Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries

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.


Introduction
Let M m,m ′ (F q ), m ≤ m ′ , be the rank metric space of all the m × m ′ matrices with entries in the finite field F q with q elements, q = p h , p a prime. The distance between two matrices by definition is the rank of their difference. An (m, m ′ , q; s)rank distance code (also rank metric code) is any subset X of M m,m ′ (F q ) such that the minimum distance between two of its distinct elements is s+1. An (m, m ′ , q; s)-rank distance code is said to be linear if it is a linear subspace of M m,m ′ (F q ).
It is known [11] that the size of an (m, m ′ , q; s)-rank distance code X is bounded by the Singleton-like bound: |X | ≤ q m ′ (m−s) .
When this bound is achieved, X is called an (m, m ′ , q; s)-maximum rank distance code, or (m, m ′ , q; s)-MRD code, for short.
Although MRD codes are very interesting by their own and they caught the attention of many researchers in recent years [1,9,32], such codes have also applications in error-correction for random network coding [18,22,37], space-time coding [38] and cryptography [17,36].
Obviously, investigations of MRD codes can be carried out in any rank metric space isomorphic to M m,m ′ (F q ). In his pioneering paper [11], Ph. Delsarte constructed linear MRD codes for all the possible values of the parameters m, m ′ , q and s by using the framework of bilinear forms on two finite-dimensional vector spaces over a finite field (Delsarte used the terminology Singleton systems instead of maximum rank distance codes).
Few years later, Gabidulin [16] independently constructed Delsarte's linear MRD codes as evaluation codes of linearized polynomials over a finite field [26]. That construction was generalized in [21] and these codes are now known as Generalized Gabidulin codes.
In the case m ′ = m, a different construction of Delsarte's MRD codes was given by Cooperstein [7] in the framework of the tensor product of a vector space over F q by itself. Very recently, Sheekey [35] and Lunardon, Trombetti and Zhou [28] provide some new linear MRD codes by using linearized polynomials over F q m .
In finite geometry, (m, m, q; m − 1)-MRD codes are known as spread sets [12]. To the extent of our knowledge the only non-linear MRD codes that are not spread sets are the (3, 3, q; 1)-MRD codes constructed by Cossidente, Marino and Pavese in [8]. They got such codes by looking at the geometry of certain algebraic curves of the projective plane PG(2, q 3 ). Such curves, called C 1 F -sets, were introduced and studied by Donati and Durante in [13]. In this paper, we construct infinite families of nonlinear (m, m, q; m − 2)-MRD codes, for q ≥ 3 and m ≥ 3. We also show that the Cossidente, Marino and Pavese non-linear MRD codes belong to these families. Our investigation will carry out in the framework of bilinear forms on a finite dimensional vector space over F q .
Let Ω = Ω(V, V ) be the set of all bilinear forms on V , where V = V (m, q) denotes an m-dimensional vector space over F q . Clearly, Ω is an Let u 1 , . . . , u m be a basis of V . For a given f ∈ Ω, the matrix (f (u i , u j )) i,j=1,...,m , is called the matrix of f in the basis u 1 , . . . , u m and the map is an isomorphism of rank metric spaces giving rk(f ) = rk(ν(f )).
The group H = GL(V ) × GL(V ) acts on Ω as a subgroup of Aut Fq (Ω): for every (g, g ′ ) ∈ H, the (g, g ′ )−image of any f ∈ Ω is defined to be the bilinear form f (g,g ′ ) given by Any θ ∈ Aut(F q ) naturally defines a semilinear transformation of V . For any f ∈ Ω and θ ∈ Aut(F q ), we can define the bilinear form is an automorphism of Ω. It turns out that the above automorphisms are all the elements in Aut Two MRD codes X 1 and X 2 are said to be equivalent if there exists ϕ ∈ Aut Fq (Ω) such that X 2 = X ϕ 1 . This paper is organized as follows. In Section 2 we introduce a cyclic model of Ω. In this model we construct infinite families of non-linear MRD codes. More precisely, for q ≥ 3, m ≥ 3 and I any subset of F q \ {0, 1}, we provide a subset F m,q;I of Ω which turns out to be a non-linear (m, m, q; m − 2)-MRD code (Theorem 2.19).
