An infinite family of non-extendable MRD codes

An inﬁnite family of non-extendable MRD co des Daniele Bartoli ∗ , Alessandro Giannoni † , Giusepp e Marino † , Alessandro Neri † Abstract In the realm of rank-metric co des, Maxim um Rank Distance (MRD) co des are optimal algebraic structures attaining the Singleton-lik e b ound. A ma jor op en problem in this ﬁeld is determining whether an MRD co de can b e extended to a longer one while preserving its optimality . This work inv estigates F q m -linear MRD co des that are non-extendable but do not attain the maxim um possible length. Geometrically , these correspond to scattered subspaces with resp ect to hyperplanes that are maximal with resp ect to inclusion but not of maxim um dimension. By exploiting this geometric connection, w e in troduce the ﬁrst inﬁnite family of non-extendable [4 , 2 , 3] q 5 /q MRD co des. F urthermore, we prov e that these co des are self-dual up to equiv alence. 1 In tro duction Rank-metric co des, originally in tro duced b y Delsarte [7] and Gabidulin [9], ha ve gathered signiﬁ- can t attention due to their applications in netw ork coding, cryptography , and space-time co ding. An F q m -linear rank-metric co de of length n , dimension k , and minimum rank distance d attains the Singleton-like b ound d ≤ n − k + 1 (assuming n ≤ m ). Co des meeting this b ound are called Maximum R ank Distanc e (MRD) c o des . A fundamental problem in coding theory is the extension of optimal co des. F or n < m , the v ast ma jority of kno wn MRD co des are obtained by puncturing maxim um-length codes (where n = m ). An MRD co de is called non-extendable if it cannot b e obtained by puncturing a longer MRD co de. Equiv alen tly , its generator matrix cannot be augmen ted with an additional column without strictly decreasing the minimum rank distance. Classifying non-extendable MRD co des is a diﬃcult algebraic task, as it requires proving the non-existence of any v alid extension. T o tac kle this, we translate the problem into ﬁnite geometry using the theory of q -systems. As established in recen t literature [16, 20], a nondegenerate F q m -linear MRD co de of dimension k naturally corresp onds to a ( k − 1)-scattered F q -subspace. Crucially , a non-extendable MRD co de corresp onds exactly to a maximal ly sc atter e d subsp ac e : a scattered subspace that is not prop erly con tained in any larger scattered subspace. While maximum scattered subspaces (corresp onding to maximum-length MRD co des) hav e b een extensiv ely studied and classiﬁed in lo w dimensions, geometric ob jects that are less struc- tured and more diﬃcult to classify emerge when studying maximality without maximum dimen- sion. Curren tly , only one sp oradic example of a maximally scattered subspace not of maxim um dimension is known [3, 12]. In this paper, we construct the ﬁrst inﬁnite family of suc h ob jects. Our main result (Theorem 3.12) provides an inﬁnite family of non-extendable [4 , 2 , 3] q 5 /q MRD co des for q = 3 2 h +1 . W e ∗ Dipartimento di Matematica e Informatica, Universit` a degli Studi di Perugia, Perugia, Italy . daniele.bartoli@unipg.it † Dipartimento di Matematica e Applicazioni “R. Cacciopp oli”, Univ ersit` a di Nap oli F ederico I I, Nap oli, Italy . alessandro.giannoni@unina.it , giuseppe.marino@unina.it , alessandro.neri@unina.it 1 ac hieve this by analyzing 4-dimensional F q -subspaces in F 2 q 5 and exploiting the recen t partial classiﬁcation of maximum scattered subspaces in F 2 q 5 obtained in [13]. Finally , in Section 4, w e explore the dual properties of our construction, proving that these co des are self-dual up to equiv alence. 2 Preliminaries 2.1 Scattered subspaces This section in tro duces the deﬁnitions and foundational results for scattered and ev asiv e sub- spaces. Deﬁnition 2.1. L et k and n b e p ositive inte gers, and let h and r b e non-ne gative inte gers satisfying h < k and h ≤ r . A n F q -subsp ac e U ⊆ F k q m := F k q m is deﬁne d as ( h, r ) -evasive if, for any h -dimensional F q m -subsp ac e H ⊆ F k q m , the ine quality dim F q ( U ∩ H ) ≤ r holds. In the sp e ciﬁc c ase wher e h = r , an ( h, h ) -evasive subsp ac e is c al le d h -sc atter e d . If h = 1 , a 1 -sc atter e d subsp ac e is simply r eferr e d to as sc atter e d . The concept of scattered subspaces was ﬁrst in troduced by Blokh uis and La vrau w in [3]. This idea was later extended to any integer h by Csa jb´ ok, Marino, Polv erino and Zullo in [6]. The broader notion of ev asiv e subspaces was formally deﬁned in [2], though related concepts had app eared earlier in works such as [8, 10, 11, 15]. A well-established result for h -scattered subspaces is the upp er b ound on their F q -dimension. Sp eciﬁcally , for an h -scattered subspace U ⊆ F k q m , we hav e: dim F q ( U ) ≤ k m h + 1 , (1) as shown in [3, 6]. Deﬁnition 2.2. A n h -sc atter e d subsp ac e that achieves Bound (1) is c al le d a maximum h - sc atter e d subsp ac e (or maximum sc atter e d subsp ac e, when h = 1 ). A n h -sc atter e d subsp ac e which is not pr op erly c ontaine d within any lar ger h -sc atter e d subsp ac e, is c al le d a maximal ly h -sc atter e d subsp ac e (or maximal ly sc atter e d subsp ac e, when h = 1 ). Consequen tly , the dimension (or rank) of a scattered subspace in F k q m is at most k m/ 2. If this maximum is met, the subspace is kno wn as a maximum scattered subspace. The ﬁrst example of a maximally scattered subspace which is not maxim um scattered – a maximally scattered 5-dimensional s ubspace of F 2 2 6 – was giv en in [12, Example 3.2] (see also [3]) and was constructed using the GAP-pack age FinInG. So far, no inﬁnite families of maximally scattered linear sets whic h are not maximum scattered are known and the aim of this pap er is to provide the ﬁrst inﬁnite family of such ob jects. On maximally scattered subspaces, the follo wing result provides a useful lo w er b ound on its dimension; see [3, Theorem 2.1]. Theorem 2.3. If U ≤ F k q m is a maximal ly sc atter e d subsp ac e, then dim F q ( U ) ≥  k m − m 2  + 1 . The ab o v e theorem can b e generalized to the case of h -scattered subspaces. 