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The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum‐rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank‐metric minimal codes. We prove some bounds on their parameters, existence results, and, via a tool that we name geometric dual, we manage to construct minimal codes with few weights. A generalization of the celebrated Ashikhmin–Barg condition is proved and used to ensure the minimality of certain constructions. [ABSTRACT FROM AUTHOR]

Guruswami and Xing introduced subspace designs in 2013 to give the first construction of positive rate rank metric codes list-decodable beyond half the distance. In this paper we provide bounds involving the parameters of a subspace design, showing they are tight via explicit constructions. We point out a connection with sum-rank metric codes, dealing with optimal codes and minimal codes with respect to this metric. Applications to two-intersection sets with respect to hyperplanes, two-weight codes, cutting blocking sets and lossless dimension expanders are also provided. [ABSTRACT FROM AUTHOR]

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. In the first part of the paper, we use methods from semifield theory to 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, in the third part of the paper 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. [ABSTRACT FROM AUTHOR]

Let $k,n,m \in {\mathbb Z} ^{+}$ be integers such that $k\leq n \leq m$ , let $\mathrm {G}_{n,k}\in {\mathbb F} _{q^{m}}^{n}$ be a Delsarte-Gabidulin code. Recently, Wachter-Zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius $\tau $ , providing $\tau $ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in ${\mathbb F}_{q^{m}}^{n}$. Some years later, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when $n\mid m$. As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by Trombetti and Zhou. By exploiting arguments such as those used by Raviv and Wachter-Zeh, when $n\mid m$ , we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +1$ , where $d$ is its minimum distance. Nonetheless, in all the examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +2$ , where $d$ is its minimum distance. In particular, this leads to show infinite families of Gabidulin codes, with underlying parameters not already covered by the result of Raviv and Wachter-Zeh, having this decodability defect. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove our main result, that is any rank-metric code of ${\mathbb F}_{q^{m}}^{n}$ of order $q^{kn}$ with $n$ dividing $m$ , such that $4n-3$ is a square in $\mathbb {Z}$ and containing $\mathrm {G}_{n,2}$ , is not efficiently list decodable at some values of the radius $\tau $. [ABSTRACT FROM AUTHOR]

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let q be a prime power, n be a positive integer and σ be a generator of Gal (F q n : F q). In this paper we provide closed formulas for the coefficients of a σ -trinomial f over F q n which ensure that the dimension of the kernel of f equals its σ -degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characterize those having σ -degree 3 and 4. Our techniques rely on the tools developed in [24]. Finally, we apply these results to investigate a class of rank metric codes introduced in [8] , to construct quasi-subfield polynomials and cyclic subspace codes, obtaining new explicit constructions to the conjecture posed in [37]. [ABSTRACT FROM AUTHOR]

Imperfect competition in the banking markets can cause non-maturity demand deposits to have positive net present values. The Jarrow-van Deventer model based on an arbitrage-flee term structure methodology can be used to determine this value. This research uses both a commercial bank's demand deposit data and Federal Reserve aggregate NOW account data to fit the model over an eight-and-a-half-year period. Robust time series models for the demand deposit balances and interest rates paid are formulated and fit to the data. The extended Vasicek model is used to model evolution of the term structure of interest rates. The premium on the bank's NOW account deposits ranges between -16% to +12% over the sample period, with an average life (of the deposit's liability) of between 7.8 and 9.8 years. For the Federal Reserve aggregate NOW account data, the premium ranges between -1.4% and +13% over the sample period, with an average life of between 8.4 and 10 years. These results are consistent with estimates obtained in previous studies of demand deposit premiums and with the premiums paid on Resolution Trust Corporation auctions of demand deposits. [ABSTRACT FROM AUTHOR]

In this paper we investigate the geometric properties of the configuration consisting of a subspace Γ and a canonical subgeometry Σ in PG (n − 1 , q n) , with Γ ∩ Σ = 0̸. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of Σ from the vertex Γ. In particular we deal with the maximum scattered linear sets of the line PG (1 , q n) found by Lunardon and Polverino (2001) and recently generalized by Sheekey (2016). Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajbók and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in PG (1 , q 6) , yielding also to new examples of MRD-codes in F q 6 × 6 with left idealizer isomorphic to F q 6 . [ABSTRACT FROM AUTHOR]

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 nonconstructive 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. [ABSTRACT FROM AUTHOR]

Abstract In [2] and [18] are presented the first two families of maximum scattered F q -linear sets of the projective line PG (1 , q n). More recently in [22] and in [5] , new examples of maximum scattered F q -subspaces of V (2 , q n) have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the F q -linear sets presented in [22] and in [5] , for n = 6 , 8 , are new. Also, for q odd, q ≡ ± 1 , 0 (mod 5) , we present new examples of maximum scattered F q -linear sets in PG (1 , q 6) , arising from trinomial polynomials, which define new F q -linear MRD-codes of F q 6 × 6 with dimension 12, minimum distance 5 and left idealiser isomorphic to F q 6 . [ABSTRACT FROM AUTHOR]

In this paper, we properly extend the family of rank-metric codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are F q 2 t -linear of dimension 2 in the space of linearized polynomials over F q 2 t , where t is any integer greater than 2, and we prove that they are maximum rank distance codes. For t ≥ 5 , we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new. [ABSTRACT FROM AUTHOR]

Scattered polynomials over a finite field F q n have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L- q t -partially scattered and R- q t -partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L- q t -partially scattered and R- q t -partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L- q t -partially scattered one. We investigate a large family F of R- q t -partially scattered polynomials, containing examples of exceptional R- q t -partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R- q t -partially scattered. Many polynomials in F are inequivalent and geometric arguments are used to determine their equivalence classes under the action of Γ L (2 n / t , q t). [ABSTRACT FROM AUTHOR]

Abstract We provide sufficient and necessary conditions for the coefficients of a q -polynomial f over F q n which ensure that the number of distinct roots of f in F q n equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q -polynomials of degree q n − 2 over F q n which have maximum kernel and for n ≤ 6 we list all q -polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary q -polynomial. Analogous results are proved for q s -polynomials as well, where gcd ⁡ (s , n) = 1. [ABSTRACT FROM AUTHOR]

Every linear set in a projective space is the projection of a subgeometry, and most known characterizations of linear sets are given under this point of view. For instance, scattered linear sets of pseudoregulus type are obtained by considering a Desarguesian spread of a subgeometry and projecting from a vertex which is spanned by all but two director spaces. In this paper we introduce the concept of linear sets of h -pseudoregulus type, which turns out to be projected from the span of an arbitrary number of director spaces of a Desarguesian spread of a subgeometry. Among these linear sets, we characterize those which are h -scattered and solve the equivalence problem between them; a key role is played by an algebraic tool recently introduced in the literature and known as Moore exponent set. As a byproduct, we classify asymptotically h -scattered linear sets of h -pseudoregulus type. [ABSTRACT FROM AUTHOR]