arXiv subjects
Publications and source records attributed to Eimear Byrne.
The connection between secret sharing and matroid theory is well established. In this paper, we generalize the concepts of secret sharing and matroid ports to $q$-polymatroids. Specifically, we introduce the notion of an access structure on a vector space, and consider properties related to duality, minors, and the relationship to $q$-polymatroids. Finally, we show how rank-metric codes give rise to secret sharing schemes within this framework.
The connections between graph theory, and more generally association schemes, and coding theory were established by Delsarte for the Hamming metric and rank-metric codes. The ambient metric space of Hamming-metric codes and rank-metric codes can be seen as Cayley graphs generated by words of weight one. The metrics considered then coincide with the geodesic distances of these distance-regular graphs. We focus on a generalisation of this framework to the space of tensors over a finite field, endowed with the tensor-rank as a metric. This space corresponds to the Cayley graph generated by rank-one tensors, which is not distance-regular for tensors of order at least 3. We show that the spectrum of this graph has a recursive expression and depends on the possible intersections between tensor subspaces of large enough dimension and the Segre variety. The spectrum of this graph for 3-order tensors can be expressed with the rank distribution of the rank-metric codes generated by these tensors. In particular, we obtain the complete spectrum of the graph for 2x3x3 tensors over any finite field. We apply this result to derive bounds on the dimension of tensor codes in the tensor-rank metric using the eigenvalue method, and in particular the ratio-type bound.
Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one tensors. Kruskal showed that the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k + d - 1$, and codes meeting this bound with equality are called minimal tensor rank (MTR) codes. It is known from algebraic complexity theory that the existence of an MTR code implies the existence of a maximum distance separable (MDS) code. In this work, we establish new results relating the tensor rank of a rank-metric code to the parameters of associated linear codes in the Hamming metric and introduce the notion of tensor rank defect. We then develop new constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes.
Tensor codes are a generalisation of matrix codes. Such codes are defined as subspaces of order-r tensors for which the ambient space is endowed with the tensor-rank as a metric. A class of these codes was introduced by Roth, who also outlined a decoding algorithm for low tensor-rank errors that can be generalised to an algorithm with exponential complexity in the decoding radius. They may be viewed as a generalisation of the well-known Delsarte-Gabidulin-Roth maximum rank distance codes. We study a generalised class of these codes. We investigate their properties and outline decoding techniques for different metrics that leverage their tensor structure. We first consider a fibre-wise decoding approach, as each fibre of a codeword corresponds to a Gabidulin codeword. We then give a generalisation of Loidreau-Overbeck's decoding method that corrects errors with properties constrained by the dimensions of the slice spaces and fibre spaces. The metrics we consider are bounded from above by the tensor-rank metric, and therefore these algorithms also decode tensor-rank weight errors.
In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More precisely, we characterize the rank function defining these polymatroids and establish sufficient conditions on the relevant parameters under which it is fully determined. We show that there are several differences in compared to the behaviour of $q$-polymatroids of unrestricted matrix codes.
We study information leakage in secure linear network coding schemes based on nested rank-metric codes. We show that the amount of information leaked to an adversary that observes a subset of network links is characterized by the conditional rank function of a representable $q$-polymatroid associated with the underlying rank-metric code pair. Building on this connection, we introduce the notions of $q$-polymatroid ports and $q$-access structures and describe their structural properties. Moreover, we extend Massey's correspondence between minimal codewords and minimal access sets to the rank-metric setting and prove a $q$-analogue of the Brickell--Davenport theorem.
We consider structural properties of $\mathcal{L}$-polymatroids, especially those defined on a finite complemented modular lattice $\mathcal{L}$. We introduce a set of cover-weight axioms and establish a cryptomorphism between these axioms and the rank axioms of an $\mathcal{L}$-polymatroid. We introduce the notion of a cyclic flat of an $\mathcal{L}$-polymatroid and study properties of its lattice of cyclic flats. We show that the weighted lattice of cyclic flats of an $\mathcal{L}$-polymatroid $\mathcal{P}$, along with the atomic weights of $\mathcal{P}$, is sufficient to define its rank function on $\mathcal{L}$. In our main result, we characterize those weighted lattices $(\mathcal{Z},λ)$ such that $\mathcal{Z}\subseteq\mathcal{L}$ is the collection of cyclic flats of an $\mathcal{L}$-polymatroid.
