Connections between scattered linear sets and MRD-codes

The aim of this paper is to survey on the known results on maximum scattered linear sets and MRD-codes. In particular, we investigate the link between these two areas. In "A new family of linear maximum rank distance codes" (2016) Sheekey showed how maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ define square MRD-codes. Later in "Maximum scattered linear sets and MRD-codes" (2017) maximum scattered linear sets in $\mathrm{PG}(r-1,q^n)$, $r>2$, were used to construct non square MRD-codes. Here, we point out a new relation regarding the other direction. We also provide an alternative proof of the well-known Blokhuis-Lavrauw's bound for the rank of maximum scattered linear sets shown in "Scattered spaces with respect to a spread in $\mathrm{PG}(n,q)$" (2000).

Related papers:

title: Linear k-blocking Sets
abstract: We point out the relationship between normal spreads and the linear k-blocking sets introduced in . We give a characterisation of linear k-blocking sets proving that the projections and the embeddings of a PG(kt,q) in are linear k-blocking sets of . Finally, we construct some new examples.

title: Linear sets in finite projective spaces
abstract: In this paper linear sets of finite projective spaces are studied and the ''dual'' of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides ''old'' results, new ones are proven and some open questions are discussed.

title: A new family of linear maximum rank distance codes
abstract: In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.

title: Classes and equivalence of linear sets in PG(1, qn)
abstract: The equivalence problem of F q -linear sets of rank n of PG ( 1 , q n ) is investigated, also in terms of the associated variety, projecting configurations, F q -linear blocking sets of Redei type and MRD-codes. We call an F q -linear set L U of rank n in PG ( W , F q n ) = PG ( 1 , q n ) simple if for any n-dimensional F q -subspace V of W, L V is P Γ L ( 2 , q n ) -equivalent to L U only when U and V lie on the same orbit of Γ L ( 2 , q n ) . We prove that U = { ( x , Tr q n / q ( x ) ) : x ∈ F q n } defines a simple F q -linear set for each n. We provide examples of non-simple linear sets not of pseudoregulus type for n > 4 and we prove that all F q -linear sets of rank 4 are simple in PG ( 1 , q 4 ) .

title: A Carlitz type result for linearized polynomials
abstract: For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$.

title: Maximum scattered linear sets and MRD-codes
abstract: The rank of a scattered $${\mathbb F}_q$$Fq-linear set of $${{\mathrm{{PG}}}}(r-1,q^n)$$PG(r-1,qn), rn even, is at most rn / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered $${\mathbb F}_q$$Fq-linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered $${\mathbb F}_q$$Fq-linear sets of $${{\mathrm{{PG}}}}(1,q^n)$$PG(1,qn) of maximum rank n yield $${\mathbb F}_q$$Fq-linear MRD-codes with dimension 2n and minimum distance $$n-1$$n-1. We generalize this result and show that scattered $${\mathbb F}_q$$Fq-linear sets of $${{\mathrm{{PG}}}}(r-1,q^n)$$PG(r-1,qn) of maximum rank rn / 2 yield $${\mathbb F}_q$$Fq-linear MRD-codes with dimension rn and minimum distance $$n-1$$n-1.

title: Scattered Spaces with Respect to a Spread in PG(n,q)
abstract: A scattered subspace of PG(n-1,q) with respect to a (t-1)-spread S is a subspace intersecting every spread element in at most a point. Upper and lower bounds for the dimension of a maximum scattered space are given. In the case of a normal spread new classes of two intersection sets with respect to hyperplanes in a projective space are obtained using scattered spaces.

title: Scattered Linear Sets and Pseudoreguli
abstract: In this paper, we show that one can associate a pseudoregulus with every scattered linear set of rank $3n$ in $\mathrm{PG}(2n-1,q^3)$. We construct a scattered linear set having a given pseudoregulus as associated pseudoregulus and prove that there are $q-1$ different scattered linear sets that have the same associated pseudoregulus. Finally, we give a characterisation of reguli and pseudoreguli in $\mathrm{PG}(3,q^3)$.

