Maximum scattered linear sets and MRD-codes

0.2 MB·

by  Bence Csajbók

Maximum scattered linear sets and MRD-codes
7
1
0
Bence Csajb´ok, Giuseppe Marino, Olga Polverino, Ferdinando Zullo ∗
2

n
a
J

4 Abstract
2
The rank of a scattered Fq-linear set of PG(r−1,qn), rn even, is

] at most rn/2 as it was proved by Blokhuis and Lavrauw. Existence
O
resultsandexplicit constructionsweregivenfor infinitely manyvalues
C of r, n, q (rn even) for scattered F -linear sets of rank rn/2. In this
q
. paper we prove that the bound rn/2 is sharp also in the remaining
h
t open cases.
a Recently Sheekey provedthat scattered F -linear sets of PG(1,qn)
m q
of maximum rank n yield F –linear MRD-codes with dimension 2n
q
[
and minimum distance n−1. We generalize this result and show that
1 scattered Fq-linear sets of PG(r−1,qn) of maximum rank rn/2 yield
v F –linearMRD-codeswithdimensionrnandminimumdistancen−1.
q
1
3
8 1 Introduction
6
0
1. Let Λ = PG(V,Fqn) = PG(r−1,qn), q = ph, p prime, V a vector space of
0 dimension r over Fqn, and let L be a set of points of Λ. The set L is said to
7 be an F –linear set of Λ of rank k if it is defined by the non-zero vectors of
q
1 an F -vector subspace U of V of dimension k, i.e.
: q
v
Xi L = LU = {huiFqn : u ∈ U \{0}}. (1)
r
a We point out that different vector subspaces can define the same linear
set. For this reason a linear set and the vector space defining it must be
considered as coming in pair.
Let Ω = PG(W,F ) bea subspaceof Λandlet L bean F -linear set of
qn U q
Λ. Then Ω∩L is an F –linear set of Ω defined by the F –vector subspace
U q q
U ∩W and, if dimFq(W ∩U)= i, we say that Ω has weight i in LU. Hence
∗The research was supported by Ministry for Education, University and Research of
Italy MIUR (Project PRIN 2012 ”Geometrie di Galois e strutture di incidenza”) and by
theItalianNationalGroupforAlgebraicandGeometricStructuresandtheirApplications
(GNSAGA - INdAM).
1
a point of Λ belongs to L if and only if it has weight at least 1 and if L
U U
has rank k, then |L | ≤ qk−1 +qk−2 +···+q +1. For further details on
U
linear sets see [40], [27], [28], [34], [35], [29], [12] and [13].
An F –linear set L of Λ of rank k is scattered if all of its points have
q U
weight 1, or equivalently, if L has maximum size qk−1 +qk−2 +···+q +
U
1. A scattered F –linear set of Λ of highest possible rank is a maximum
q
scattered F –linear set of Λ; see [4]. Maximum scattered linear sets have a
q
lot of applications in Galois Geometry, such as translation hyperovals [19],
translation caps in affine spaces [2], two-intersection sets ([4], [5]), blocking
sets ([41], [31], [32] [7], [1]), translation spreads of the Cayley generalized
hexagon ([9], [6], [37]), finite semifields (see e.g. [33], [10], [38], [15], [34],
[24], [25], [26]), coding theory and graph theory [8]. For a recent survey on
the theory of scattered spaces in Galois Geometry and its applications see
[23].
Therank of a scattered F -linear set of PG(r−1,qn), rneven, is at most
q
rn/2 ([4, Theorems 2.1, 4.2 and 4.3]). For n = 2 scattered F -linear sets of
q
PG(r−1,q2) of rank r are the Baer subgeometries. When r is even there
always exist scattered F –linear sets of rank rn in PG(r − 1,qn), for any
q 2
n ≥ 2 (see [22, Theorem 2.5.5] for an explicit example). Existence results
were proved for r odd, n−1 ≤ r, n even, and q > 2 in [4, Theorem 4.4], but
no explicit constructions were known for r odd, except for the case r = 3,
n = 4, see [1, Section 3]. Very recently families of scattered linear sets of
rank rn/2 in PG(r−1,qn), r odd, n even, were constructed in [2, Theorem
1.2] for infinitely many values of r, n and q.