2 Theorem 2.4. If U ≤ F k q m is a maximal ly h -sc atter e d subsp ac e, then dim F q ( U ) ≥  k m − hm h + 1  + h. Pr o of. Let s be the dimension of an h -scattered subspace U in F k q m . The subspace U can be extended to a larger h -scattered subspace U ′ if and only if there exists a vector v ∈ F k q m not con tained in an y space generated by U and an F q m -subspace of F k q m whose in tersection with U has dimension h (i.e., of weigh t h in U ). First w e ha ve to determine an upp er b ound on the n umber of such F q m -subspaces. The num ber of h -dimensional F q m -subspaces of weigh t h in U is precisely the num ber of h -dimensional F q -subspaces of U , which is  s h  q . The total n um b er of v ectors of F k q m con tained in an y of the spaces generated b y U and an h -dimensional F q m -subspace of weigh t h is at most ( q hm + s − h − q s )  s h  q + q s . Th us, U can b e extended to a larger h -scattered subspace if q km > ( q hm + s − h − q s )  s h  q + q s and in particular if k m > hm + s − h + h ( s − h ) , i.e., s < k m − hm h + 1 + h. This is equiv alen t to say that if U is maximally scattered then s ≥  k m − hm h + 1  + h. 2.2 Linearized p olynomials and Dickson matrices A key to ol for describing F q -subspaces of F k q m is the class of linearized p olynomials (or q - p olynomials) ov er F q m . Let ˜ L m,q denote the F q -algebra of linearized p olynomials ov er F q m reduced mo dulo x q m − x . Ev ery p olynomial in ˜ L m,q has the form L ( x ) = m − 1 X i =0 a i x q i ∈ F q m [ x ] . It is a well-kno wn fact that ˜ L m,q is isomorphic to the F q -algebra of F q -linear endomorphisms of F q m , namely End F q ( F q m ). Consequently , for any L ∈ ˜ L m,q , its k ernel is a w ell-deﬁned F q - subspace of F q m , given by k er( L ) := { x ∈ F q m : L ( x ) = 0 } . Similarly , the rank of L ( x ) is deﬁned as the F q -dimension of its image. By the Rank-Nullit y Theorem, we hav e rank( L ) := m − dim F q (k er( L )) . The rank of suc h an op erator is strictly related to the prop erties of its asso ciated Dickson matrix. 3 Deﬁnition 2.5. L et L ( x ) = P m − 1 i =0 a i x q i ∈ ˜ L m,q . The Dickson matrix asso ciate d with L ( x ) is the m × m matrix D ( L ) =      a 0 a 1 . . . a m − 1 a q m − 1 a q 0 . . . a q m − 2 . . . . . . . . . . . . a q m − 1 1 a q m − 1 2 . . . a q m − 1 0      . It is a w ell-kno wn fact that the rank of L ( x ) equals the rank of D ( L ) o ver F q m . This algebraic c haracterization allows us to translate geometric conditions on scattered subspaces in to matrix rank conditions. 2.3 Rank metric co des It is w orth noting that linear F q -subspaces of F k q m ha ve a geometric connection to rank metric co des with some additional linear prop erty . W e start by introducing the theory of rank metric co des. Deﬁnition 2.6. A r ank metric c o de is a subset C ⊂ F r × s q of matric es, e quipp e d with the r ank distanc e d ( M , N ) = rank( M − N ) for any M , N ∈ C . The minimum r ank distanc e of C is the inte ger d rk ( C ) := min { d rk ( M , N ) : M , N ∈ C , M  = N } . If in addition C is an F q -subsp ac e, then C is said to b e F q -line ar , and in this c ase d rk ( C ) = min { rank( M ) : M ∈ C \ { 0 }} . The parameters of a rank-metric co de C are constrained by the Singleton-like b ound: |C | ≤ q max { r,s } (min { r ,s }− d rk ( C )+1) , as established by Delsarte in [7]. Co des that attain this b ound are known as maxim um rank distance (MRD) co des , and they represent the most studied family of rank-metric co des. W e will see later the most prominent family of MRD co des, which was given indep endently by Delsarte [7] and Gabidulin [9]. These constructions of MRD co des exploit an additional notion of linearity , which w e now in tro duce. Recall that a ﬁeld F q m is isomorphic to F m q as F q -v ector space and any isomorphism can be deriv ed by choosing an ordered F q -basis B of F q m . In the same wa y , the space F n q m is isomorphic to F m × n q as F q -v ector space. Hence, one can deﬁne the rank distance on F n q m as the one induced b y the rank distance in F m × n q after writing every entry in co ordinates with resp ect to an F q -basis B of F q m . Since using a diﬀerent F q -basis acts as an inv ertible matrix on F m × n q – that is, the change of basis matrix – the rank distance do es not dep end on the choice of the F q -basis B . Actually , it is well-kno wn that the rank distance can b e intrinsically deﬁned on F n q m as follows: d rk ( u, v ) = dim F q ( ⟨ u 1 − v 1 , . . . , u n − v n ⟩ F q ) , for u = ( u 1 , . . . , u n ) , v = ( v 1 , . . . , v n ) ∈ F n q m . In this w ay , one has a natural notion of F q m -linearit y that is hidden in the matrix representation. F rom now on, we will only consider co des p ossessing suc h a linearity . 4 Deﬁnition 2.7. A n [ n, k , d ] q m /q (r ank-metric) c o de C is a k -dimensional F q m -subsp ac e of F n q m , endowe d with the r ank distanc e, wher e d = d rk ( C ) is the minimum r ank distanc e of C . If the minimum r ank distanc e of C is not known nor r elevant, we wil l simply r efer to C as an [ n, k ] q m /q c o de. A generator matrix for an [ n, k ] q m /q co de is a matrix G ∈ F k × n q m suc h that C = rowsp( G ) ≤ F n q m , where rowsp( · ) denotes the F q m -v ector space generated by the rows. If the columns of one (and hence all) generator matrix G of C are F q -linearly indep enden t, then the co de is said to b e nondegenerate . The terminology app ears evident b ecause this prop ert y on the generator matrix is equiv alent to say that C cannot b e isometrically embedded in a smaller ambien t space ( F n ′ q m , d rk ), with n ′ < n ; see e.g. [1]. There is a strong connection b et w een [ n, k ] q m /q nondegenerate rank-metric co des and n - dimensional F q -subspaces of F k q m that are not contained in an y F q m -h yp erplane. These F q - subspaces are now ada ys kno wn as q -systems , and their connection to rank-metric co des was explicitly introduced for the ﬁrst time in [16]. Given an n -dimensional F q -subspace U of F k q m with the prop ert y that ⟨ U ⟩ F q m = F k q m – namely , a q -system – and given an F q -basis { g 1 , . . . , g n } of U , one can construct a matrix G ∈ F k × n q m whose columns are these basis vectors, that is, G =   | | g 1 · · · g n | |   . This matrix G can b e view ed as the generator of an [ n, k ] q m /q nondegenerate rank-metric code C . In other words, C = rowsp( G ) ≤ F n q m . Vice-v ersa, if C is an [ n, k ] q m /q nondegenerate co de, then for a given generator matrix G ∈ F k × n q m , we can select its columns and consider the F q -subspace U that they generate inside F k q m . By the fact that C is nondegenerate, the F q -dimension of U is n . Moreo v er, since the rank of G is k , then ⟨ U ⟩ F q m = F k q m , implying that U is indeed a q -system. The t w o procedures describ ed ab o ve are not w ell-deﬁned, since the ﬁrst dep ends on the c hoice of the basis { g 1 , . . . , g n } of U , while the second dep ends on the c hoice of the generator matrix G of C (or, equiv alen tly , on the choice of a basis of C ). How ev er, if w e consider F q -subspaces of F k q m up to GL( k , q m )-equiv alence, and the [ n, k ] q m /q co des up to GL( n, q )-equiv alence, the tw o pro cedures are one the in verse of the other, and w e obtain a bijection betw een equiv alence classes of [ n, k ] q m /q nondegenerate codes, and equiv alence classes of F q -subspaces of F k q m of F q -dimension n ; see [1, 16]. In the remainder of the pap er, we will say