In Section 3 we give a geometric description of such codes. If a given rank distance code X is considered as a subset of V (m 2 , q), then one can consider the corresponding set of projective points in PG(m 2 − 1, q) under the canonical homomorphism ψ : GL(V (m 2 , q)) → PGL(m 2 , q). We prove (Theorem 3.5) that the projective set defined by F m,q;I , with |I| = k, is a subset of a Desarguesian m-spread of PG(m 2 − 1, q) [34] consisting of two spread elements, k pairwise disjoint Segre varieties S m,m (F q ) [20] and q − 1 − k hyperreguli [30]. Additionally, if one consider the projective space PG(m 2 − 1, q) as the field reduction of PG(m − 1, q m ) over F q , then the projective set defined by F m,q;I is, in fact, the field reduction of the union of two projective points, k mutually disjoint (m−1)-dimensional F q -subgeometries and q − 1 − k scattered F q -linear sets of pseudoregulus type of PG(m − 1, q m ) [13,24,29]. The main tool we use to get the above geometric description is the field reduction of V (m, q m ) over F q in the cyclic model for the tensor product F q m ⊗ V as described in [7].

The non-linear MRD codes in the cyclic model of bilinear forms
In the paper [7], the cyclic model of the m-dimensional vector space V = V (m, q) over F q was introduced by taking eigenvectors, say v 1 , . . . , v m , of a given Singer cycle σ of V , where a Singer cycle of V is an element of GL(V ) of order q m − 1. Since the vectors v 1 , . . . , v m have distinct eigenvalues over F q m , they form a basis of the extension V = V (m, q m ) of V . In this basis the vector space V is represented by We call v 1 , . . . , v m a Singer basis of V and the above representation is called the cyclic model for V [19,15].
The set of all 1−dimensional F q −subspaces of V spanned by vectors in the cyclic model for V is called the cyclic model for the projective space PG(V ). Note that the above cyclic model corresponds to the cyclic model of PG(V ) where the points are identified with the elements of the group Z q m−1 +q m−2 +···+q+1 [19, pp. 95-98] [15]. Very recently, the cyclic model for V (3, q) has been used to give an alternative model for the triality quadric Q + (7, q) [2].
Let V * be the dual vector space of V with basis v * 1 , . . . , v * m , the dual basis of the Singer basis v 1 , . . . , v m . Then the dual vector space of V is A linear transformation from V to itself is called an endomorphism of V . We will denote the set of all endomorphisms of V by End(V ).
An m × m Dickson matrix (or q-circulant matrix) over F q m is a matrix of the form with a i ∈ F q m . We say that the above matrix is generated by the array (a 0 , a 1 , . . . , a m−1 ).
Let D m (F q m ) denote the Dickson matrix algebra formed by all m × m Dickson matrices over F q m . The set B m (F q m ) of all invertible Dickson m × m matrices is known as the Betti-Mathieu group [6].
is called a linearized polynomial (or q-polynomial) over F q m . It is known that every endomorphism of F q m over F q can be represented by a unique q−polynomial [33].
Let L m (F q m ) be the set of all q-polynomials over F q m . In the paper [39], it was showed that the map is an isomorphism between the non-commutative F q −algebras L m (F q m ) and D m (F q m ). From Proposition 2.1 we see that any Singer basis of V realizes this isomorphism. Proof. Let D a be an m×m Dickson matrix generated by the m-ple a = (a 0 , a 1 , . . . , a m−1 ) over F q m . Let f a be the bilinear mapping on V × V defined by where subscripts are taken modulo m, and then extended over V by linearity. Set L a (x) = m−1 i=0 a i x q i and let Tr denote the trace function from F q m onto F q : It is easily seen that the action of f a on V × V is given by with The assertion follows from consideration on the size of D m (F q m ). ✷ For any m-ple a = (a 0 , . . . , a m−1 ) over F q m , f a will denote the bilinear form having matrix D a in the Singer basis v 1 , . . . , v m . For any set A of m−ples over F q m we put is an isomorphism of rank metric spaces giving rk(f a ) = rk(D (a 0 ,...,a m−1 ) ).
and it corresponds to the operator ⊤.
Remark 2.6. Since a change of basis in V × V preserves the rank of bilinear forms, for any given f ∈ Ω we can consider its matrix representation in the Singer basis v 1 , . . . , v m . Therefore, we can assume f = f a for some m-ple a over F q m , so that We are now in position to construct non-linear MRD codes as subsets of Ω.