title: On linear sets on a projective line
abstract: Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, ), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, qh) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, qh).

title: Maximum scattered linear sets of pseudoregulus type and the Segre variety $\mathcal{S}_{n,n}$
abstract: In this paper we study a family of scattered $\mathbb{F}_{q}$-linear sets of rank tn of the projective space PG(2n−1,qt) (n≥1, t≥3), called of pseudoregulus type, generalizing results contained in Lavrauw and van de Voorde,  (2013) and in Marino et al.  (2007). As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twisted Fields and the 2-dimensional Knuth semifields.

title: Scattered linear sets generated by collineations between pencils of lines
abstract: Let $$A$$A and $$B$$B be two points of $$\mathrm{{PG}}(2,q^n)$$PG(2,qn), and let $$\Phi $$Φ be a collineation between the pencils of lines with vertices $$A$$A and $$B$$B. In this paper, we prove that the set of points of intersection of corresponding lines under $$\Phi $$Φ is either the union of a scattered $$\mathrm{{GF}}(q)$$GF(q)-linear set of rank $$n+1$$n+1 with the line $$AB$$AB or the union of $$q-1$$q-1 scattered $$\mathrm{{GF}}(q)$$GF(q)-linear sets of rank $$n$$n with $$A$$A and $$B$$B. We also determine the intersection configurations of two scattered $$\mathrm{{GF}}(q)$$GF(q)-linear sets of rank $$n+1$$n+1 of $$\mathrm{{PG}}(2,q^n)$$PG(2,qn) both meeting the line $$AB$$AB in a $$\mathrm{{GF}}(q)$$GF(q)-linear set of pseudoregulus type with transversal points $$A$$A and $$B$$B.

title: On scattered linear sets of pseudoregulus type in PG(1, qt)
abstract: Scattered linear sets of pseudoregulus type in PG ( 1 , q t ) have been defined and investigated in 19. The aim of this paper is to continue such an investigation. Properties of a scattered linear set of pseudoregulus type, say L , are proved by means of three different ways to obtain L : (i) as projection of a q-order canonical subgeometry 20, (ii) as a point set whose image under the field reduction map is the hypersurface of degree t in PG ( 2 t - 1 , q ) studied in 10, (iii) as exterior splash, by the correspondence described in 15. In particular, given a canonical subgeometry Σ of PG ( t - 1 , q t ) , necessary and sufficient conditions are given for the projection of Σ with center a ( t - 3 ) -subspace to be a linear set of pseudoregulus type. Furthermore, the q-order sublines are counted and geometrically described.

title: Blocking Sets and Derivable Partial Spreads
abstract: We prove that a GF(q)-linear Rédei blocking set of size qt + qt−1 + ··· + q + 1 of PG(2,qt) defines a derivable partial spread of PG(2t − 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size qt + qt−1 + ··· + q + 1 in PG(2,qt), if t ≥ 4.

title: New maximum scattered linear sets of the projective line
abstract: In  and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in , new examples of maximum scattered $\mathbb{F}_q$-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 $\mathbb{F}_q$-linear sets presented in  and in , for $n=6,8$, are new. Also, for $q$ odd, $q\equiv \pm 1,\,0 \pmod 5$, we present new examples of maximum scattered $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^6)$, arising from trinomial polynomials, which define new $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$ with dimension $12$, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to $\mathbb{F}_{q^6}$.

title: A new family of maximum scattered linear sets in PG(1, q6)
abstract: We generalize the example of linear set presented by the last two authors in “Vertex properties of maximum scattered linear sets of PG (1,  q n ) " (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are new. This solves an open problem posed in “Vertex properties of maximum scattered linear sets of PG (1,  q n ) " (2019). As a consequence of Sheekey’s results in “A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters (6, 6,  q ; 5) .