In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of $q$-polymatroids. We give a Critical Theorem for representable $q$-polymatroids and we provide a lower bound on the critical exponent. We show that $q$-polymatroids arising from certain families of rank-metric codes attain this lower bound.
We introduce the concept of a sum-rank saturating system and outline its correspondence to a covering properties of a sum-rank metric code. We consider the problem of determining the shortest sum-rank-$ρ$-saturating systems of a fixed dimension, which is equivalent to the covering problem in the sum-rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.
We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats. We show that the free product of two $q$-matroids $M_1$ and $M_2$ is maximal with respect to the weak order on $q$-matroids having $M_1$ as a restriction and $M_2$ as the complementary contraction. We characterise $q$-matroids that are irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of irreducibles is unique up to isomorphism. We discuss the representability of the free product, with a particular focus on rank one uniform $q$-matroids and show that such a product is represented by clubs on the projective line.
We describe a construction of the Tutte polynomial for both matroids and $q$-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition. We show that such partitions in the matroid case include the class of partitions arising in Crapo's definition of the Tutte polynomial, while not representing a direct $q$-analogue of such partitions. We propose axioms of $q$-Tutte-Grothendiek invariance and show that this yields a $q$-analogue of Tutte-Grothendiek invariance. We establish the connection between the rank polynomial and the Tutte polynomial, showing that one can be obtained from the other by convolution.
We introduce the concept of a rank saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of $s_{q^m/q}(k,ρ)$, which is the minimum $\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank $ρ$-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $s_{q^m/q}(k,ρ)$ and evaluate it for certain values of $k$ and $ρ$. We give constructions of rank $ρ$-saturating systems suggested from geometry.
In this paper we develop the theory of cyclic flats of $q$-matroids. We show that the lattice of cyclic flats, together with their ranks, uniquely determines a $q$-matroid and hence derive a new $q$-cryptomorphism. We introduce the notion of $\mathbb{F}_{q^m}$-independence of an $\mathbb{F}_q$-subspace of $\mathbb{F}_q^n$ and we show that $q$-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.
The Assmus-Mattson theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs. In this work we present a further two-fold generalisation: first from matroids to polymatroids and also from sets to vector spaces. To achieve this, we introduce the characteristic polynomial of a $q$-polymatroid and outline several of its properties.
A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do so, we first establish a new cryptomorphic definition for q-matroids. We show that q-Steiner systems are examples of q-PMD's and we use this q-matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to the only known q-Steiner system, which has parameters S(2,3,13;2), and hence establish the existence of a new subspace design with parameters 2-(13,4,5115;2).
In this paper we focus on modules over a finite chain ring $\mathcal{R}$ of size $q^s$. We compute the density of free modules of $\mathcal{R}^n$, where we separately treat the asymptotics in $n,q$ and $s$. In particular, we focus on two cases: one where we fix the length of the module and one where we fix the rank of the module. In both cases, the density results can be bounded by the Andrews-Gordon identities. We also study the asymptotic behaviour of modules generated by random matrices over $\mathcal{R}$. Since linear codes over $\mathcal{R}$ are submodules of $\mathcal{R}^n$ we get direct implications for coding theory. For example, we show that random codes achieve the Gilbert-Varshamov bound with high probability.
In 1991, Roth introduced a natural generalization of rank metric codes, namely tensor codes. The latter are defined to be subspaces of $r$-tensors where the ambient space is endowed with the tensor rank as a distance function. In this work, we describe the general class of tensor codes and we study their invariants that correspond to different families of anticodes. In our context, an anticode is a perfect space that has some additional properties. A perfect space is one that is spanned by tensors of rank 1. Our use of the anticode concept is motivated by an interest in capturing structural properties of tensor codes. In particular, we indentify four different classes of tensor anticodes and show how these gives different information on the codes they describe. We also define the generalized tensor binomial moments and the generalized tensor weight distribution of a code and establish a bijection between these invariants. We use the generalized tensor binomial moments to define the concept of an $i$-tensor BMD code, which is an extremal code in relation to an inequality arising from them. Finally, we give MacWilliams identities for generalized tensor binomial moments.
In this paper we investigate known Singleton-like bounds in the Lee metric and characterize optimal codes, which turn out to be very few. We then focus on Plotkin-like bounds in the Lee metric and present a new bound that extends and refines a previously known, and out-performs it in the case of non-free codes. We then compute the density of optimal codes with regard to the new bound. Finally we fill a gap in the characterization of Lee-equidistant codes.