that a q -system U is an [ n, k , d ] q m /q system asso ciated with a nondegenerate [ n, k , d ] q m /q co de C if it can b e obtained from C by the ab ov e pro cedure on one of its generator matrices. Similarly , we will say that C is asso ciated with U . The imp ortance of q -system is given by the fact that they represent the geometrical counter- parts of [ n, k ] q m /q co des, and their scatteredness prop erties reveal the prop ert y of the asso ciated co des of b eing MRD. This was sho wn in a series of pap ers [18, 20]. W e reformulate these ﬁndings in the following more compact wa y . Theorem 2.8 (see [18, Corollary 5.7], [20, Theorem 3.2]) . L et C b e a nonde gener ate [ n, k ] q m /q c o de, and let U b e any of its asso ciate d [ n, k ] q m /q systems. Then, C is MRD if and only if one of the fol lowing holds: 5 (a) n ≤ m and U is ( k − 1) -sc atter e d, or (b) n > m , n divides km and U is maximum h -sc atter e d, with h = km n − 1 . F or a full o verview of the relation b etw een scattered and ev asive q -system and the parameters of their asso ciated rank-metric codes, we refer the interested reader to [14]. MRD codes hav e been studied since the late 70’s, when Delsarte pro vided their ﬁrst construc- tion [7]. This construction was rediscov ered a few y ears later by Gabidulin [9]. These co des are no w known as Delsarte-Gabidulin co des and can b e describ ed in the follo wing wa y . Let v = ( v 1 , . . . .v n ) ∈ F n q m b e such that dim F q ( ⟨ v 1 , . . . , v n ⟩ F q ) = n , and let k ≤ n . The Delsarte-Gabidulin co de C k ( v ) is the [ n, k ] q m /q co de whose generator matrix is giv en by the k × n Mo ore matrix M k ( v ) =      v 1 v 2 · · · v n v q 1 v q 2 · · · v q n . . . . . . . . . v q k − 1 1 v q k − 1 2 · · · v q k − 1 n      . Using the pro cedure describ ed previously , it is easy to see that one of its asso ciated [ n, k ] q m /q systems is V k ( X ) := { ( x, x q , . . . , x q k − 1 : x ∈ X } , where X = ⟨ v 1 , . . . , v n ⟩ F q . Apart from Delsarte-Gabidulin co des, only few other systematic constructions of [ n, k ] q m /q MRD co des are kno wn; see e.g. [17]. All such constructions ha ve an algebraic ﬂav or, and rely on the representation of the space of square matrices F m × m q as End F q ( F q m ). Hence, these con- structions are all designed for the square case. F or what concerns the rectangular case, esp e- cially for n < m , the kno wn examples of [ n, k , n − k + 1] q m /q MRD are obtained from a square [ m, k , m − k + 1] q m /q MRD co de by puncturing . Deﬁnition 2.9. L et C b e an [ n, k ] q m /q c o de and let A ∈ F n × n ′ q b e such that rank( A ) = n ′ . The puncturing of C on A is the [ n ′ , k ′ ] q m /q c o de π ( C , A ) := { cA : c ∈ C } . In terms of asso ciated q -systems, the follo wing result is straightforw ard. Theorem 2.10. L et C b e an [ n, k ] q m /q c o de, and let U b e any of its asso ciate d [ n, k ] q m /q systems. Then, an [ n ′ , k ] q m /q c o de C ′ is a puncturing of C if and only if ther e exists an [ n ′ , k ] q m /q system U ′ asso ciate d with C ′ such that U ′ ≤ U . Pr o of. First, supp ose that U ′ is an [ n ′ , k ] q m /q system contained in U . Let { g 1 , . . . , g n ′ } b e an F q -basis of U ′ , and let G ′ ∈ F k × n ′ q m b e the matrix whose i th column is g i . Clearly , we hav e that C ′ = ro wsp( G ′ ) is an [ n ′ , k ′ ] q m /q co de asso ciated with U ′ . Let us complete { g 1 , . . . , g n ′ } to an F q -basis { g 1 , . . . , g n } of U and let G ∈ F k × n q m b e the matrix whose i th column is g i . for i ∈ { 1 , . . . , n } . In this wa y , C = rowsp( G ). Then, if w e consider the matrix A = ( I n ′ | 0) ⊤ , w e ha ve G ′ = GA , and hence C ′ = ro wsp( G ′ ) = ro wsp( GA ) = π ( C , A ) . On the other hand, let us assume that C ′ is a puncturing of C . Then, there exists A ∈ F n × n ′ q of rank n ′ suc h that C ′ = π ( C , A ). Hence, if G is a generator matrix of C , then G ′ = GA is 6 a generator matrix of C ′ . By deﬁnition of asso ciated q -system, this means that, there exists a generator matrix G of C such that U is the F q -span of the columns of G . Moreo ver, the F q -span U ′ of the columns of G ′ is clearly an F q subspace of U , concluding the pro of. As mentioned ab o v e, all the known constructions of [ n, k ] q m /q MRD co des with n < m are obtained b y puncturing an [ m, k ] q m /q MRD co de. It is natural to ask whether one can construct new [ n, k ] q m /q MRD co des that cannot b e obtained by puncturing a longer MRD co de. In terms of the asso ciated q -systems, by Theorem 2.10, w e can characterize such co des as follows. Corollary 2.11. L et C b e an [ n, k ] q m /q c o de with n ≤ m , and let U b e any of its asso ciate d [ n, k ] q m /q systems. Then, C is an MRD c o des that c annot b e obtaine d by puncturing a [ n +1 , k ] q m /q c o de if and only if U is maximal ly ( k − 1) -sc atter e d. Pr o of. This directly follows from Theorem 2.8 and Theorem 2.10. Th us, constructing maximally ( k − 1)-scattered [ n, k ] q m /q systems that are not maximum co- incides with constructing [ n, k ] q m /q MRD co des with n < m that are not obtained b y puncturing a longer MRD co de. W e will call such co des non-extendable , follo wing the terminology of the analogous co des in the classical Hamming metric. This represen ts the main motiv ation of our w ork. 2.4 A few things on algebraic v arieties In this section, we summarize several concepts and results related to algebraic v arieties. W e use the notations P r ( F q ) and A r ( F q ) (or simply F r q ) to denote the pro jectiv e and the aﬃne space of dimension r ∈ N ov er the ﬁnite ﬁeld F q , resp ectiv ely . Let F q denote the algebraic closure of F q . A v ariet y V is deﬁned as the set of common zeros of a ﬁnite collection of p olynomials. Sp ecif- ically , an aﬃne F q -rational v ariet y (or an aﬃne v ariet y deﬁned ov er F q ) is a set V ⊂ A r ( F q ) for whic h there exist p olynomials F 1 , . . . , F s in the p olynomial ring F q [ X 1 , . . . , X r ] such that: V = { ( a 1 , . . . , a r ) ∈ A r ( F q ) | F i ( a 1 , . . . , a r ) = 0 for all i = 1 , . . . , s } . This set is also denoted as V ( F 1 , . . . , F s ). Similarly , a pro jectiv e F q -rational v ariet y (or a pro jec- tiv e v ariet y deﬁned o ver F q ) in P r ( F q ) is deﬁned using p olynomials F 1 , . . . , F s ∈ F q [ X 0 , X 1 , . . . , X r ], with the additional requirement that eac h p olynomial F i m ust b e homogeneous. The set of F q - rational p oin ts of an F q -rational v ariet y V is given by the intersection V ∩ A r ( F q ) or V ∩ P r ( F q ), and it is usually denoted b y V ( F q ). A hypersurface is a v ariet y deﬁned by a single p olynomial. W e say that a v ariet y V is absolutely irr e ducible if it cannot b e expressed as the union of tw o prop er subv arieties deﬁned ov er the algebraic