title: On embeddings of minimum dimension of PG(n, q) × PG(n, q)
abstract: A construction is given of an embedding of \({\mathrm{PG}}(n-1,q)\times {\mathrm{PG}}(n-1,q)\) into \({\mathrm{PG}}(2n-1,q)\), i.e. of minimum dimension, and it is shown that the image is a nonsingular hypersurface of degree \(n\). The construction arises from a scattered subspace with respect to a Desarguesian spread in \({\mathrm{PG}}(2n-1,q)\). By construction there are two systems of maximum subspaces (in this case \((n-1)\)-dimensional) which cover this hypersurface. However, unlike the standard Segre embedding, the minimum embedding constructed here allows another \(n-2\) systems of maximum subspaces which cover this embedding. We describe these systems and study the stabiliser of these embeddings. The results can be considered as a generalization of the properties of the hyperbolic quadric \(Q^+(3,q)\).

title: Rank-metric codes, linear sets, and their duality
abstract: In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of Sheekey ( 2016, Sect. 5) on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in Lunardon ( 2017). Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $$\mathrm {PG}(1,q^n)$$ PG ( 1 , q n ) .

title: A special class of scattered subspaces
abstract: Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an $\mathbb{F}_q$-subspace $U$ of $V$ $h$-scattered if $U$ meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $h$. For $h=1$ and $\dim_q U=rn/2$ these subspaces were first studied by Blokhuis and Lavrauw in 2000 and are called maximum scattered. They have been investigated intensively because of their relations with two-weight codes and strongly regular graphs. In the recent years maximum scattered and $n$-dimensional $(r-1)$-scattered $\mathbb{F}_q$-subspaces have been linked also to MRD-codes with a maximum idealiser. 
In this paper we prove the upper bound $rn/(h+1)$ for the dimension of $h$-scattered subspaces, $h>1$, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.

title: Bilinear Forms over a Finite Field, with Applications to Coding Theory
abstract: Let Ω be the set of bilinear forms on a pair of finite-dimensional vector spaces over GF(q). If two bilinear forms are associated according to their q-distance (i.e., the rank of their difference), then Ω becomes an association scheme. The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. The main emphasis is put on subsets of Ω and their q-distance structure. Certain q-ary codes are attached to a given X ⊂ Ω ; the Hamming distance enumerators of these codes depend only on the q-distance enumerator of X. Interesting examples are provided by Singleton systems X ⊂ Ω , which are defined as t-designs of index 1 in a suitable semilattice (for a given integer t). The q-distance enumerator of a Singleton system is explicitly determined from the parameters. Finally, a construction of Singleton systems is given for all values of the parameters.

title: Isometries for rank distance and permutation group of Gabidulin codes
abstract: The rank distance was introduced by E.M. Gabidulin (see Probl.  1985). He determined an upper bound for the minimum rank distance of a code. Moreover, he constructed a class of codes which meet this bound: the so-called Gabidulin codes. We first characterize the linear and semilinear isometries for the rank distance. Then we determine the isometry group and the permutation group of Gabidulin codes of any length. We give a characterization of equivalent Gabidulin codes. Finally, we prove that the number of equivalence classes of Gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in GF(p/sup m/) under some particular relations.

title: Equivalence for Rank-Metric and Matrix Codes and Automorphism Groups of Gabidulin Codes
abstract: For a growing number of applications, such as cellular, peer-to-peer, and sensor networks, efficient error-free transmission of data through a network is essential. Toward this end, Kötter and Kschischang propose the use of subspace codes to provide error correction in the network coding context. The primary construction for subspace codes is the lifting of rank-metric or matrix codes, a process that preserves the structural and distance properties of the underlying code. Thus, to characterize the structure and error-correcting capability of these subspace codes, it is valuable to perform such a characterization of the underlying rank-metric and matrix codes. This paper lays a foundation for this analysis through a framework for classifying rank-metric and matrix codes based on their structure and distance properties. To enable this classification, we extend work by Berger on equivalence for rank-metric codes to define a notion of equivalence for matrix codes, and we characterize the group structure of the collection of maps that preserve such equivalence. We then compare the notions of equivalence for these two related types of codes and show that matrix equivalence is strictly more general than rank-metric equivalence. Finally, we characterize the set of equivalence maps that fix the prominent class of rank-metric codes known as Gabidulin codes. In particular, we give a complete characterization of the rank-metric automorphism group of Gabidulin codes, correcting work by Berger, and give a partial characterization of the matrix-automorphism group of the expanded matrix codes that arise from Gabidulin codes.