Let N denote the norm map from F q m onto F q : For every nonzero element α ∈ F q m , let If S is the Singer cyclic group generated by σ, then the set F πa is the (S × S)-orbit of the bilinear form f a , with a = (1, α, α 1+q , . . . , α 1+...+q m−2 ). It turns out that the bilinear forms in F πa have constant rank.
if and only if λx q i = ρy q i , for i = 0, . . . , m − 1. If we compare the equalities with i = 0 and i = 1, we get x q−1 = y q−1 . For every fixed x ∈ F q m there are exactly q − 1 elements y in F q m such that y q−1 = x q−1 .
After subtracting Equation (5) side-by-side from Equation (6) multiplied by α, we get for i = 1, 2. Then, the m−ple is a solution of the linear system . . , m and d 3 = −∆ q . We then write By plugging (8) in the right-hands of the above equalities we get and From (12) it turns out that the value of x 1 must satisfy From (8), we have x 1 = (a − 1) + (λ − λ q αβ q+...+q m−1 )x. Therefore, we get By plugging this value in b, we get We claim that the bilinear form (f a − f b ) has maximum rank m. Indeed, suppose there exists a nonzero z ∈ F q m such that L a−b (z) = 0. By plugging (13) in Equation (7) we get Therefore, the following equation holds: given β q 2 +...+q m−1 y + β 1+q 2 +...+q m−1 y q + y q 2 + β q 2 y q 3 + . . . + β q 2 +...+q m−2 y q m−1 = 0.
By subtracting Equation (14) from (15) multiplied by β q we get b = 1, a contradiction. ✷ For every nonzero element α ∈ F q m , let Remark 2.11. Note that the set F Jα is the (S × S)-orbit of the bilinear form f a , with a = (1, 0, . . . , 0, −α). It turns out that the bilinear forms in F Jα have constant rank.
By arguing similarly to the proof of Proposition 2.8 and Lemma 2.9, we get the following result. We will write J a instead of J α , if α is an element of F q m with N(α) = a. Proof. The bilinear form f a , is equivalent to the bilinear form fâ, withâ = (x, y q , 0, . . . , 0), via the automorphism ⊤. The result then follows from Remark 2.5 and Theorem 6.3 in [11].  Proof. By Remark 2.7 we can assume a = (1, α, . . . , α 1+...+q m−2 ) with N(α) = a. By arguing as in the proof of Lemma 2.10 we see that the triple is a solution of the linear system for some z 1 , z 2 ∈ F q m linearly independent over F q with ∆ = z 1 z q 1 z 2 z q 2 = 0. Any solution (x 1 , x 2 , x 3 ) of (17) satisfies . As a solution of (17), the triple (16) must satisfies aN(λ)N(x) = bN(λ)N(x) giving either λx = 0 or a = b, a contradiction. ✷ Proof. The first part can be easily proved by taking the Dickson matrix D a with a ∈ A i . The second part follows from Lemma 2.13.
where 0 is the zero m−ple. Then the subset F m,q;I = {f a : a ∈ A m,q;I } of Ω is a non-linear (m, m, q; m − 2)-MRD code.
Proof. By Lemmas 2.9, 2.12 we get that A m,q;I has size q 2m . By Lemmas 2.10, 2.13, 2.15, 2.16, 2.17 and Corollary 2.14, we see that F m,q;I has minimum distance m − 1, i.e. it is a (m, m, q; m − 2)-MRD code. To show the non-linearity of F m,q;I , it suffices to find two distinct elements in it whose F q -span is not contained in F m,q;I .
Let f a ∈ F A 2 and f b ∈ F πa , a ∈ I. By corollary 2.3, we can work with the Dickson matrices D a and D b , or equivalently, with m-ples a and b as arrays in V (m, q m ). Let a = (0, . . . , 0, µ) and b = (λx, λαx q , . . . , λα 1+...+q m−2 x q m−1 ). Suppose a+ b ∈ π b , for some b ∈ F q . Then giving µ = 0. Therefore, the subspace spanned by a and b meets trivially every π b if b = a, or just in the 1-dimensional subspace spanned by b if b = a. The result then follows. ✷ : v ∈ X, v = 0}. The set [X] is said to be an F q -linear set of rank r if X is an r-dimensional F q -linear subspace of V (t, q s ). An F q -linear set [X] of rank r is said to be scattered if the size of [X] equals |PG(r − 1, q)|; see [31] for more details on F q -linear sets and [27] for a relationship between linear MRD-codes and F q -linear sets.