closure F q . That is, there are no v arieties V ′ and V ′′ deﬁned ov er F q and diﬀerent from V suc h that V = V ′ ∪ V ′′ . In the speciﬁc case of a hyper- surface V = V ( F ) ⊂ A r ( F q ) (or pro jectiv e space), it is absolutely irr e ducible if and only if its deﬁning polynomial F is irreducible ov er the algebraic closure. That is, there are no non-constan t p olynomials G, H ∈ F q [ X 1 , . . . , X r ] (or homogeneous p olynomials in F q [ X 0 , . . . , X r ]) such that F = GH . The dimension of a v ariet y can b e deﬁned as the maximal in teger s for which there exists a chain of distinct, nonempty , absolutely irreducible v arieties con tained in V : ∅ = V 0 ⊊ V 1 ⊊ · · · ⊊ V s +1 = V . W e sa y that an s -dimensional pro jectiv e v ariet y V has degree d , written deg( V ) = d , if d is the n umber of intersection points with a general pro jective subspace of complementary dimension. That is, d = #( V ∩ H ) , 7 where H ⊆ P r ( F q ) is a general pro jective subspace of dimension r − s . Algebraic v arieties V ⊂ A r ( F q ) (or V ⊂ P r ( F q )) of dimension 1, 2, and r − 1 are called curves, surfaces, and h yp ersurfaces, resp ectively . Determining the degree of a v ariet y is generally not straightforw ard; ho wev er, an upp er b ound to deg ( V ) is given by Q s i =1 deg( F i ) . W e also recall that the F robenius map Φ q : x 7→ x q is an automorphism of F q k and generates the group Gal ( F q k / F q ) of automorphisms of F q k that ﬁxes F q p oin t wise. The F rob enius automorphism also induces a collineation of A r ( F q ) and an automorphism of F q [ X 1 , . . . , X r ] . F rom now on, with a slight abuse of notation, w e will write V ⊂ A r ( F q ) or V ⊂ P r ( F q ) to indicate that the v ariet y V is F q -rational. 3 An inﬁnite family of maximally scattered subspaces in F 2 q 5 This section presents the pap er’s main result, the ﬁrst known inﬁnite family of maximally scat- tered subspaces. Let start this section by sho wing a non-existence result of maximally ( k − 1)-scattered q - systems in a sp ecial case, by using their corresp ondence with MRD co des. Prop osition 3.1. L et q b e a prime p ower and let k < m . The only maximal ly ( k − 1) -sc atter e d [ k + 1 , k ] q m /q systems ar e al l e quivalent to V k ( X ) = { ( x, x q , . . . , x q k − 1 ) : x ∈ X } , for some F q -subsp ac e X ≤ F q m of dimension k + 1 . In p articular, if k + 2 ≤ m , then ther e ar e no maximal ly ( k − 1) -sc atter e d [ k + 1 , k ] q m /q systems. Pr o of. By Theorem 2.8, if U is a ( k − 1)-scattered [ k + 1 , k ] q m /q system, then an y of its associated co des m ust b e a [ k + 1 , k , 2] q m /q MRD co de C . Moreov er, by (1), this implies that k + 1 ≤ m . It is well known that ev ery MRD co de of co dimension 1 is a Delsarte-Gabidulin co de [9], whic h means that U is equiv alent to { ( x, x q , . . . , x q k − 1 ) : x ∈ X } , for some F q -subspace X ≤ F q m of dimension k + 1. In addition, if k + 1 ≤ m − 1, up to equiv alence, U is prop erly contained in the ( k − 1)- scattered [ m, k ] q m /q system { ( x, x q , . . . , x q k − 1 ) : x ∈ F q m } , which has dimension m > k + 1. Th us, U cannot b e maximally ( k − 1)-scattered. W e fo cus on the case k = 2. The follo wing result sho ws that every maximally scattered subspace in F 2 q m is also maximum scattered, when m ≤ 4. Prop osition 3.2. L et q b e a prime p ower. F or m ≤ 4 , any maximal ly sc atter e d subsp ac e in F 2 q m is maximum. Pr o of. By Theorem 2.3, if m ≤ 3 then a maximally scattered subspace has dimension at least m , which is the dimension of a m axim um scattered subspace. Let us consider the case m = 4. By Theorem 2.3, a maximally scattered subspace in F 2 q 4 has dimension at least 3. By Proposition 3.1 a maximally 1-scattered subspace with these parameters cannot exist. Th us, for our purp oses, w e fo cus on the ﬁrst open case, that is F 2 q 5 . By Theorem 2.3, a maximally scattered subspace in F 2 q 5 has dimension 4. 8 Let N q 5 /q and T r q 5 /q denote the norm and the trace functions from F q 5 to F q , resp ectiv ely . Our inv estigation exploits the partial classiﬁcation of maximum scattered subspaces in [13]. Theorem 3.3 (see [13, Section 6]) . Any maximum sc atter e d subsp ac e in F 2 q 5 is up to e quivalenc e in ΓL(2 , q 5 ) one of the fol lowing (C1) PR s := { ( x, x q s ) : x ∈ F q 5 } , s ∈ { 1 , . . . , 4 } ; (C2) LP s,η := { ( x, x q s + η x q 5 − s ) : x ∈ F q 5 } , s ∈ { 1 , 2 } , N q 5 /q ( η )  = 0 , 1 ; (C3) W η ,ρ := { ( η ( x q − x )+ T r q 5 /q ( ρx ) , x q − x q 4 ) : x ∈ F q 5 } , η  = 0 , T r q 5 /q ( η ) = 0  = T r q 5 /q ( ρ ) ; (C4) Z k := { ( x, k ( x q + x q 3 ) + x q 2 + x q 4 ) : x ∈ F q 5 } , N q 5 /q ( k ) = 1 . The classes of sets of typ es (C3) and (C4) might b e empty, as they actual ly ar e for q ≤ 25 . The purp ose of the rest of this section is to provide the ﬁrst inﬁnite family of maximally scattered subspaces in F 2 q 5 whic h are not maximum scattered. Curren tly , the family is prov en to b e maximally scattered only for ﬁnite ﬁelds of order q = 3 2 h +1 (where h ≥ 1) and for sp eciﬁc choices of the parameter δ . The pro of relies on the partial classiﬁcation of maximum scattered subspaces in F 2 q 5 from [13]. While the formal pro of is restricted to this case, our exp erimen tal results suggest that the construction holds in an y characteristic. Notably , generalizing this result primarily depends on proving that the families designated as (C3) and (C4) in the aforementioned c lassiﬁcation are empty . Let q b e an odd prime p ow er and let X ≤ F q 5 b e an F q -h yp erplane. Every such hyperplane can b e describ ed as X λ := { x ∈ F q 5 : T r q 5 /q ( λx ) = 0 } for some λ ∈ F ∗ q 5 . Consider δ ∈ F q 5 satisfying N q 5 /q ( δ ) = 1. W e deﬁne the corresp onding family of subspaces as U δ ( X λ ) := { ( x, x q + δ x q 4 ) : x ∈ X λ } ≤ F 2 q 5 . Prop osition 3.4. The subsp ac e U δ ( X λ ) is e quivalent via the diagonal matrix D = diag ( λ, λ q ) ∈ GL(2 , q 5 ) to the subsp ac e U δ ′ ( X 1 ) = { ( y , y q + δ ′ y q 4 ) : T r q 5 /q ( y ) = 0 } , wher e δ ′ = δ λ q − q 4 . F urthermor e, N q 5 /q ( δ ′ ) = N q 5 /q ( δ ) = 1 . Pr o of. Applying the diagonal matrix D = diag( λ, λ q ) to a generic element of U δ ( X λ ), we obtain D  x x q + δ x q 4  =  λx λ q x q + λ q δ x q 4  . By setting y = λx , the condition x ∈ X λ b ecomes y ∈ X 1 , that is, T r q 5 /q ( y ) = 0. The second co ordinate translates to y q + λ q δ ( λ − q 4 y q 4 ) = y q + δ ′ y q 4 . This shows D ( U δ ( X λ )) = U δ ′ ( X 1 ). Since N q 5 /q ( λ ) ∈ F q , w e hav e N q 5 /q ( λ ) q − q 4 = 1. Therefore, w e hav e N q 5 /q ( δ ′ ) = N q 5 /q ( δ ) N q 5 /q ( λ ) q − q 4 = N q 5 /q ( δ ) = 1. 