The existence of scattered F –linear sets of rank 3n in PG(2,qn), n ≥ 6
q 2
even, n ≡ 0 (mod 3), q 6≡ 1 (mod 3) and q > 2 was posed as an open
problem in [2, Section 4]. As it was pointed out in [2], the existence of such
planar linear sets and the construction method of [2, Theorem 3.1] would
imply that the bound rn for the maximum rank of a scattered F –linear set
2 q
in PG(r−1,qn) is also tight when r is odd and n is even. In Theorem 2.3
we construct linear sets of rank 3n/2 of PG(2,qn), n even, and hence we
prove the sharpness of the bound also in the remaining open cases. Our
construction relies on the existence of non-scattered linear sets of rank 3t of
PG(1,q3t) (with t = n/2) defined by binomial polynomials.
In [42, Section 4] Sheekey showed that maximum scattered F -linear sets
q
of PG(1,qn) correspond to F -linear maximum rank distance codes (MRD-
q
codes) of dimension 2n and minimumdistance n−1. In Section 3 we extend
thisresultshowingthatMRD-codescanbeconstructedfromeveryscattered
linear set of rank rn/2 of PG(r − 1,qn), rn even, and we point out some
2
relations with Sheekey’s construction. Finally, we exhibit the MRD-codes
arising from maximum scattered linear sets constructed in Theorem 2.3 and
those constructed in [2, Theorems 2.2 and 2.3]
2 Maximum scattered linear sets in PG(r − 1,qn)
AsitwaspointedoutintheIntroduction,theexistenceofscatteredF –linear
q
sets of rank 3n in PG(2,qn), n ≥ 6 even, n ≡ 0 (mod 3), q 6≡ 1 (mod 3) and
2
q > 2 would imply that the bound rn for the rank of a maximum scattered
2
F –linear set in PG(r −1,qn) is tight in the remaining open cases (cf. [2,
q
Remark 2.11 and Section 4]).
In this section we show that binomials of the form f(x) = axqi +bx2t+i
definedover F can beusedtoconstructmaximum scattered F –linear sets
q3t q
in PG(2,q2t) for any t ≥ 2 and for any prime power q.
Consider the finite field F as a 3–dimensional vector space over its
q6t
subfieldF ,t ≥ 2,andletP = PG(F ,F ) =PG(2,q2t)betheassociated
q2t q6t q2t
projective plane. From [2, Section 2.2], the F -subspace
q
U := {ωx+f(x): x ∈ F }, (2)
q3t
ofF withω ∈ F \F ,f(x) = axqi+bxq2t+i,a,b ∈ F∗ ,1≤ i ≤ 3t−1and
q6t q2t qt q3t
gcd(i,2t) = 1, defines a maximum scattered F -linear set in the projective
q
plane P of rank 3t if f(x) ∈/ F for each x ∈ F∗ (cf. [2, Prop. 2.7]). The
x qt q3t
q-polynomial f(x) also defines an Fq-linear set Lf := {h(x,f(x))iFq3t : x ∈
F∗ } of the projective line PG(F ,F ) = PG(1,q3t). In what follows we
q3t q6t q3t
determine some conditions on L in order to obtain maximum scattered
f
F -linear sets in P of rank 3t.
q
If h | n, then by N (α) we will denote the norm of α ∈ F over the
qn/qh qn
subfield F , that is, N (α) = α1+qh+...+qn−h. We will need the following
qh qn/qh
preliminary result.
Lemma 2.1. Let f := f : x ∈ F 7→ axqi + bxq2t+i ∈ F , with
i,a,b q3t q3t
a,b ∈ F∗ , N (a) 6= −N (b) and gcd(i,t) = 1. If
q3t q3t/qt q3t/qt
Lf := {h(x,f(x))iFq3t: x ∈ F∗q3t} (3)
is not a scattered F –linear set of PG(1,q3t), then there exists c ∈F∗ such
q q3t
that
f (x)
g (x) := i,ca,cb ∈/ F for each x ∈F∗ . (4)
c x qt q3t
3
qi−1 q2t+i−1
Proof. First we show that 0 ∈/ Img for each c. If cax = −cbx
c 0 0
for some x ∈ F∗ , then −a/b = xqi(q2t−1), where the right hand side is a
0 q3t 0
(qt−1)-th power and hence N (−a/b) = 1, a contradiction.
q3t/qt
The non-zero elements of the one-dimensional F -spaces of F∗ yield a
qt q3t
partition of F∗ into q2t+qt+1 subsets of size qt−1. More precisely, if µ
q3t
is a primitive element of F , then
q3t
q2t+qt
F∗ = µkF∗ .
q3t [ qt
k=0
Let G := µkF∗ . We show that, for each k, either Img ∩ G = ∅, or
k qt 1 k
|Img ∩G |≥ (qt−1)/(q −1).
