arXiv subjects

Giuseppe Cotardo

Publications and source records attributed to Giuseppe Cotardo.

14 recordsLinked to original sources

arXiv2026

Constructions of Rank-Metric Codes of Small Tensor Rank

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.

cs.IT↗

arXiv2026

Symmetric Tensor Decompositions over Finite Fields

We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing symmetric tensor decompositions to be reformulated as spanning problems by rank-one symmetric linearized polynomials. We translate these spanning conditions into explicit linear systems over finite fields and use the Frobenius automorphism to obtain computationally effective criteria. As applications, we recover known values of the symmetric bilinear complexity for small extension degrees and obtain explicit symmetric decompositions for several parameters. We also introduce the symmetric tensor-rank of a symmetric rank-metric code and show that, for the natural one-dimensional Gabidulin code associated with finite field multiplication, this invariant coincides with the symmetric tensor rank of the multiplication map.

math.CO↗

arXiv2026

Weak Composition Lattices and Ring-Linear Anticodes

Lattices and partially ordered sets have played an increasingly important role in coding theory, providing combinatorial frameworks for studying structural and algebraic properties of error-correcting codes. Motivated by recent works connecting lattice theory, anticodes, and coding-theoretic invariants, we investigate ring-linear codes endowed with the Lee metric. We introduce and characterize optimal Lee-metric anticodes over the ring $\mathbb{Z}/p^s\mathbb{Z}$. We show that the family of such anticodes admits a natural partition into subtypes and forms a lattice under inclusion. We establish a bijection between this lattice and a lattice of weak compositions ordered by dominance. As an application, we use this correspondence to introduce new invariants for Lee-metric codes via an anticode approach.

cs.IT↗

arXiv2025

Quantum Anticodes

This work introduces a symplectic framework for quantum error correcting codes in which local structure is analyzed through an anticode perspective. In this setting, a code is treated as a symplectic space, and anticodes arise as maximal symplectic subspaces whose elements vanish on a prescribed set of components, providing a natural quantum analogue of their classical counterparts. This framework encompasses several families of quantum codes, including stabilizer and subsystem codes, provides a natural extension of generalized distances in quantum codes, and yields new invariants that capture local algebraic and combinatorial features. The notion of anticodes also naturally leads to operations such as puncturing and shortening for symplectic codes, which in turn provide algebraic interpretations of key phenomena in quantum error correction, such as the cleaning lemma and complementary recovery and yield new descriptions of weight enumerators.

quant-ph↗

arXiv2025

Multishot Capacity of Networks with Restricted Adversaries

We investigate adversarial network coding and decoding, focusing on the multishot regime and when the adversary is restricted to operate on a vulnerable region of the network. Errors can occur on a proper subset of the network edges and are modeled via an adversarial channel. The paper contains both bounds and capacity-achieving schemes for the Diamond Network, the Mirrored Diamond Network, and generalizations of these networks. We also initiate the study of the capacity of 3-level networks in the multishot setting by computing the multishot capacity of the Butterfly Network, considered in [IEEE Transactions on Information Theory, vol. 69, no. 6, 2023], which is a variant of the network introduced by Ahlswede, Cai, Li and Yeung in 2000.

cs.IT↗

arXiv2024

Whitney Numbers of Rank-Metric Lattices and Code Enumeration

We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.

math.CO↗

arXiv2023

The Diagonals of a Ferrers Diagram

We propose and develop a theory of Ferrers diagrams and their $q$-rook polynomials solely based on their diagonals. We show that the cardinalities of the diagonals of a Ferrers diagram are equivalent information to their rook numbers, $q$-rook polynomials, and the rank distribution of matrices supported on the diagram. Our approach is based on the concept of \textit{canonical form} of a Ferrers diagrams, and on two simple diagram operations as the main proof tools. In the second part of the paper we develop the same theory for symmetric Ferrers diagrams, considering symmetric and alternating matrices supported on them. As an application of our results, we establish some combinatorial identities linking symmetric and alternating matrices, which do not appear to have an obvious bijective proof, and which generalize some curious results in enumerative combinatorics.

math.CO↗

arXiv2023

Multishot Adversarial Network Decoding

We investigate adversarial network coding and decoding focusing on the multishot regime. Errors can occur on a proper subset of the network edges and are modeled via an adversarial channel. The paper contains both bounds and capacity-achieving schemes for the Diamond Network and the Mirrored Diamond Network. We also initiate the study of the generalizations of these networks.

cs.IT↗

arXiv2022

Zeta Functions for Tensor Codes

In this work we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes, namely the $j$-tensor maximum rank distance codes. We show that it extends the family of $j$-maximum rank distance codes and contains the $j$-tensor binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the generalized zeta function for tensor codes. We establish connections between this object and the weight enumerator of a code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application we derive connections on the generalized tensor weights related to the Delsarte and Ravagnani-type anticodes.

cs.IT↗

arXiv2022

Rank-Metric Lattices

We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on the smallest rank-metric lattice whose characteristic polynomial we cannot compute, and provide a formula for it under a polynomiality assumption on its Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.

math.CO↗

arXiv2022

Parameters of Codes for the Binary Asymmetric Channel

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. In turn, these lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality.

cs.IT↗

arXiv2021

Tensor Codes and their Invariants

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.

cs.IT↗

arXiv2021

Bilinear Complexity of 3-Tensors Linked to Coding Theory

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of non-scalar multiplications required to evaluate it. This quantity is also described as its tensor rank, which is the smallest number of rank one matrices whose span contains its first slice space. In this paper we derive upper bounds on the tensor ranks of certain classes of $3$-tensors and give explicit constructions of sets of rank one matrices containing their first slice spaces. We also show how these results can be applied in coding theory to derive upper bounds on the tensor rank of some rank-metric codes. In particular, we compute the tensor rank of some families of $\mathbb{F}_{q^m}$-linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound.

cs.IT↗

arXiv2019

Rank-Metric Codes, Generalized Binomial Moments and their Zeta Functions

In this paper we introduce a new class of extremal codes, namely the $i$-BMD codes. We show that for this family several of the invariants are determined by the parameters of the underlying code. We refine and extend the notion of an $i$-MRD code and show that the $i$-BMD codes form a proper subclass of the $i$-MRD codes. Using the class of $i$-BMD codes we then obtain a relation between the generalized rank weight enumerator and its corresponding generalized zeta function. We also establish a MacWilliams identity for generalized rank weight distributions.

cs.IT↗