title: Automorphism groups of Gabidulin-like codes
abstract: Let K/k be a cyclic Galois extension of degree $${\ell}$$ℓ and $${\theta }$$θ a generator of Gal(K/k). For any $${v=(v_1, \ldots, v_m)\in K^{m}}$$v=(v1,…,vm)∈Km such that v is linearly independent over k, and any $${1\leq d  < m }$$1≤d<m the Gabidulin-like code $${\mathcal{C}(v,\theta, d) \leq k^{\ell \times m }}$$C(v,θ,d)≤kℓ×m is a maximum rank distance code of dimension $${\ell d}$$ℓd over k. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.

title: On kernels and nuclei of rank metric codes
abstract: For each rank metric code $$\mathcal {C}\subseteq \mathbb {K}^{m\times n}$$C⊆Km×n, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When $$\mathcal {C}$$C is $$\mathbb {K}$$K-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When $$\mathbb {K}$$K is a finite field $$\mathbb {F}_q$$Fq and $$\mathcal {C}$$C is a maximum rank distance code with minimum distance $$d<\min \{m,n\}$$d<min{m,n} or $$\gcd (m,n)=1$$gcd(m,n)=1, the kernel of the associated translation structure is proved to be $$\mathbb {F}_q$$Fq. Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over $$\mathbb {F}_q$$Fq must be a finite field; its right nucleus also has to be a finite field under the condition $$\max \{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1$$max{d,m-d+2}⩾n2+1. Let $$\mathcal {D}$$D be the DHO-set associated with a bilinear dimensional dual hyperoval over $$\mathbb {F}_2$$F2. The set $$\mathcal {D}$$D gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to $$\mathbb {F}_2$$F2. Also, its middle nucleus must be a finite field containing $$\mathbb {F}_q$$Fq. Moreover, we also consider the kernel and the nuclei of $$\mathcal {D}^k$$Dk where k is a Knuth operation.

title: Algebraic structures of MRD codes
abstract: Based on results in finite geometry we prove the existence of MRD codes in (F_q)_(n,n) with minimum distance n which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.

title: Generalized Twisted Gabidulin Codes
abstract: Let C be a set of m by n matrices over F q such that the rank of A − B is at least d for all distinct A , B ∈ C . Suppose that m ⩽ n . If # C = q n ( m − d + 1 ) , then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1 d m − 1 . One was found by Delsarte (1978)  and Gabidulin (1985)  independently, and it was later generalized by Kshevetskiy and Gabidulin (2005)  . We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016)  , and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.

title: New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions
abstract: It is well-known that maximum rank distance (MRD) codes can be constructed as generalized Gabidulin codes. However, it was unknown until recently whether other constructions of linear MRD codes exist. In this paper, we derive a new criterion for linear MRD codes as well as an algebraic criterion for testing whether a given linear MRD code is a generalized Gabidulin code. We then use the criteria to construct examples of linear MRD codes which are not generalized Gabidulin codes.

title: Additive Rank Metric Codes
abstract: We give an infinite family of maximum rank distance (MRD) codes, which covers properly the largest known linear MRD code family. Our family contains infinite families of non-linear MRD codes, which are the first non-linear examples for most of the parameters. We also give explicit examples and a table that demonstrates the proportion of linear and non-linear families for some small parameters.

title: New semifields and new MRD codes from skew polynomial rings
abstract: In this article we construct a new family of semifields, containing and extending two well-known families, namely Albert's generalised twisted fields and Petit's cyclic semifields (also known as Johnson-Jha semifields). The construction also gives examples of semifields with parameters for which no examples were previously known. In the case of semifields two dimensions over a nucleus and four-dimensional over their centre, the construction gives all possible examples. 
Furthermore we embed these semifields in a new family of maximum rank-distance codes, encompassing most known current constructions, including the (twisted) Delsarte-Gabidulin codes, and containing new examples for most parameters.

