9 research outputs found
Common Complements of Linear Subspaces and the Sparseness of MRD Codes
We consider the problem of estimating the number of common complements of a
family of subspaces over a finite field, in terms of the cardinality of the
family and its intersection structure. We derive upper and lower bounds for
this number, along with their asymptotic versions as the field size tends to
infinity. We use these bounds to describe the general behavior of common
complements with respect to sparsity and density, showing that the decisive
property is whether or not the number of spaces to be complemented is
negligible with respect to the field size. The proof techniques are based on
the study of isolated vertices in certain bipartite graphs. By specializing our
results to matrix spaces, we answer an open question in coding theory, proving
that MRD codes in the rank metric are sparse for all parameter sets as the
field grows, with only very few exceptions. We also investigate the density of
MRD codes as their column length tends to infinity, obtaining a new asymptotic
bound. Using properties of the Euler function from number theory, we then show
that our bound improves on known results for most parameter sets. We conclude
the paper by establishing two structural properties of the density function of
rank-metric codes
Densities of Codes of Various Linearity Degrees in Translation-Invariant Metric Spaces
We investigate the asymptotic density of error-correcting codes with good
distance properties and prescribed linearity degree, including sublinear and
nonlinear codes. We focus on the general setting of finite
translation-invariant metric spaces, and then specialize our results to the
Hamming metric, to the rank metric, and to the sum-rank metric. Our results
show that the asymptotic density of codes heavily depends on the imposed
linearity degree and the chosen metric
Common Complements of Linear Subspaces and the Sparseness of MRD Codes
Motivated by applications to the theory of rank-metric codes, we study the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We then use these bounds to describe the general behavior of common complements with respect to sparseness and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. By specializing our results to matrix spaces, we obtain upper and lower bounds for the number of maximum-rank-distance (MRD) codes in the rank metric. In particular, we answer an open question in coding theory, proving that MRD codes are sparse for all parameter sets as the field size grows, with only very few exceptions. We also investigate the density of MRD codes as their number of columns tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing general structural properties of the density function of rank-metric codes
Rank-Metric Codes, Semifields, and the Average Critical Problem
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
Generalised Evasive Subspaces
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
Convolutional codes over finite chain rings, MDP codes and their characterization
In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field. Finally, we provide a construction of (reverse) MDP convolutional codes over finite chain rings generalizing the notion of (reverse) superregular matrices