arXiv subjects

Alberto Ravagnani

Publications and source records attributed to Alberto Ravagnani.

At least 73 records · Page 4Linked to original sources

arXiv2019

An Assmus-Mattson Theorem for Rank Metric Codes

A $t$-$(n,d,λ)$ design over ${\mathbb F}_q$, or a subspace design, is a collection of $d$-dimensional subspaces of ${\mathbb F}_q^n$, called blocks, with the property that every $t$-dimensional subspace of ${\mathbb F}_q^n$ is contained in the same number $λ$ of blocks. A collection of matrices in over ${\mathbb F}_q$ is said to hold a subspace design if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a subspace design.

math.CO↗

arXiv2018

Partition-Balanced Families of Codes and Asymptotic Enumeration in Coding Theory

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the $\mathbb{F}_{q^m}$-linear MRD codes, the $\mathbb{F}_q$-linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of $\mathbb{F}_q$-linear MRD codes in $\mathbb{F}_q^{n \times m}$ in the set of all matrix codes of the same dimension is asymptotically at most $1/2$, both as $q \to +\infty$ and as $m \to +\infty$. We also prove that MDS and $\mathbb{F}_{q^m}$-linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of $\mathbb{F}_q$-linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code.

cs.IT↗

arXiv2017

Duality of codes supported on regular lattices, with an application to enumerative combinatorics

We introduce a general class of regular weight functions on finite abelian groups, and study the combinatorics, the duality theory, and the metric properties of codes endowed with such functions. The weights are obtained by composing a suitable support map with the rank function of a graded lattice satisfying certain regularity properties. A regular weight on a group canonically induces a regular weight on the character group, and invertible MacWilliams identities always hold for such a pair of weights. Moreover, the Krawtchouk coefficients of the corresponding MacWilliams transformation have a precise combinatorial significance, and can be expressed in terms of the invariants of the underlying lattice. In particular, they are easy to compute in many examples. Several weight functions traditionally studied in Coding Theory belong to the class of weights introduced in this paper. Our lattice-theory approach also offers a control on metric structures that a regular weight induces on the underlying group. In particular, it allows to show that every finite abelian group admits weight functions that, simultaneously, give rise to MacWilliams identities, and endow the underlying group with a metric space structure. We propose a general notion of extremality for (not necessarily additive) codes in groups endowed with semi-regular supports, and establish a Singleton-type bound. We then investigate the combinatorics and duality theory of extremal codes, extending classical results on the weight and distance distribution of error-correcting codes. Finally, we apply the theory of MacWilliams identities to enumerative combinatorics problems, obtaining closed formulae for the number of rectangular matrices over a finite having prescribed rank and satisfying some linear conditions.

cs.IT↗

arXiv2017

Codes Endowed With the Rank Metric

We review the main results of the theory of rank-metric codes, with emphasis on their combinatorial properties. We study their duality theory and MacWilliams identities, comparing in particular rank-metric codes in vector and matrix representation. We then investigate the combinatorial structure of MRD codes and optimal anticodes in the rank metric, describing how they relate to each other.

cs.IT↗

arXiv2017

Adversarial Network Coding

A combinatorial framework for adversarial network coding is presented. Channels are described by specifying the possible actions that one or more (possibly coordinated) adversaries may take. Upper bounds on three notions of capacity (the one-shot capacity, the zero-error capacity, and the compound zero-error capacity) are obtained for point-to-point channels, and generalized to corresponding capacity regions appropriate for multi-source networks. A key result of this paper is a general method by which bounds on these capacities in point-to-point channels may be ported to networks. This technique is illustrated in detail for Hamming-type channels with multiple adversaries operating on specific coordinates, which correspond, in the context of networks, to multiple adversaries acting on specific network edges. Capacity-achieving coding schemes are described for some of the considered adversarial models.

cs.IT↗

arXiv2016

An algebraic framework for end-to-end physical-layer network coding

We propose an algebraic setup for end-to-end physical-layer network coding based on submodule transmission. We introduce a distance function between modules, describe how it relates to information loss and errors, and show how to compute it. Then we propose a definition of submodule error-correcting code, and investigate bounds and constructions for such codes.