title: On the number of roots of some linearized polynomials
abstract: Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7,10,24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form \[ ax+b_0x^{\sigma}+b_1x^{\sigma q^n}+b_2x^{\sigma q^{2n}}+\ldots+b_{t-1}x^{\sigma q^{n(t-1)}} \in \mathbb{F}_{q^{nt}}[x], \] with $\sigma$ a generator of the Galois group $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a $q^n$-polynomial. Finally, as an applications of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in $\mathrm{PG}(1,q^6)$ presented in .

title: Maximum rank-distance codes with maximum left and right idealisers.
abstract: Left and right idealisers are important invariants of linear rank-distance codes. In case of maximum rank-distance (MRD for short) codes in $\mathbb{F}_q^{n\times n}$ the idealisers have been proved to be isomorphic to finite fields of size at most $q^n$. Up to now, the only known MRD codes with maximum left and right idealisers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in $\mathbb{F}_q^{n\times n}$ for $n\leq 9$ with maximum left and right idealisers and connect them to Moore type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealisers for $n=7$, $q$ odd and for $n=8$, $q\equiv 1 \pmod 3$. These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for $n\geq 9$.

title: On the number of inequivalent Gabidulin codes
abstract: Maximum rank-distance (MRD) codes are extremal codes in the space of $$m\times n$$m×n matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct MRD codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of MRD codes that are pairwise inequivalent, provided that $$2\leqslant m\leqslant n-2$$2⩽m⩽n-2.

title: Covering Radius of Matrix Codes Endowed with the Rank Metric
abstract: 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.

title: Puncturing maximum rank distance codes
abstract: We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models, we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes, and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey ( 2016).

title: On the similarities between generalized rank and Hamming weights and their applications to network coding
abstract: Rank weights and generalized rank weights have been proven to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical error and erasure correction, and information leakage in wire-tap channels of type II and code-based secret sharing. Although many similarities between both cases have been established and proven in the literature, many other known results in the Hamming case, such as bounds or characterizations of weight-preserving maps, have not been translated to the rank case yet, or in some cases have been proven after developing a different machinery. The aim of this paper is to further relate both weights and generalized weights, show that the results and proofs in both cases are usually essentially the same, and see the significance of these similarities in network coding. Some of the new results in the rank case also have new consequences in the Hamming case.

title: Non-linear maximum rank distance codes
abstract: By exploring some geometry of Segre varieties and Veronese varieties, new families of non-linear maximum rank distance codes and optimal constant rank codes are provided.

title: Non-Linear Maximum Rank Distance Codes in the Cyclic Model for the Field Reduction of Finite Geometries
abstract: In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.

title: A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and its associated non-linear MRD codes
abstract: Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and let $$\Phi $$Φ be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if $$d=2$$d=2 or $$d\ge 3$$d≥3 and the lines $$\Phi ^{-1}(AB), AB, \Phi (AB) $$Φ-1(AB),AB,Φ(AB) are not in a common plane, then the set $$\mathcal{C}$$C of points of intersection of corresponding lines under $$\Phi $$Φ is the union of $$q-1$$q-1 scattered $${\mathbb {F}}_{q}$$Fq-linear sets of rank n together with $$\{A,B\}$${A,B}. As an application we will construct, starting from the set $$\mathcal{C}$$C, infinite families of non-linear $$(d+1, n, q;d-1)$$(d+1,n,q;d-1)-MRD codes, $$d\le n-1$$d≤n-1, generalizing those recently constructed in Cossidente et al. ( 2016) and Durante and Siciliano (Electron J Comb, 2017).

title: Some new non-additive maximum rank distance codes
abstract: In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters.

title: MRD-codes and linear sets
abstract: In this paper we point out the relationship between linear MRD-codes and various geometric objects as linearized polynomials, linear sets of P G ( n − 1 , q n ) , and generalized Segre varieties. We introduce and characterize a class of particular linear MRD-codes called F q n -linear codes with distance n − k + 1 proving that such codes are equivalent to ( n − k + 1 ) -embeddings of a canonical subgeometry.