
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 an…
View article: Two-weight rank-metric codes
Two-weight rank-metric codes Open
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…
View article: On the geometry of tensor products over finite fields
On the geometry of tensor products over finite fields Open
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.…
View article: Cyclic 2-Spreads in $V(6,q)$ and Flag-Transitive Affine Linear Spaces
Cyclic 2-Spreads in $V(6,q)$ and Flag-Transitive Affine Linear Spaces Open
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…
View article: On Translation Hyperovals in Semifield Planes
On Translation Hyperovals in Semifield Planes Open
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…
View article: On MSRD codes, h-designs and disjoint maximum scattered linear sets
On MSRD codes, h-designs and disjoint maximum scattered linear sets Open
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 …
View article: Rank-Metric Codes, Semifields, and the Average Critical Problem
Rank-Metric Codes, Semifields, and the Average Critical Problem Open
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…
View article: Symplectic 4-dimensional semifields of order $$8^4$$ and $$9^4$$
Symplectic 4-dimensional semifields of order $$8^4$$ and $$9^4$$ Open
We classify symplectic 4-dimensional semifields over $$\mathbb {F}_q$$ , for $$q\le 9$$ , thereby extending (and confirming) the previously obtained classifications for $$q\le 7$$ . The classification is obtained by classifyi…
View article: Divisible linear rank metric codes
Divisible linear rank metric codes Open
A subspace of matrices over $\mathbb{F}_{q^e}^{m\times n}$ 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 t…
View article: Generalised Evasive Subspaces
Generalised Evasive Subspaces Open
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 an…
View article: Symplectic 4-dimensional semifields of order $8^4$ and $9^4$
Symplectic 4-dimensional semifields of order $8^4$ and $9^4$ Open
We classify symplectic 4-dimensional semifields over $\mathbb{F}_q$, for $q\leq 9$, thereby extending (and confirming) the previously obtained classifications for $q\leq 7$. The classification is obtained by classifying all symplectic semi…
View article: Rank-Metric Codes, Semifields, and the Average Critical Problem
Rank-Metric Codes, Semifields, and the Average Critical Problem Open
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…
View article: Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$
Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$ Open
The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $${\mathbb {C}}$$ and $$\mathbb {R}$$ in 1906–1907. The analogous p…
View article: Combinatorial invariants for nets of conics in $\text{PG}(2,q)$
Combinatorial invariants for nets of conics in $\text{PG}(2,q)$ Open
The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for …
View article: ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE
ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE Open
This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$ , for which elements $z$ in $\mathbb{L}$ , and $a$ , $b$ in…
View article: Nets of conics of rank one in PG(2,q), q odd
Nets of conics of rank one in PG(2,q), q odd Open
We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent pro…
View article: A note on depth-$b$ normal elements
A note on depth-$b$ normal elements Open
In this paper we study elements $β\in \mathbb{F}_{q^n}$ having normal $α$-depth $b$; that is, elements for which $β, β- α, \ldots, β-(b-1)α$ are simultaneously normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$. In [1], the author…
View article: New semifields and new MRD codes from skew polynomial rings
New semifields and new MRD codes from skew polynomial rings Open
In this article we construct a new family of semifields, containing and\nextending two well-known families, namely Albert's generalised twisted fields\nand Petit's cyclic semifields (also known as Johnson-Jha semifields). The\nconstruction…
View article: 13. MRD codes: constructions and connections
13. MRD codes: constructions and connections Open
This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics}, K.-U. Schmidt and A. Winterhof (eds.)…
View article: Tensor Representation of Rank-Metric Codes
Tensor Representation of Rank-Metric Codes Open
We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We defin…
View article: Binary additive MRD codes with minimum distance n-1 must contain a semifield spread set
Binary additive MRD codes with minimum distance n-1 must contain a semifield spread set Open
In this paper we prove a result on the structure of the elements of an additive {\it maximum rank distance (MRD) code} over the field of order two, namely that in some cases such codes must contain a semifield spread set. We use this resul…
View article: Rank-Metric Codes and Zeta Functions
Rank-Metric Codes and Zeta Functions Open
We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metri…
View article: Further Generalisations of Twisted Gabidulin Codes
Further Generalisations of Twisted Gabidulin Codes Open
We present a new family of maximum rank distance (MRD) codes. The new class contains codes that are neither equivalent to a generalised Gabidulin nor to a twisted Gabidulin code, the only two known general constructions of linear MRD codes.