1 k
Suppose g (x )∈ G . Then for each γ ∈ F∗ we have
1 0 k qt
g (γx )= γqi−1g (x ).
1 0 1 0
Since gcd(i,t) = 1, it follows that
{g (γx ): γ ∈ F∗ } = g (x ){x ∈ F : N (x) = 1} ⊆ G
1 0 qt 1 0 qt qt/q k
and hence |Img ∩G |≥ (qt−1)/(q −1).
1 k
Next we show that there exists G such that Img ∩G = ∅. Supposeto
d 1 d
the contrary Img ∩G 6= ∅ for each j ∈ {0,1,...,q2t+qt}. Then |Img |≥
1 j 1
(q2t+qt+1)(qt−1)/(q−1) = (q3t−1)/(q−1) and since |Img |= |L | we
1 f
get a contradiction.
Supposethat Img ∩G =∅ and let c = µ−d. Then Img ∩F = ∅.
1 d c qt
Hence, by the previous lemma and by [2, Prop. 2.7], the existence of
a non-scattered linear set in PG(1,q3t) of form (3) implies the existence
of a binomial polynomial producing maximum scattered F -linear set in
q
PG(2,q2t) of rank 3t.
Lemma 2.2. Let f := f : x ∈ F 7→ axqi +bxq2t+i ∈ F , with a,b ∈
i,a,b q3t q3t
F∗ and 1 ≤ i ≤ 3t−1. For any prime power q ≥ 2 and any integer t ≥ 2
q3t
there exist a,b ∈ F∗ , with
q3t
N (b) 6= −N (a), (5)
q3t/qt q3t/qt
such that
Lfi,a,b := {h(x,fi,a,b(x))iFq3t: x ∈ F∗q3t},
is a non–scattered F –linear set in PG(1,q3t) of rank 3t.
q
4
Proof. First supposed := gcd(i,t) > 1. Then f is F -linear and hence each
qd
point of L has wight at least d, i.e. L cannot be scattered. Since qt ≥ 4
f f
we can always choose a,b such that (5) holds. From now on we assume
gcd(i,t) = 1.
The linear set L of PG(1,q3t) is not scattered if there exists a point
f
Px0 = h(x0,f(x0))iFq3t of rank greater than 1, i.e. if there exist x0 ∈ Fq3t∗
and λ ∈ F \ F such that f(λx ) = λf(x ). The latter condition is
q3t q 0 0
equivalent to
axqi(λ−λqi)= bxq2t+i(λq2t+i −λ). (6)
0 0
Since gcd(2t+i,3t),gcd(i,3t) ∈ {1,3}, the expressions in the two sides of
(6) arenon-zero whenλ ∈/ F . We firstprove that there exists λ¯ ∈ F \F
q3 q3t q3
such that
N (α )6= −1, (7)
q3t/qt λ¯
where α = λ¯−λ¯qi .
λ¯ λ¯q2t+i−λ¯
By way of contradiction, suppose that N (α ) = −1 for each λ¯ ∈
q3t/qt λ¯
F \F . Then the polynomial
q3t q3
g(x) := (x−xqi)(xqt−xqt+i)(xq2t−xqi+2t)+(xq2t+i−x)(xqi−xqt)(xqt+i−xq2t)
(8)
vanishes on F \F . It also vanishes on F , thus it has at least q3t−q3+q
q3t q3 q
roots. Put i = c+mt, with m ∈ {0,1,2} and 1 ≤ c < t, the degree of g(x)
is
q2t+c +q2t+qt (9)
when m = 0 and
q2t+c+q2t+qt+c (10)
when m ∈ {1,2}. Since qt−2 ≥ qc we obtain
q2t+c +q2t+qt+c = qc(q2t+qt)+q2t ≤ (qt−2)(q2t +qt)+q2t = q3t−2qt.
