Ferdinando Zullo

Jonathan Mannaert, Marta Messia, Ferdinando Zullo

Linearized Reed-Solomon (LRS) codes form an important family of maximum sum-rank distance (MSRD) codes that generalize both Reed--Solomon codes and Gabidulin codes. In this paper we study the equivalence problem for LRS codes and determine the number of inequivalent codes within this family. Using the correspondence between sum-rank metric codes and systems of $\mathbb{F}_q$-subspaces, we analyze the stabilizer of the Gabidulin system and derive a characterization of equivalence between LRS codes. In particular, we prove that two LRS codes are equivalent if and only if the sets of norms that define the codes coincide up to multiplication by an element of $\mathbb{F}_q^\ast$. This description allows us to reduce the classification problem to the action of $\mathbb{F}_q^\ast$ on subsets of $\mathbb{F}_q^\ast$. As a consequence, we derive formulas for the number of inequivalent linearized Reed-Solomon codes and illustrate the results with explicit examples.

Giuseppe Cotardo, Ferdinando Zullo

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.

Aida Abiad, Antonina P. Khramova, Sven C. Polak et al.

The sum-rank metric provides a unifying framework that generalizes both the celebrated Hamming and rank metrics, and has found applications in areas such as network coding, distributed storage, and space-time coding. A central problem is to determine the maximum size of a code with prescribed minimum distance. In this paper, we derive new sharp upper bounds on the size of a sum-rank-metric code using spectral and optimization techniques, including a semidefinite programming (SDP) bound that can outperform the best existing bounds based on computational experiments. Furthermore, we compare the Delsarte linear programming (LP) bound and a recent eigenvalue LP bound, and show equivalences between them, with particular emphasis on extremal regimes of the sum-rank metric. Finally, we show how to use the several SDP, LP and eigenvalue bounds to prove non-existence results for certain optimal and perfect sum-rank metric codes. Our results suggest that the combination of spectral and optimization methods effectively captures the hybrid nature of the sum-rank metric, providing new techniques that overcome the limitations of classical coding-theoretic approaches.

Martino Borello, Olga Polverino, Ferdinando Zullo

Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric intersecting codes exhibit substantially different behavior. It was previously shown that a nondegenerate $[n,k,d]_{q^m/q}$ rank-metric intersecting code must satisfy $2k-1 \le n \le 2m-3$, and the tightness of the upper bound was left open. Using the geometric interpretation of rank-metric codes via $q$-systems, we prove that the dual subspace associated with a rank-metric intersecting code must satisfy strong evasiveness properties. This connection allows us to derive new restrictions on the parameters of such codes and to show that the bound $n=2m-3$ can be attained only when $k=3$ and $m\ge 6$. More generally, we show that $n \leq 2m-\lfloor(k+4)/2\rfloor$. Moreover, we obtain a geometric characterization of these extremal codes in terms of scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$. As a consequence, the existence problem for $[2m-3,3,d]_{q^m/q}$ rank-metric intersecting codes is reduced to the existence of scattered subspaces of dimension $m+3$. Using known constructions of maximum scattered subspaces, we derive existence results when $m$ is even. Finally, we prove that $[6,3,3]_{q^5/q}$ rank-metric intersecting codes do not exist for any prime power $q$, thus resolving an open problem posed by Bartoli et al. in 2025.