cs.IT↗

arXiv2016

Covering Radius of Matrix Codes Endowed with the Rank Metric

In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. We apply the various bounds to the classes of maximal rank distance and quasi maximal rank distance codes.

math.CO↗

arXiv2015

Generalized weights: an anticode approach

In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with Frobenius-closed spaces. Then we characterize both generalized Hamming and rank weights of a code in terms of the intersection of the code with optimal anticodes in the respective metrics. Inspired by this description, we propose a new algebraic invariant, which we call "Delsarte generalized weights", for Delsarte rank-metric codes based on optimal anticodes of matrices. We show that our invariant refines the generalized rank weights for Gabidulin codes proposed by Kurihara, Matsumoto and Uyematsu, and establish a series of properties of Delsarte generalized weights. In particular, we characterize Delsarte optimal codes and anticodes in terms of their generalized weights. We also present a duality theory for the new algebraic invariant, proving that the Delsarte generalized weights of a code completely determine the Delsarte generalized weights of the dual code. Our results extend the theory of generalized rank weights for Gabidulin codes. Finally, we prove the analogue for Gabidulin codes of a theorem of Wei, proving that their generalized rank weights characterize the worst-case security drops of a Gabidulin rank-metric code.

cs.IT↗

arXiv2015

Rank distribution of Delsarte codes

In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for Delsarte rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to Gabidulin codes.

cs.IT↗

arXiv2015

Equidistant subspace codes

In this paper we study equidistant subspace codes, i.e. subspace codes with the property that each two distinct codewords have the same distance. We provide an almost complete classification of such codes under the assumption that the cardinality of the ground field is large enough. More precisely, we prove that for most values of the parameters, an equidistant code of maximum cardinality is either a sunflower or the orthogonal of a sunflower. We also study equidistant codes with extremal parameters, and establish general properties of equidistant codes that are not sunflowers. Finally, we propose a systematic construction of equidistant codes based on our previous construction of partial spread codes, and provide an efficient decoding algorithm.

cs.IT↗

arXiv2015

Rank-metric codes and their duality theory

We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes first proved by Delsarte employing the theory of association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, MRD codes). In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive both a recursive and an explicit formula for the number of $k \times m$ matrices over a finite field with given rank and $h$-trace.

cs.IT↗

arXiv2014

Subspace codes from Ferrers diagrams

In this paper we give new constructions of Ferrer diagram rank metric codes, which achieve the largest possible dimension. In particular, we prove several cases of a conjecture by T. Etzion and N. Silberstein. We also establish a sharp lower bound on the dimension of linear rank metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters.

cs.IT↗

arXiv2013

A zero-dimensional approach to Hermitian codes

We study the algebraic geometry of a family of evaluation codes from plane smooth curves defined over any field. In particular, we provide a cohomological characterization of their dual minimum distance. After having discussed some general results on zero-dimensional subschemes of the plane, we focus on the interesting case of Hermitian $s$-point codes, describing the geometry of their dual minimum-weight codewords.

math.AG↗

arXiv2013

Projective normality of Artin-Schreier curves

In this paper we study the projective normality of certain Artin-Schreier curves $Y_f$ defined over a field $\F$ of characteristic $p$ by the equations $y^q+y=f(x)$, $q$ being a power of $p$ and $f\in \F[x]$ being a polynomial in $x$ of degree $m$, with $(m,p)=1$. Many $Y_f$ curves are singular and so, to be precise, here we study the projective normality of appropriate projective models of their normalizations.

math.AG↗

arXiv2013

The dual geometry of Hermitian two-point codes

In this paper we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.

math.AG↗

arXiv2013

On the duals of geometric Goppa codes from norm-trace curves

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppa's construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitly computed.

math.AG↗

arXiv2013

Classification of binary systematic codes of small defect

In this paper non-trivial non-linear binary systematic AMDS codes are classified in terms of their weight distributions, employing only elementary techniques. In particular, we show that their length and minimum distance completely determine the weight distribution.

cs.IT↗

arXiv2013

On the minimum distance and the minimum weight of Goppa codes from a quotient of the Hermitian curve

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form $y^q+y=x^m$, $q$ being a prime power and $m$ a positive integer which divides $q+1$. The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.

math.AG↗