For t > 2 this is a contradiction since q3t−2qt < q3t−q3+q. If t = 2, then
gcd(i,t) = 1 yields c = 1 and hence we obtain
degg ≤ q5+q4+q3 < q6−q3+q,
again a contradiction. It follows that there always exists an element λ¯ ∈
F \F which is not a root of g(x), and α satisfies Condition (7).
q3t q3 λ¯
5
Choose a,b ∈ F∗ such that N (b) = N (α ), then there exists
q3t q3t/qt a q3t/qt λ¯
an element x ∈ F∗ such that
0 q3t
q2t+i−qi a
x = α ,
0 b λ¯
and hence x is a non-zero solution of the equation f(λ¯x) = λ¯f(x), i.e. with
0
these choices of a and b the linear set L is not scattered.
fi,a,b
Now we are able to prove the following result.
Theorem 2.3. Let w ∈ F \F . For any prime power q and any integer
q2t qt
t ≥ 2, there exist a,b ∈ F∗ and an integer 1 ≤ i ≤ 3t−1 such that the F -
q3t q
linear set L of rank 3t of the projective plane PG(F ,F ) = PG(2,q2t),
U q6t q2t
where
U = {axqi +bxq2t+i +wx: x ∈ F },
q3t
is scattered.
Proof. According to Lemma 2.2 for any prime power q and any integers
t ≥ 2, 1 ≤ i ≤ 3t− 1 with gcd(i,2t) = 1 we can choose a¯,¯b ∈ F∗ , with
q3t
N (¯b) 6= −N (a¯)suchthatthelinearsetL ofthelinePG(F ,F ) =
q3t/qt q3t/qt f q6t q3t
PG(1,q3) with f(x) = a¯xqi +¯bxq2t+i is non-scattered. Then by Lemma 2.1
there exists c∈ F∗ such that
q3t
a¯cxqi +¯bcxq2t+i
∈/ F
qt
x
for each x ∈ F∗ . Then the theorem follows from [2, Proposition 2.7] with
q3t
a = a¯c and b =¯bc.
As it was pointed out in [2], the existence of maximum scattered F –
q
linear sets of rank 3n in the projective plane PG(2,q2t) (proved in Theorem
2.3) and the construction method of [2, Theorem 3.1] imply the following.
Theorem 2.4. For any integers r,n ≥ 2, rn even, and for any prime power
q ≥ 2 the rank of a maximum scattered F -linear set of PG(r−1,qn) is rn/2.
q
Taking into account the previous result, from now on, a scattered F –
q
linear set L of PG(W,F ) = PG(r −1,qn) of rank rn (rn even) will be
U qn 2
simply called a maximum scattered linear set and the F -subspace U will be
q
called a maximum scattered subspace.
We complete this section by showing a connection between scattered F -
q
linear sets of PG(1,qrn/2), r even, and scattered F -linear sets of PG(r −
q
1,qn).
6
Proposition 2.5. Every maximum scattered F -linear set of PG(1,qrn/2),
q
r even, gives a maximum scattered F -linear set of PG(r−1,qn).
q
Proof. Let L be a maximum scattered F -linear set of PG(W,F ) =
U q qrn/2
PG(1,qrn/2). Then for each v ∈ W the one dimensional F -subspace
qrn/2
hviF meets U in an Fq-subspace of dimension at most one. Since Fqn
qrn/2
is a subfield of Fqrn/2 (recall r even) the same holds for the subspace hviFqn
and hence U also defines a scattered F -linear set in PG(W,F )= PG(r−
q qn
1,qn).
Note that the converse of the above result does not hold.
3 Maximum scattered subspaces and MRD-codes
The set of m×n matrices Fm×n over F is a rank metric F -space with rank
q q q
metric distance defined by d(A,B) = rk(A−B) for A,B ∈ Fm×n. A subset
q
C ⊆ Fm×n is called a rankdistance code (RD-code for short). Theminimum
q
distance of C is
d(C)= min {d(A,B)}.
A,B∈C, A6=B
When C is an F -linear subspace of Fm×n, we say that C is an F -linear
q q q
code and the dimension dim (C) is defined to be the dimension of C as a
q
subspace over F . If d is the minimum distance of C we say that C has
q
parameters (m,n,q;d).
The Singleton bound for an m×n rank metric code C with minimum
rank distance d is
#C ≤ qmax{m,n}(min{m,n}−d+1).
If this bound is achieved, then C is an MRD-code. MRD-codes have various
applications in communications and cryptography; for instance, see [17, 21].
More properties of MRD-codes can be found in [14, 16, 18, 39].