9 F or this reason, up to equiv alence via diag ( λ, λ q ), we can restrict our analysis to the sp eciﬁc h yp erplane X 1 = k er(T r q 5 /q ). F rom now on, w e simply write U δ := { ( x, x q + δ x q 4 ) : T r q 5 /q ( x ) = 0 } , N q 5 /q ( δ ) = 1 . W e ﬁrst classify those δ suc h that U δ is scattered, and to do this the following prop osition will b e helpful. Prop osition 3.5. Ther e is no δ ∈ F q 5 such that N q 5 /q ( δ ) = 1 and δ q 2 +1 − δ + 1 = 0 . Pr o of. Consider δ q 2 = ( δ − 1) /δ , and its q i -th p o w er with i = 1 , 2      δ q 2 = ( δ − 1) /δ δ q 3 = ( δ q − 1) /δ q δ q 4 = ( δ q 2 − 1) /δ q 2 . Using N q 5 /q ( δ ) = 1 the last equation reads δ − 1 − q − q 2 − q 3 = ( δ q 2 − 1) /δ q 2 that is equiv alen t to 1 + δ q 3 δ q δ ( − δ q 2 + 1) = 0 and substituting the information we hav e on δ q 2 , δ q 3 b y the ﬁrst and second equation ab o v e we obtain 0 = 1 + δ q − 1 δ q δ q δ  − δ − 1 δ + 1  = 1 + ( δ q − 1)( − δ + 1 + δ ) = δ q , a contradiction to N q 5 /q ( δ ) = 1. Let f δ ( Y ) = Y 2 + ( δ q 3 + q 2 + q +1 − δ q 2 + q +1 + δ q +1 − δ − 1) Y − δ q 3 + q 2 + q +2 − δ q +1 + δ ∈ F q 5 [ Y ]. Prop osition 3.6. The subsp ac e U δ is a sc atter e d subsp ac e of dimension 4 if and only if f δ ( Y ) has no r o ots in F q 5 . Pr o of. T o prov e that U δ is scattered w e need to pro ve that rank( mx + x q + δ x q 4 ) ≥ 4 for ev ery m ∈ F q 5 . Let g m ( x ) = mx + x q + δ x q 4 = mx + x q + δ ( − x − x q − x q 2 − x q 3 ). Consider the system g m ( x ) = ( g m ( x )) q = ( g m ( x )) q 2 = ( g m ( x )) q 3 = ( g m ( x )) q 4 = 0 . It can b e written as       m − δ − δ + 1 − δ − δ δ q m q 1 0 0 δ q 2 m q 2 1 − 1 − 1 δ q 3 − 1 m q 3 − 1 − m q 4 + 1 − m q 4 − m q 4 − m q 4 + δ − q 3 − q 2 − q − 1           x x q x q 2 x q 3     = M     x x q x q 2 x q 3     =     0 0 0 0     , and we hav e that dim F q ( { g m ( x ) : x ∈ F q 5 } ) = 4 − rank( M ). W e w ant to prov e that rank( M ) ≥ 3 if and only if f δ ( Y ) has no solutions in F q 5 . W e will indicate with M c 1 ,c 2 ,...,c t r 1 ,r 2 ,...,r s the submatrix of M chosen selecting the columns { c 1 , c 2 , . . . , c t } and the rows { r 1 , r 2 , . . . , r s } . Notice that det( M 1 , 4 1 , 2 ) = δ q +1  = 0. W e hav e that rank( M ) = 2 if and only if det ( M 1 , 4 ,j 1 , 2 ,i ) = 0 for every i = 3 , 4 , 5 and j = 2 , 3. Let g j i ( m ) := det( M 1 , 4 ,j 1 , 2 ,i ) and consider the system E i,j,ℓ : ( g j i ( m )) q ℓ = 0 with i = 3 , 4 , 5, j = 2 , 3 and ℓ = 0 , 1 , 2 , 3 , 4. In particular, g 2 3 ( m ) = m q ( δ − m ) + δ q ( δ q 2 +1 − δ + 1). First notice that for m = δ , rank( M ) ≥ 3, since g 2 3 ( δ ) = 0 is incompatible with N q 5 /q ( δ ) = 1 by Prop osition 3.5. 10 • Supp ose that f δ ( m ) = 0 for some m ∈ F q 5 . Such an m ∈ F q 5 is not δ , since f δ ( δ ) = − δ q +1 ( δ q 2 +1 − δ + 1) = 0 is incompatible with N q 5 /q ( δ ) = 1 b y Prop osition 3.5. Consider no w δ  = m . F rom g 2 3 ( m ) = 0 we obtain m q = s 1 ( m ), for some s 1 ( Y ) ∈ F q 5 ( Y ). Therefore, m q 2 = ( s 1 ( m )) q and using m q = s 1 ( m ) w e will obtain m q 2 = s 2 ( m ), for some s 2 ( Y ) ∈ F q 5 ( Y ), and analogously m q 3 = s 3 ( m ), m q 4 = s 4 ( m ), for some s 3 ( Y ) , s 4 ( Y ) ∈ F q 5 ( Y ). Substituting these in the system we obtain that every equation is of the type E i,j,ℓ : f δ ( m ) h i,j,ℓ ( m ) = 0, for some p olynomial h i,j,ℓ ( Y ) ∈ F q 5 [ Y ]. This shows that if f δ ( m ) = 0 then det( M 1 , 4 ,i 1 , 2 ,j ) = 0 for each i = 3 , 4 , 5 and j = 2 , 3. • Supp ose now that det( M 1 , 4 ,i 1 , 2 ,j ) = 0 for each i = 3 , 4 , 5 and j = 2 , 3 for some m ∈ F q 5 . One of the polynomials h i,j,ℓ ( m ) is δ q and th us if det( M 1 , 4 ,i 1 , 2 ,j ) = 0 for eac h i = 3 , 4 , 5 and j = 2 , 3 then f δ ( m ) = 0. T o sum up, we hav e that the system has a solution in F q 5 if and only if f δ ( Y ) has a ro ot in F q 5 . In the follo wing propositions, w e prov e that for sp eciﬁc c hoices of δ , the subspace U δ is not ΓL(2 , q 5 )-equiv alen t to an y of the maxim um scattered subspaces from the families (C1)–(C4) listed in Theorem 3.3. In particular, Propositions 3.7 and 3.8 work for an y q and any δ with N q 5 /q ( δ ) = 1. Prop osition 3.7. The subsp ac e U δ := { ( x, x q + δ x q 4 ) : T r q 5 /q ( x ) = 0 } ≤ F 2 q 5 , N q 5 /q ( δ ) = 1 , is not c ontaine d up to ΓL(2 , q 5 ) -e quivalenc e in any sc atter e d subsp ac e of the typ e PR s := { ( x, x q s ) : x ∈ F q 5 } , s = 1 , . . . , 4 . Pr o of. Since an y PR s is Γ L -equiv alen t to PR 1 or PR 2 , we can divide the pro of in tw o cases. Case 1. s = 1 W e hav e that U δ ≤ PR 1 if and only if there exist A, B , C , D ∈ F q 5 , AD  = B C such that for all x ∈ F q 5 with T r q 5 /q ( x ) = 0 there exists y ∈ F q 5 satisfying ( Ax + B ( x q + δ x q 4 ) = y C x + D ( x q + δ x q 4 ) = y q . This yields C x + D ( x q + δ x q 4 ) = A q x q + B q ( x q 2 + δ q x ) for all x ∈ F q 5 with T r q 5 /q ( x ) = 0, and so          C − D δ = B q δ q D − D δ = A q − D δ = B q − D δ = 0 . This implies D = 0 = B = A = C , a con tradiction. Case 2. s = 2 11 W e hav e that U δ ≤ PR 2 if and only if there exist A, B , C , D ∈ F q 5 , AD  = B C such that for all x ∈ F q 5 with T r q 5 /q ( x ) = 0 there exists y ∈ F q 5 satisfying ( Ax + B ( x q + δ x q 4 ) = y C x + D ( x q + δ x q 4 ) = y q 2 . This yields C x + D ( x q + δ x q 4 ) = A q 2 x q 2 + B q 2 ( x q 3 + δ q 2 x q ) for all x ∈ F q 5 with T r q 5 /q ( x ) = 0, and so          C − D δ = 0 D − D δ = B q 2 δ q 2 − D δ = A q 2 − D δ = B q 2 . If D = 0, then we obtain A = B = C = D = 0, a con tradiction. If D  = 0 w e obtain D δ = C = − A q 2 = − B q 2 , and so from the second equation we obtain 1 − δ = − δ q 2 +1 that is incompatible with N q 5 /q ( δ ) = 1 by Prop osition 3.5, so we hav e a contradiction. Prop osition 3.8. The subsp ac e U δ := { ( x, x q + δ x q 4 ) : T r q 5 /q ( x ) = 0 } ≤ F 2 q 5 , with N q 5 /q ( δ ) = 1 , is not c ontaine d up to ΓL(2 , q 5 ) -e quivalenc e in any sc atter e d subsp ac e of the typ e LP s,η := { ( x, x q s + η x q 5 − s ) : x ∈ F q 5 } , s = 1 , 2 , N q 5 /q ( η )  = 1 . Pr o of. Consider ﬁrst the case s = 1. W e hav e that U δ ≤ LP 1 ,η if and only if there exist A, B , C , D ∈ F q 5 , AD  = B C such that for all x ∈ F q 5 with T r q 5 /q ( x ) = 0 there exists y ∈ F q 5 satisfying ( Ax + B ( x q + δ x q 4 ) = y C x + D ( x q + δ x q 4 ) = y q + η y q 4 . This yields C x + D ( x q + δ x q 4 ) = A q x q + B q ( x q 2 + δ q x ) + η ( A q 4 x q 4 + B q 4 ( x + δ q 4 x q 3 )) (2) for all x ∈ F q 5 with T r q 5 /q ( x ) = 0, and so          C − D δ = B q δ q − η A q 4 + η B q 4 D − D δ = A q − η A q 4 − D δ = B q − η A q 4 − D δ = − η A q 4 + η B q 4 δ q 4 . Com bining the last tw o equations one gets η B q 4 δ q 4 = B q and thus, since N q 5 /q ( η δ q 4 ) = N q 5 /q ( η ) N q 5 /q ( δ )  = 1 , w e obtain B = 0. 