A geometric description for the non-linear MRD codes
Consider the set A m,q;I defined in Theorem 2.19 as a subset of V = V (m, q m ), by setting a 0 v 1 + a 1 v 2 + . . . + a m−1 v m , for any a = (a 0 , . . . , a m−1 ) ∈ A m,q;I ; here, v 1 , . . . , v m is the Singer basis of V defined in Section 2. Therefore, [π 1 ] = [V ] is a scattered F q -linear set of rank m of PG(m − 1, q m ) isomorphic to the projective space PG(m − 1, q).
For any α ∈ F q m \ {0}, the endomorphism Let W be the span of v 1 and v m in V . For any a ∈ F q \ {0}, [J a ] is a scattered F q -linear set of rank m of [W ]. In particular [J a ] is a maximum scattered F q -linear set of pseudoregulus type of [W ] [24,29].
Summarizing we have the following result. We now investigate the geometry in PG(m 2 − 1, q) of the projective set defined by each MRD code F m,q;I viewed as a subset of V (m 2 , q). q). The rank of a vector v = a 1 u 1 + a 2 u 2 + . . . + a m u m ∈ V by definition is the maximum number of linearly independent coordinates a i over F q .
If we consider F q m as the m-dimensional vector space V , then every α ∈ F q m can be uniquely written as α = x 1 u 1 + x 2 u 2 + . . . + x m u m , with x i ∈ F q . Hence, V can be viewed as V ⊗ V , the tensor product of V with itself, with basis {u (i,j) = u i ⊗ u j : i, j = 1, . . . , m}. Elements of V ⊗ V are called tensors and those of the form v ⊗ v ′ , with v, v ′ ∈ V are called fundamental tensors. In PG(V ⊗V ), the set of fundamental tensors correspond to the Segre variety S m,m (F q ) of PG(V ⊗ V ) [20].
Let φ be the map defined by We call this map the field reduction of V over F q with respect to the basis u 1 , . . . , u m . The projective space PG(V ⊗ V ) is the the field reduction of PG( V ) over F q with respect to the basis u 1 , . . . , u m .
is a partition of the nonzero vectors of V ⊗ V . In particular K is a Desarguesian partition, i.e. the stabilizer of K in GL(V ⊗ V ) contains a cyclic subgroup acting regularly on the components of K [34], [14]. [34], [14].
In addition, the projective set of PG(V ⊗ V ) corresponding to the φ-image of the 1-dimensional subspaces spanned by non-zero vectors in V is the Segre variety S m,m (F q ).
Let ν be the map defined by For every v = α 1 u 1 + . . .+ α m u m ∈ V , the k-th column of the matrix ν(φ(v)) is the m-ple (x 1k , . . . , x mk ) of the coordinates of α k with respect to the basis u 1 , . . . , u m of F q m . From [16], the rank of v equals the rank of ν(φ(v)), for all v ∈ V . In addition, the ν-image of fundamental tensors is precisely the set of rank 1 matrices.
Remark 3.2. Evidently, ν is an isomorphism of rank metric spaces which also provides an isomorphism between the field reduction V ⊗ V of V with respect to u 1 , . . . , u m and the metric space Ω of all bilinear forms on V = u 1 , . . . , u m Fq . Now embed V ⊗ V into V ⊗ V by extending the scalars from F q to F q m . By taking a Singer basis v 1 , . . . , v m of V defined by the Singer cycle σ, Cooperstein [7] defined a cyclic model for where the subscript j −1 + i is taken modulo m. As an F q -space, Φ(j) has dimension m and by consideration on dimension we have [7]. We call this representation the cyclic representation of the tensor product V ⊗ V .
Then Im( φ) is linearly equivalent to Im(φ) in V ⊗ V .
Let τ be the change of basis map of V from the basis u 1 , . . . , u m to the Singer basis v 1 , . . . , v m .
. On the other hand we have

✷
We call the map φ the field reduction of V over F q with respect to the Singer basis v 1 , . . . , v m and its image the cyclic model for the field reduction of V over F q . The projective space whose points are the 1-dimensional F q −subspaces generated by the elements of φ( V ) is the cyclic model for the field reduction of PG( V ) over F q .