Delsarte[14]andGabidulin[16]constructed,independently,linearMRD-
codes over F for any values of m and n and for arbitrary value of the min-
q
imum distance d. In the literature these are called Gabidulin codes, even
if the first construction is due to Delsarte. These codes were later general-
ized by Kshevetskiy and Gabidulin in [20], they are the so called generalized
Gabidulin codes.
A generalized Gabidulin code is defined as follows: under a given basis
of F over F , each element a of F can be written as a (column) vector
qn q qn
7
v(a) in Fn. Let α ,...,α be a set of linearly independent elements of F
q 1 m qn
over F , where m ≤ n. Then
q
(v(f(α )),...,v(f(α )))T :f ∈ G (11)
n 1 m k,so
is the original generalized Gabidulin code, where
Gk,s = {f(x) = a0x+a1xqs+...ak−1xqs(k−1) : a0,a1,...,ak−1 ∈ Fqn}, (12)
with n,k,s ∈ Z+ satisfying k < n and gcd(n,s) = 1.
All members of G are of the form f(x) = n−1a xqi, where a ∈
Fqn. A polynomial ofkt,shis form is called a linearizPedi=po0lyniomial (also ai q-
polynomialbecauseitsexponentsareallpowersofq). Theyareequivalentto
Fq-lineartransformationsfromFqn toitself,i.e.,elementsofE = EndFq(Fqn).
We refer to [30, Section 4] for their basic properties.
In the literature, there are different definitions of equivalence for rank
metric codes; see [3,39]. If C andC′ aretwo sets of GL(U,F ), whereU is an
q
F -space of dimension n, then up to an isomorphism we may consider U as
q
the finite field F and it is natural to define equivalence in the language of
qn
q-polynomials, see[42]. For F -linear mapsbetween vector spaces of distinct
q
dimensions we will use the following definition of equivalence.
Definition 3.1. Let U(n,q) and V(m,q) be two F -spaces, n 6= m, and let
q
C and C′ be two sets of F -linear maps from U to V. They are equivalent
q
if there exist two invertible F -linear maps L ∈ GL(V,F ), L ∈ GL(U,F )
q 1 q 2 q
and ρ ∈ Aut(F ) such that C′ = {L ◦ fρ ◦ L : f ∈ C}, where fρ(x) =
q 1 2
f(xρ−1)ρ.
Very recently, Sheekey made a breakthrough in the construction of new
linear MRD-codes using linearized polynomials [42] (see also [36]).
In [42, Section 4], theauthor showed that maximum scattered linear sets
of PG(1,qn) correspond to F -linear MRD-codes of dimension 2n and min-
q
imum distance n−1. The number of non-equivalent MRD-codes obtained
fromamaximumscatteredlinearsetofPG(1,qn)wasstudiedin[11,Section
5.4].
HereweextendthisresultshowingthatMRD-codes ofdimensionrnand
minimum distance n−1 can be constructed from every maximum scattered
F –linear set of PG(r−1,qn), rn even, and we exhibit some relations with
q
Sheekey’s construction when r is even.
Tothisaim,recallthatanFq-subspaceU ofFqrn isscatteredwithrespect
to F if itdefinesascattered F -linear setinPG(F ,F ) = PG(r−1,qn),
qn q qrn qn
i.e. dimFq(U ∩hxiFqn) ≤ 1 for each x ∈ F∗qrn.
8
Theorem3.2. LetU beanrn/2-dimensionalF -subspaceofther-dimensional
q
Fqn-space V = V(r,qn), rn even, and let i = max{dimFq(U ∩hviFqn): v ∈
V}. For any F -linear function G: V → W, with W = V(rn/2,q) such that
q
kerG = U, if i < n, then the pair (U,G) determines an RD-code C (cf.
U,G
(13)) of dimension rn and with parameters (rn/2,n,q;n − i). Also, C
U,G
is an MRD-code if and only if U is a maximum scattered F -subspace with
q
respect to F .
qn
Proof. For v ∈ V the set
Rv := {λ ∈Fqn: λv ∈ U}
is an Fq-subspace with dimension the weight of the point hviFqn in the Fq-
linear set LU of PG(V,Fqn). Since i is the maximum weight of the points in
LU, it follows that dimFqRv ≤ i for each v. Also, let τv denote the map
λ ∈ Fqn 7→ λv ∈ V.