12 Th us D = η δ A q 4 , and the second equation reads  η (1 − δ ) δ + η  A q 4 = A q ⇐ ⇒ η δ A q 4 = A q , yielding again A = 0. This is a con tradiction to AD  = B C . W e consider now the case s = 2. Arguing as for s = 1 one gets, similarly to Equation (2), C x + D ( x q + δ x q 4 ) = A q 2 x q 2 + B q 2 ( x q 3 + δ q 2 x q ) + η ( A q 3 x q 3 + B q 3 ( x q 4 + δ q 3 x q 2 )) (3) and thus          C − D δ = − η B q 3 D − D δ = B q 2 δ q 2 − η B q 3 − D δ = A q 2 − η B q 3 + η B q 3 δ q 3 − D δ = B q 2 + η A q 3 − η B q 3 . If A = 0, combining the last tw o equations we obtain η δ q 3 B q 3 = B q 2 , yielding again, by the assumptions on η and δ , B = 0, a contradiction to AD  = B C . Supp ose that A  = 0. Com bining the last three equations w e obtain δ A q 2 − δ η A q 3 − δ B q 2 + δ q 3 +1 η B q 3 = 0 = ( η − η δ ) A q 3 + ( δ q 2 +1 − δ + 1) B q 2 − η B q 3 . If δ = 1 then η B q 3 = B q 2 , whic h is equiv alent to B = 0 due to our assumptions on η . This yields A q 2 = η A q 3 and so A = 0, again a contradiction to AD  = B C . F rom now on we can supp ose that δ  = 1. Th us A q 2 = δ q 2 +1 B q 2 + η ( − δ q 3 +1 + δ q 3 − 1) B q 3 δ − 1 , A q 3 = δ q 3 + q B q 3 + η q ( − δ q 4 +1 + δ q 4 − 1) B q 4 δ q − 1 . F rom δ A q 2 − δ η A q 3 − δ B q 2 + δ q 3 +1 η B q 3 = 0, ( δ − 1) q δ ( δ q 2 +1 − δ + 1) B q 2 + ηδ ( δ q 3 + q +1 − δ q 3 + q + δ q − 1) B q 3 + η q +1 ( δ q 4 + q − δ q 4 + 1) δ ( δ − 1) B q 4 . The determinan t of the Dickson matrix of the ab o ve linearized p olynomial, when considering also N q 5 /q ( δ ) = 1 factorizes as (N q 5 /q ( η ) − 1) 2 δ q 3 +2 q 2 + q +1 ( δ − 1) q 3 + q 2 + q +1 ( δ q 2 +1 − δ + 1) q 3 + q +1 · ( δ q 3 + q 2 + q +1 + δ q − 1)( δ q 3 + q 2 + q +1 − 1)( δ q 3 + q 2 + q +1 − δ q 3 + q +1 − 1) = 0 , and we hav e a non-zero solution B if and only if one of these factors is v anishing. The ﬁrst three are trivially non-zero. (a) δ q 3 + q 2 + q +1 = 1. This would yield δ = 1, a contradiction. (b) δ q 2 +1 − δ + 1 = 0. This would yield δ q 4 = 1 1 − δ , δ q = δ ∈ F q , and thus δ 5 = 1. Now, δ 5 = 1 and δ 2 − δ + 1 = 0 pro vide a contradiction since gcd( x 5 − 1 , x 2 − x + 1) = 1. 13 (c) δ q 3 + q 2 + q +1 + δ q − 1 = 0. This would yield δ q 4 + q 3 + q 2 + q = 1 − δ q 2 and thus 1 = δ − δ q 2 +1 , that is δ q 2 = δ − 1 δ , a contradiction as ab o ve. (d) δ q 3 + q 2 + q +1 − δ q 3 + q +1 − 1 = 0. This would yield δ q 4 + q 3 + q 2 + q − δ q 4 + q 2 + q − 1 = 0 and then, m ultiplying by δ , 1 − δ q 4 + q 2 + q +1 − δ = 0. Raising it to the pow er q , one obtains 1 − δ q 3 + q 2 + q +1 − δ q = 0 and a contradiction as ab o ve. Th us the only p ossibilit y is B = 0, a contradiction since from δ A q 2 − δ η A q 3 − δ B q 2 + δ q 3 +1 η B q 3 = 0 and our assumptions on η one gets A = 0. Note that when q = 3 2 h +1 , by Prop osition 3.6, U 1 is scattered and of dimension 4, since 5 is not a square in F q . In the following we pro v e that, with this particular choice of q and with δ = 1, U 1 is not contained up to ΓL(2 , q 5 )-equiv alence in any subspace W η ,ρ . Prop osition 3.9. L et q = 3 2 h +1 . The subsp ac e U 1 := { ( x, x q + x q 4 ) : T r q 5 /q ( x ) = 0 } ≤ F 2 q 5 , is not c ontaine d in any of the subsp ac es in the ΓL(2 , q 5 ) -orbit of W η ,ρ := { ( η ( x q − x ) + T r q 5 /q ( ρx ) , x q − x q 4 ) : x ∈ F q 5 } , for every η , ρ ∈ F q 5 with T r q 5 /q ( η ) = 0  = T r q 5 /q ( ρ ) . Pr o of. The subspace U 1 is contained in any subspace within the GL-orbit of W η ,ρ if and only if there exist A, B , C , D ∈ F q 5 with AD  = B C , suc h that for ev ery x ∈ F q 5 with T r q 5 /q ( x ) = 0, there exists a y ∈ F q 5 for which the following holds:  A B C D   x x q + x q 4  =  η ( y q − y ) + T r q 5 /q ( ρy ) y q − y q 4  . This matrix equation yields tw o separate equations: Ax + B ( x q + x q 4 ) = η ( y q − y ) + T r q 5 /q ( ρy ) , (4) C x + D ( x q + x q 4 ) = y q − y q 4 . (5) Let us ﬁrst focus on Equation (5), which can be rewritten as C x + C ( x q + x q 4 ) − y q + y q 4 = 0. W e can then consider the system of equations ( C x + C ( x q + x q 4 ) − y q + y q 4 ) q ℓ = 0 for ℓ = 0 , . . . , 4. F rom the equations corresp onding to ℓ = 1 and ℓ = 4, w e obtain expressions for y q 2 and y q 3 , resp ectiv ely . These can b e written as y q 2 = f 2 ( y ) and y q 3 = f 3 ( y ), for some p olynomials f 2 ( Y ) , f 3 ( Y ) ∈ F q 5 [ Y ]. W e can then substitute these expressions in to the other equations. Next, from the equation with ℓ = 3, we obtain y q 4 = f 4 ( y ), where f 4 ( Y ) ∈ F q 5 [ Y ]. Once these substitutions are made, the equations for ℓ = 0 and ℓ = 2 b ecome ( 2 C x + 2 C q x q + 2 C q 3 x q 3 + 2 D ( x q + x q 4 ) + 2 D q ( x + x q 2 ) + 2 D q 3 ( x q 2 + x q 4 ) + 2 y + y q = 0 , 2 C q 2 x q 2 + 2 C q 4 x q 4 + 2 D q 2 x q + 2 D q 2 x q 3 + 2 D q 4 x + 2 D q 4 x q 3 + y + 2 y q = 0 . 14 F rom the second equation, we can obtain y q = f 1 ( y ), where f 1 ( Y ) ∈ F q 5 [ Y ]. By substituting this into the ﬁrst equation, we get the follo wing: (2 C q 4 + 2 D + 2 D q 3 ) x q 4 + (2 C q 3 + 2 D q 2 + 2 D q 4 ) x q 3 + (2 C q 2 + 2 D q + 2 D q 3 ) x q 2 + (2 C q + 2 D + 2 D q 2 ) x q + (2 C + 2 D q + 2 D q 4 ) x = 0 . By substituting x q 4 = − x q 3 − x q 2 − x q − x , w e obtain a polynomial in x of degree q 3 . Since this p olynomial has at least q 4 solutions, it must b e the zero p olynomial, whic h means all of its co eﬃcien ts must b e equal to zero. This leads to the following system of equations:          C − C q 4 − D + D q − D q 3 + D q 4 = 0 C q − C q 4 + D q 2 − D q 3 = 0 C q 2 − C q 4 − D + D q = 0 C q 3 − C q 4 − D + D q 2 − D q 3 + D q 4 = 0 . (6) W e can apply a similar pro cess to Equation (4). Consider the expression Ax + B ( x q + x q 4 ) − η ( y q − y ) − T r q 5 /q ( ρy ) = 0 and so the system of equations ( Ax + B ( x q + x q 4 ) − η ( y q − y ) − T r q 5 /q ( ρy )) q ℓ = 0 for ℓ = 0 , . . . , 4. By substituting the expressions for y q i = f i ( y ) with i = 1 , . . . , 4, the third equation becomes: ( A q 4 + C q 4 ρ q + C q 4 ρ q 3 + D q 3 η q 4 + 2 D q 3 ρ q 4 ) x q 4 + ( B q 4 + C q 3 η q 4 + 2 C q 3 ρ q 4 + D q 2 ρ q + D q 4 ρ q + D q 4 ρ q 3 ) x q 3 + ( C q 2 ρ q + D q η q 4 + 2 D q ρ q 2 + 2 D q ρ q 4 + D q 3 η q 4 + 2 D q 3 ρ q 4 ) x q 2 + ( C q η q 4 + 2 C q ρ q 2 + 2 C q ρ q 4 + D q 2 ρ q ) x q + ( B q 4 + D q η q 4 + 2 D q ρ q 2 + 2 D q ρ q 4 + D q 4 ρ q + D q 4 ρ q 3 ) x + y (T r q 5 /q ( ρ )) = 0 . Since T r q 5 /q ( ρ )  = 0, w e can express y as a function of x , i.e., y = f ( x ). W e can then substitute this expression for y into the other equations. T his will giv e us four equations that dep end solely on x . As b efore, b y substituting x q 4 = − x q 3 − x q 2 − x q − x , we will obtain four p olynomials in x . Since these p olynomials must b e the zero polynomial, every co eﬃcien t of each p olynomial must 15 b e zero, leading to the following system of equations:                                                                    2 A q 4 + B q 3 + B q 4 + 2 C q 4 η q 3 + D q η q 3 + D q η q 4 + 2 D q 3 η q 3 + 2 D q 3 η q 4 + D q 4 η