Alessandro Neri

We extend the existing skew polynomial representations of matrix algebras that are direct sum of matrix spaces over division rings. In this representation, the sum‐rank distance between two tuples of matrices is captured by a weight function on their associated skew polynomials, defined through degrees and greatest common right divisors with the polynomial that defines the representation. We exploit this representation to construct new families of maximum sum‐rank distance (MSRD) codes over finite and infinite fields, and over division rings. These constructions generalize many of the known existing constructions of MSRD codes as well as of optimal codes in the rank and in the Hamming metric. As a by‐product, in the case of finite fields, we obtain new families of maximum distance separable codes that are linear over a subfield and whose length is close to the field size.

We derive eigenvalue bounds for the distance-[ ] chromatic number of a graph, which is a generalization of the classical chromatic number. We apply such bounds to hypercube graphs, providing alternative spectral proofs for results by Ngo, Du and Graham [New bounds on a hypercube coloring problem, Inform. Process. Lett. 84(5) (2002) 265–269], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [The [ ]-distance coloring of the Cartesian product of cycles using optimal Lee codes, Discr. Appl. Math. 159(18) (2011) 2222–2228]. Finally, we provide a complete characterization for the existence of perfect Lee codes of minimum distance [ ]. In order to prove our results, we use a mix of spectral and number theory tools. Our results, which provide the first application of spectral methods to Lee codes, illustrate that spectral graph theory succeeds to capture the nature of the Lee metric.

Over fields of characteristic unequal to 2, we can identify symmetric matrices with homogeneous polynomials of degree 2. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper, we generalize the construction of symmetric Delsarte–Gabidulin codes to polynomials of degree [Formula: see text] over fields of characteristic 0 or [Formula: see text]. To do so, we equip the space of homogeneous polynomials of degree [Formula: see text] with the metric induced by the essential rank, which is the minimal number of linear forms needed to express a polynomial. We provide bounds on the minimum distance and dimension of the essential-rank-metric codes we construct and provide an efficient decoding algorithm. Finally, we show how essential-rank-metric codes can be seen as special instances of rank-metric codes and compare our construction to known rank-metric codes with the same parameters.

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the ( k − 1 ) (k-1) -dimensional projective space over F q \mathbb {F}_q that have size at most c q k c q k for some universal constant c c . Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of F q \mathbb {F}_q -linear minimal codes of length n n and dimension k k , for every prime power q q , for which n ≤ c q k n \leq c q k . This solves one of the main open problems on minimal codes.

In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.

The concept of scattered polynomials is generalized to those of exceptional scattered sequences which are shown to be the natural algebraic counterpart of $\mathbb{F}_{q^n}$-linear MRD codes. The first infinite family in the first nontrivial case is also provided and equivalence issues are considered. As a byproduct, a new infinite family of MRD codes is obtained.