Direct computation shows that the kernel of G◦τv is Rv for each v ∈ V
and hence it has rank at least n−i. It remains to show that G◦τv 6= G◦τw
for v 6= w. Suppose, contrary to our claim, that there exist v,w ∈ V with
v 6= w and with G(λv) = G(λw) for each λ ∈ Fqn. Note that v 7→ G◦τv is
an F -linear map and hence G(λ(v−w)) = 0 for each λ ∈ F . This means
q qn
dimFq(kerG◦τv−w) = n= i, a contradiction. Hence
CU,G = {G◦τv: v ∈ V} (13)
isanF -linearRD-codewithdimensionrnandwithparameters(rn/2,n,q;n−
q
i). The second part is obvious since L is scattered if and only if i = 1.
U
Now we will show that different choices of the function G give rise to
equivalent RD-codes. Let’s start by proving the following result.
Lemma3.3. LetU beanrn/2-dimensionalF -subspaceofther-dimensional
q
F -space F . Then there exists ω ∈ F \F such that
qn qrn qrn qrn/2
U = {x+ωf(x): x ∈ F }
qrn/2
where f(x) is a q-polynomial over F .
qrn/2
Proof. Observe that F∗qrn = Sa∈F∗qrn aF∗qrn/2 and for any a,b ∈ F∗qrn either
aF∗ ∩bF∗ = ∅ or aF∗ = bF∗ and the latter case happens if and
qrn/2 qrn/2 qrn/2 qrn/2
onlyif a ∈ F∗ . SinceU∗∩aF∗ iseitheremptyorcontainsatleastq−1
b qrn/2 qrn/2
9
elements and since |U∗| = qr2n −1, there exist a,b ∈ F∗qrn, with ab ∈/ Fqrn/2
such that U∗ ∩aF∗ = U∗ ∩bF∗ = ∅. We may assume a ∈/ F∗ and
qrn/2 qrn/2 qrn/2
putω := a. ThenU∩ωF = {0}andtakingintoaccountthatU hasrank
qrn/2
r2n and{1,ω}isanFqrn/2–basisofFqrn,wehaveU = {x+ωf(x): x ∈ Fqrn/2}
for some q-polynomial f over F .
qrn/2
Hence, we are able to prove the following
Proposition 3.4. Let U be an rn/2-dimensional F -subspace of the r-
q
dimensional Fqn-space V = V(r,qn), rn even, and let G and G be two
F -linear functions determining two RD-codes C and C as in Theorem
q U,G U,G
3.2. Then C and C are equivalent.
U,G U,G
Proof. Uptoanisomorphism,wecanalways assumeV = F ×F and
qrn/2 qrn/2
W = Fqrn/2. Thenby Lemma3.3 wehave U = {(x,f(x)) : x ∈ Fqr2n}, where
f(x) is a q-polynomial over F . Then G,G : F ×F → F are
qrn/2 qrn/2 qrn/2 qrn/2
two F -linear maps such that U = kerG = kerG. We want to show that
q
there exist two permutation q-polynomials H and L over Fqrn/2 and Fqn,
respectively, and σ ∈ Aut(F ) such that
q
CU,G = {H ◦(G◦τv)σ ◦L : v ∈ Fqrn}.
Let G ,G ,G ,G : F → F be F −linear maps such that
0 1 0 1 qrn/2 qrn/2 q
G(x,y) = G (x)−G (y) and G(x,y) = G (x)−G (y),
0 1 0 1
for all x,y ∈ F . Since kerG = kerG = U it can be easily seen that
qrn/2
G = G ◦f, G = G ◦f and that G and G are invertible maps. Hence,
0 1 0 1 1 1
putting H = G1◦G−11, σ = idFq and L = idFqn, we have
H ◦G◦τv = G◦τv,
for each v = (x,y) ∈ V, and hence the assertion follows.
Firstweshowsomeresultsinthecaser even. Startingwiththefollowing
example for r = 2, we examine further the codes defined in Theorem 3.2.
Later, in Theorem 3.7 we will also give a different construction of MRD-
codes.
Example 3.5. Let U = {(x,f(x)): x ∈ F } be a maximum scattered F -
f qn q
subspace of the two-dimensional F -space V = F × F , where f is a
qn qn qn
q-polynomial over F . Let
qn
G: (a,b) ∈ V 7→ f(a)−b ∈ F .
qn
10