Let ν be the map defined by Then, for any v = α 1 v 1 + . . . + α m v m ∈ V , the matrix ν( φ(v)) is the Dickson matrix D (α 1 ,...,αm) . Since the cyclic model for the field reduction of V is obtained from the field reduction φ( V ) by changing a basis in V ⊗ V , we get that the rank of ν( φ(v)) equals the rank of ν(φ(v)), for any v ∈ V .
In addition, the element k v = φ( v ) of the m-partition K is In particular, v∈V \{0} ν(k v ) is the set of all rank 1 matrices in D m (F q m ).
From the arguments above, we see that the set F m,q;I can be considered, via the isomorphism (3), as the field reduction of the set A m,q;I with respect to the Singer basis v 1 , . . . , v m .
As [π 1 ] = [V ], then the set F π 1 = φ(π 1 ) defines the Segre variety S m,m (F q ) of PG(V ⊗ V ) and F πa defines a Segre variety projectively equivalent to S m,m (F q ) under the element of PGL(V ⊗ V ) corresponding to the linear transformation τ α with N(α) = a.
Remark 3.4. Note that, whenever a = 1, elements in F πa have rank bigger then 1 by Lemma 2.10. This is explained by the fact that the linear transformation of V ⊗ V = V (m 2 , q) corresponding to τ α is not in Aut Fq (V ⊗ V ). [3] and widely investigated in [4,5] and recently in [10,23]. For any x ∈ F q m } is a so-called hyper-regulus of PG( W ) [30]. It turns out, that under the linear transformation τ α with N(α) = a, also J a defines a hyper-regulus of [ φ(W )].
The following result, which summarizes all above arguments, gives a geometric description of the MRD codes F m,q;I . Theorem 3.5. Let q > 2 be a prime power and m > 2 a positive integer. Let I be any nonempty subset of F q \ {0, 1} with k = |I|. The projective image of the MRD code F m,q;I in PG(m 2 − 1, q) is a subset of a Desarguesian spread which is union of two spread elements, k mutually disjoint Segre varieties S m,m (F q ) and q − 1 − k mutually disjoint hypereguli all contained in the (2m − 1)-dimensional projective subspace generated by the two spread elements.
In PG(2, q 3 ), q ≥ 3, let C be the set of points whose coordinates satisfy the equation X 1 X q 2 − X q+1 3 = 0, that is a C 1 F -set of PG(2, q 3 ) as introduced and studied in [13]. The set C is the projective image of a subset of V (3, q 3 ) which is the union For any nonzero a ∈ F q , let α ∈ F q 3 with N(α) = a and set Z a = {(λx, −λαx q , 0) : λ, x ∈ F q 3 \ {0}}. Let I be any non-empty subset of F q \ {0, 1} and put A ′ (q; I) = a∈I γ a b∈Fq\(I∪{0}) Up to an endomorphism of V ⊗ V viewed as the vector space V (9, q), the image of set A ′ (q; I) under ν • φ is a non-linear (3, 3, q; 1)-MRD code [8, Proposition 3.8].
Lemma 4.1. Let θ be the semilinear transformation of V (3, q 3 ) defined by with associated automorphism x → x q 2 . Then θ maps γ a into π a −1 and Z a into J a −1 , for any nonzero element a of F q .
The last part of the statement follows from straightforward calculations. ✷ Let L be any line of PG(2, q 3 ) disjoint from a subgeometry PG(2, q). The set of points of L that lie on some proper subspace spanned by points of PG(2, q) is called the exterior splash of PG(2, q) on L [25]. Proof. First we note that [W ] is disjoint from [π 1 ]. The F q m -span of some hyperplane in the cyclic model of V is a hyperplane of V with equation m i=1 α q i−1 X i = 0, for some nonzeroα ∈ F q m . As the Singer cycle σ acts on the hyperplanes of V by mapping the hyperplane with equation m i=1 α q i−1 X q i−1 = 0 to the hyperplane with equation m i=1 (µα) q i−1 X q i−1 = 0, then σ maps the hyperplane of V with equation In [8], the splash of [γ a ] was erroneusly given as the set [Z a ]. Note that, [Z a ] never coincides with [γ a ], unless a = 1.