q 3 = 0 2 A q 4 + 2 B q + C q 2 η q + 2 C q 4 η q + D q η q + D q η q 4 = 0 2 A q 4 + B q 3 + C q η q 3 + C q η q 4 + 2 C q 4 η q 3 + 2 D q 3 η q 3 + 2 D q 3 η q 4 = 0 2 A + 2 A q 4 + B + B q 4 + C q 4 η + D q η q 4 + 2 D q 3 η q 4 + 2 D q 4 η = 0 2 A q 2 + 2 A q 4 + C q 4 η q 2 + 2 D q η q 2 + D q η q 4 = 0 2 A q + 2 A q 4 + C q η q + C q η q 4 + 2 C q 4 η q + D q 2 η q + 2 D q 3 η q 4 = 0 2 A q 4 + C q η q 4 + C q 4 η + 2 D q 2 η + 2 D q 3 η q 4 = 0 2 A q 4 + 2 B q + B q 4 + 2 C q 4 η q + D q η q + D q η q 4 + 2 D q 3 η q 4 + D q 4 η q = 0 2 A q 4 + 2 B q 2 + 2 C q η q 2 + C q η q 4 + C q 4 η q 2 + 2 D q 3 η q 4 = 0 2 A q 4 + B q 4 + C q 3 η q 4 + 2 C q 4 η q + D q 2 η q + 2 D q 3 η q 4 + D q 4 η q = 0 2 A q 4 + B q 4 + C q 4 η q 2 + 2 D q η q 2 + D q η q 4 + 2 D q 3 η q 4 + 2 D q 4 η q 2 = 0 2 A q 4 + B + B q 4 + C q 3 η q 4 + C q 4 η + 2 D q 2 η + 2 D q 3 η q 4 + 2 D q 4 η = 0 2 A q 4 + B + 2 C q 2 η + C q 4 η + D q η q 4 = 0 2 A q 4 + 2 C q 4 η q 3 + D q η q 3 + D q η q 4 = 0 2 A q 4 + 2 B q 2 + B q 4 + C q 3 η q 4 + C q 4 η q 2 + 2 D q 3 η q 4 + 2 D q 4 η q 2 = 0 2 A q 3 + 2 A q 4 + B q 3 + B q 4 + C q 3 η q 3 + C q 3 η q 4 + 2 C q 4 η q 3 + 2 D q 3 η q 3 + 2 D q 3 η q 4 + D q 4 η q 3 = 0 . (7) F rom the equations in (6), we obtain e xpressions for C q i in terms of C q 4 for i = 0 , 1 , 2 , 3. W e can substitute these into (7). Let η = 0. Then A = 0 from the third-to-last equation in (7). F rom the fourth-to-last equation, we obtain B = 0. This con tradicts the assumption that AD  = B C . Therefore, from no w on we make the following assumption (A1) η  = 0. Again from the third-to-last equation in (7), we can express C q 4 in terms of D q , and then substitute this into the other equations. The second equation in (7) b ecomes: A q 4 + B q + 2 D η q + 2 D q η q 4 = 0 , (8) from which we ﬁnd A = 2 B q 2 + D q η q 2 + D q 2 η . W e can substitute this in to the other equations. The fourth-to-last equation in (7) b ecomes B + B q + 2 D η + 2 D η q + D q η = 0. Similarly , we can ﬁnd expressions for B q i in terms of B q 4 for i = 0 , 1 , 2 , 3, and substitute these. The eigh th equation in (7) b ecomes: B q 4 ( η q + 2 η q 3 ) + D ( η q +1 + η q 3 + q + η q 4 + q ) + 2 D q η q +1 D q 3 η q 4 + q 3 + D q 4 (2 η q +1 + 2 η q 3 + q + 2 η q 4 + q ) = 0 . (9) Noting that η q + 2 η q 3  = 0 since η  = 0, we can express B q 4 in terms of the v ariables represented b y D , and substitute this in to the other equations. After remo ving the dep enden t equations, we 16 are left with                                                                D ( η + η q 3 + η q 4 )( η + η q + η q 4 ) + D q η ( η + η q + η q 4 ) + D q 2 ( η + η q 4 )( η q + 2 η q 3 )+ + D q 3 η q 4 ( η + η q 3 + η q 4 ) + D q 4 ( η + η q 3 + η q 4 )( η + η q + η q 4 ) = 0 , D ( η q 3 +1 + η q 4 +1 + 2 η 2 q + 2 η q + q 3 + 2 η q 4 + q 3 + η 2 q 4 ) + D q ( η q 3 +1 + η q 4 +1 + 2 η 2 q + η q 3 + q ) + D q 2 ( η q + 2 η q 4 )( η q + 2 η q 3 ) + D q 3 η q 4 ( η q + η q 3 + 2 η q 4 ) + D q 4 ( η q 3 +1 + η q 4 +1 + η q + q 3 + 2 η q 4 + q 3 + η 2 q 4 ) = 0 , D ( η q +1 + η q 2 +1 + η q 3 +1 + η q 2 + q + 2 η q 3 + q + η q 4 + q + η q 4 + q 2 + η q 4 + q 3 ) + D q η ( η q + η q 2 + η q 3 ) + D q 3 η q 4 ( η q + η q 2 + η q 3 ) + D q 4 ( η q +1 + η q 2 +1 + η q 3 +1 + 2 η q 3 + q + η q 4 + q + η q 3 + q 2 + η q 4 + q 2 + η q 4 + q 3 ) = 0 , D ( η q 3 +1 + 2 η q 4 +1 + η 2 q + 2 η q 4 + q + η q 4 + q 3 + 2 η 2 q 4 ) + D q ( η q 3 +1 + 2 η q 4 +1 + η 2 q + η q 2 + q + 2 η q 3 + q + 2 η q 3 + q 2 ) + D q 2 ( η q + 2 η q 3 )( η q + η q 2 + 2 η q 4 ) + D q 3 ( η q 2 + q + 2 η q 3 + q 2 + 2 η q 4 + q 3 + η 2 q 4 ) + D q 4 ( η q 3 + 2 η q 4 )( η + η q + η q 4 ) = 0 , D ( η q +1 + 2 η q 3 +1 + 2 η q 4 +1 + 2 η q 4 + q 3 + 2 η 2 q 4 ) + D q η ( η q + 2 η q 3 + 2 η q 4 ) + D q 2 ( η q 3 + η q 4 )( η q + 2 η q 3 ) + + D q 3 ( η q 3 + q + η q 4 + q + 2 η 2 q 3 + 2 η q 4 + q 3 + 2 η 2 q 4 )+ + D q 4 ( η q +1 + 2 η q 3 +1 + 2 η q 4 +1 + η q 3 + q + η 2 q 3 + 2 η q 4 + q 3 + 2 η 2 q 4 ) = 0 . Notice that if D = 0, then from (9) w e obtain B = 0, whic h contradicts the assumption that AD  = B C . Thus, w e can consider D  = 0. This would yield a solution ( D, D q , D q 2 , D q 3 , D q 4 )  = 0 to the system of equations in (3). If we consider (3) to b e a linear system with unknowns ( D , D q , D q 2 , D q 3 , D q 4 ), the asso ciated matrix must b e singular. Therefore, we can require the determinan t to b e zero. Let ω b e the generator of F ∗ 9 , where w 2 + 2 w + 2 = 0. W e ha ve that the determinan t is equal to 2 c 1 c 2 c 3 , where c 1 = η q + 2 η q 3 , c 2 = η q 2 + q +1 + ω 2 η q 3 + q +1 + η q 4 + q +1 + ω 2 η q 3 + q 2 +1 + ω 2 η q 4 + q 2 +1 + η q 4 + q 3 +1 + η q 3 + q 2 + q + + ω 2 η q 4 + q 2 + q + ω 2 η q 4 + q 3 + q + η q 4 + q 3 + q 2 , c 3 = η q 2 + q +1 + ω 6 η q 3 + q +1 + η q 4 + q +1 + ω 6 η q 3 + q 2 +1 + ω 6 η q 4 + q 2 +1 + η q 4 + q 3 +1 + η q 3 + q 2 + q + + ω 6 η q 4 + q 2 + q + ω 6 η q 4 + q 3 + q + η q 4 + q 3 + q 2 . W e ﬁrst note that c 1  = 0. W e can also see that c q 5 2 = c 3 , whic h allows us to consider a case where b oth coeﬃcients are zero. In this case, we can substitute the expression for η q 4 giv en b y η q 4 = 2 η q 3 + 2 η q 2 + 2 η q + 2 η into both equations and then pro ceed to consider b oth the follo wing expressions to b e v anishing t 1 := c 2 + c 3 2 = η q +1 ( η + η q ) + η q 3 ( η 2 + 2 η q +1 + 2 η q 2 +1 + η 2 q 2 ) + η 2 q 3 ( η + η q 2 ) ∈ F q 5 , t 2 := c 2 − c 3 ω 6 = η q 2 ( η + η q )( η + η q + η q 2 ) + η q 3 ( η q )( η q + 2 η q 2 ) + η 2 q 3 η q ∈ F q 5 . Let t 3 := ( η + η q 2 ) t 2 − η q t 1 = 2( η + η q )( η q 2 +2 + 2 η 2 q +1 + η q 2 + q +1 + 2 η 2 q 2 +1 + η 2 q 2 + q + η 3 q 2 ) + η q 3 η q ( η + 2 η q 2 )( η + η q + η q 2 ). Since η q ( η + 2 η q 2 )( η + 2 η q 2 )  = 0 from the assumption on η , w e can express η q 3 as a function of η , η q , and η q 2 , i.e., η q 3 = h ( η , η q , η q 2 ). W e can then substitute this into the expression for t 2 to get: η q 2 + q +1 ( η + η q ) q +1 ( η + ω η q + η q 2 ) 2 ( η + ω 3 η q + η q 2 ) 2 = 0 . 17 The ﬁrst t wo factors are clearly non-zero. As b efore, w e can observe that the third and fourth factors are expressions in F q 10 . By setting both to zero and then combining them, we arrive at the conclusion that η = 0, whic h is a contradiction to our assumption (A1). In order to deal with the case (C4), w e ﬁrst obtain necessary conditions on k ∈ F q 5 , q = 3 2 h +1 , for U 1 to b e contained in Z k . Prop osition 3.10. L et q = 3 2 h +1 . If U 1 is c ontaine d, up to ΓL(2 , q 5 ) e quivalenc e, in Z k , N q 5 /q ( k ) = 1 , then u 1 ( k ) u 2 ( k ) = 0 and t ( k ) = 0 , wher e u 1 ( k ) := k 2+2 q +2 q 2 + 2 k 1+2 q +2 q 2 + 2 k 1+2 q + q 2 + k 1+ q +2 q 2 + 2 k 1+ q 2 + k q + q 2 + 2 k q 2 + 1; u 2 ( k ) := k 2+2 q +2 q 2 + 2 k 2+2 q + q 2 + k 2+ q + q 2 + 2 k 1+2 q + q 2 + k 1+ q + 2 k 1+ q 2 + 2 k + 1; t ( k ) := k 1+ q + q 2 ( k 3+3 q +3 q 2 + q 3 + 2 k 3+3 q +2 q 2 + q 3 + 2 k 2+3 q +3 q 2 + q 3 + k 2+2 q + q 2 + q 3 + k 2+2 q + q 2 +2 k 2+ q +2 q 2 + q 3 + k 2+ q + q 2 + q 3 + 2 k 2+ q + q 2 + k 1+3 q +2 q 2 + q 3 + k 1+2 q +3 q 2 + q 3 + 2 k 1+2 q +2 q 2 +2 k 1+2 q + q 2 + q 3 + 2 k 1+ q +3 q 2 + q 3 + k 1+ q +2 q 2 + 2 k 1+ q + k 1+2 q 2 + q 3 + 2 k 1+ q 3 + k +2 k 2 q +2 q 2 + q 3 + k q +2 q 2 + q 3 + k q + q 2 + 2 k q 2 + q 3 + 2 k q 2 + k q 3 ) . Pr o of. Let us consider k  = 1. The subspace U 1 is contained in any of the subspaces in the ΓL-orbit of Z k if and only if there exist A, B , C , D ∈ F q 5 with AD  = B C , such that for every x ∈ F q 5 , T r q 5 /q ( x ) = 0, there exists y ∈ F q 5 for which ( y = Ax + B ( x q + x q 4 ) , k ( y q + y q 3 ) + y q 2 + y q 4 = C x + D ( x q + x q 4 ) . F rom the system ab o ve we obtain          A q 4 + 2 B q k + B q 3 k + 2 B q 4 + C + 2 D = 0 , 2 A q k + A q 4 + 2 B q 2 + B q 3 k = 0 , 2 A q 2 + A q 4 + 2 B q k + 2 D = 0 , 2 A q 3 k + A q 4 + 2 B q 2 + B q 3 k + 2 B q 4 + 2 D = 0 , and thus A q 2 + 2 A q 3 k + B q k + 2 B q 2 + B q 3 k + 2 B q 4 = 0 = 2 A q k + A q 4 + 2 B q 2 + B q 3 k . In particular, considering the q -F rob enius of the ab ov e quantities and combining them,                                  2 A q k + A q 4 + 2 B q 2 + B q 3 k = 0 , 2 Ak q 4 + A q 3 + 2 B q + B q 2 k q 4 = 0 , Ak q 4 +1 + 2 A q 2 + 2 B q 2 k q 4 +1 + B q 2 + 2 B q 3 k + B q 4 = 0 , 2 Ak 2 q 4 +1 + A q + B k q 4 + 2 B q + B q 2 k 2 q 4 +1 + B q 3 k q 4 +1 + 2 B q 3 + 2 B q 4 k q 4 = 0 , Ak q 4 + q +1 + 2 A + 2 B q 2 k q 4 + q +1 + B q 2 k q + 2 B q 3 k q +1 + B q 3 = 0 , 2 B k 2 q 4 + q 3 + q +2 + B k q 4 + q +1 + B k q 4 + q 3 +1 + 2 B + B q k q 4 + q 3 + q +2 +2 B q k q 4 + q 3 + q +1 + 2 B q k q 3 +1 + B q k q 3 + 2 B q 2 k 2 q 4 + q 3 + q +2 + B q 2 k 2 q 4 + q 3 +2 + B q 2 k q 4 + q 3 + q +1 + 2 B q 2 k q 4 +1 + 2 B q 2 k q 3 + B q 2 + 2 B q 3 k 2 q 4 + q 3 +2 + B q 3 k q 4 + q 3 +2 + B q 3 k q 4 +1 + 2 B q 3 k + B q 4 k 2 q 4 + q 3 + q +2 + 2 B q 4 k q 4 + q +1 + 2 B q 4 k q 4 + q 3 +1 + B q 4 = 0 . 18 Also, the determinant of the Dickson matrix with resp ect to B of the last linearized p olynomial in the ab o v e system factorizes as  N q 5 /q ( k 1+ q + q 2 − 1)  N q 5 /q ( k ) + 1  k 1+ q + q 2 + q 3 + q 4 + ξ k 1+ q + q 2 + q 3 + ξ k 1+ q + q 2 + q 4 + 2 k 1+ q + q 2 + ξ k 1+ q + q 3 + q 4 + 2 k 1+ q + q 4 + k 1+ q + ξ k 1+ q 2 + q 3 + q 4 + 2 k 1+ q 3 + q 4 + k 1+ q 4 + ξ 5 k + ξ k q + q 2 + q 3 + q 4 +2 k q + q 2 + q 3 + k q + q 2 + ξ 5 k q + 2 k q 2 + q 3 + q 4 + k q 2 + q 3 + ξ 5 k q 2 + k q 3 + q 4 + ξ 5 k q 3 + ξ 5 k q 4 + 2) 1+ q 5  , where ξ ∈ F 9 satisﬁes ξ 2 + 2 ξ + 2 = 0. Supp ose that the ab ov e quan tity is diﬀerent from zero. Then the unique solution is B = 0 and from Ak q 4 + q +1 + 2 A + 2 B q 2 k q 4 + q +1 + B q 2 k q + 2 B q 3 k q +1 + B q 3 = 0 w e obtain A = 0 since k q 4 + q +1 = 1 and N q 5 /q ( k ) = 1 would yield k = 1, a contradiction. This means that the determinant of the Dic kson matrix must v anish. The ﬁrst 2 factors are diﬀerent from 0 since, together with N q 5 /q ( k ) = 1, they yield a con tradiction to our assumptions on k . Since F 9 ≤ F q 5 , if the last factor, sa y g ( k ) for some g ( Y ) ∈ F q 5 [ Y ], is zero then g ( k ) + ( g ( k )) q 5 = 0 = g ( k ) − ( g ( k )) q 5 , i.e.,                2 k 1+ q + q 2 + q 3 + q 4 + k 1+ q + q 2 + q 3 + k 1+ q + q 2 + q 4 + k 1+ q + q 2 + k 1+ q + q 3 + q 4 + k 1+ q + q 4 + 2 k 1+ q + k 1+ q 2 + q 3 + q 4 + k 1+ q 3 + q 4 + 2 k 1+ q 4 + 2 k + k q + q 2 + q 3 + q 4 + k q + q 2 + q 3 + 2 k q + q 2 + 2 k q + k q 2 + q 3 + q 4 + 2 k q 2 + q 3 + 2 k q 2 + 2 k q 3 + q 4 + 2 k q 3 + 2 k q 4 + 1 = 0 , k 1+ q + q 2 + q 3 + k 1+ q + q 2 + q 4 + k 1+ q + q 3 + q 4 + k 1+ q 2 + q 3 + q 4 + 2 k + k q + q 2 + q 3 + q 4 +2 k q + 2 k q 2 + 2 k q 3 + 2 k q 4 = 0 . Com bining them with N q 5 /q ( k ) = 1 we obtain                s 1 ( k ) := 2 k 2+2 q +2 q 2 +2 q 3 + 2 k 2+2 q +2 q 2 + q 3 + k 2+2 q + q 2 + q 3 + k 2+ q + q 2 + q 3 + 2 k 1+2 q +2 q 2 +2 q 3 + k 1+2 q +2 q 2 + q 3 + k 1+2 q + q 2 + q 3 + k 1+ q +2 q 2 +2 q 3 + k 1+ q +2 q 2 + q 3 + k 1+ q + q 2 +2 q 3 + 2 k 1+ q + q 2 +2 k 1+ q + q 3 + 2 k 1+ q + 2 k 1+ q 2 + q 3 + 2 k 1+ q 3 + k + 2 k q + q 2 + q 3 + 2 k q 2 + q 3 + k q 3 + 1 = 0 s 2 ( k ) := 2 k 2+2 q +2 q 2 +2 q 3 + k 2+ q + q 2 + q 3 + k 1+2 q + q 2 + q 3 + k 1+ q +2 q 2 + q 3 + k 1+ q + q 2 +2 q 3 + 2 k 1+ q + q 2 +2 k 1+ q + q 3 + 2 k 1+ q 2 + q 3 + 2 k q + q 2 + q 3 + 1 = 0 , and thus eliminating k 2 q 3 from the tw o equations ab ov e, we obtain t ( k ) = 0. Getting k q 3 from t ( k ) = 0 and substituting it in s 2 ( k ) we obtain ( k − 1) 1+ q 2 ( k q +1 − 1) q +1 ( k q 2 + q +1 − 1) u 1 ( k ) u 2 ( k ) = 0 , and the claim follows, since ( k − 1) 1+ q 2 ( k q +1 − 1) q +1 ( k q 2 + q +1 − 1) = 0 implies k = 1. The following prop osition deals with the case (C4). Prop osition 3.11. L et q = 3 2 h +1 , h > 1 . If U 1 is sc atter e d and of dimension 4 and it is c ontaine d in Z k := { ( x, k ( x q + x q 3 ) + x q 2 + x q 4 ) : x ∈ F q 5 } , with N q 5 /q ( k ) = 1 , then Z k is not sc atter e d. 19 Pr o of. By Prop osition 3.10, we can assume that u 1 ( k ) u 2 ( k ) = t ( k ) = 0. Note that for k = 1, the set Z k is not scattered. In what follows, we consider k  = 1. W e will pro ve that for all k  = 1 satisfying u 1 ( k ) u 2 ( k ) = t ( k ) = 0, the set Z k is not scattered. T o this end, w e will show that there exists an element m ∈ F q 5 suc h that rank( mx + k ( x q + x q 3 ) + x q 2 + x q 4 ) = 3 . (10) Consider the Dickson matrix M associated with the linearized p olynomial in (10): M =       m k 1 k 1 1 m q k q 1 k q k q 2 1 m q 2 k q 2 1 1 k q 3 1 m q 3 k q 3 k q 4 1 k q 4 1 m q 4       . W e hav e that (10) holds if and only if det( M ) = 0 and det( M 2 , 3 , 4 , 5 1 , 2 , 3 , 4 ) = 0 [5]. These conditions, together with their F rob enius conjugates, yield six equations in the v ariables m i ∈ F q , where m = P 4 i =0 m i ξ q i and { ξ , ξ q , . . . , ξ q 4 } is a normal basis of F q 5 o ver F q . These equations deﬁne a v ariety ov er F q , parametrized by k ∈ F q 5 . Our aim is to prov e, for eac h k satisfying our constrain ts, the existence of a solution ( m 0 , m 1 , m 2 , m 3 , m 4 ) ∈ F 5 q to these six equations. Our pro of is divided in to sev eral steps. (a) First, we show that there is a smaller set of equations, { h 1 , h 2 , h 3 , h 4 } , whose solutions are also solutions to the original system in terms of ( m 0 , m 1 , m 2 , m 3 , m 4 ) ∈ F 5 q . (b) Let V be the v ariet y deﬁned by { h 1 , h 2 , h 3 , h 4 } . W e consider the aﬃne transformation ϕ ( m 0 , m 1 , m 2 , m 3 , m 4 ) =  4 X i =0 ξ q i m i , . . . , 4 X i =0 ξ q i +4 m i +4 (mo d 5)  , (11) and the corresp onding v ariet y W . This pro jectivit y preserves the absolute irreducibility of v arieties but is not deﬁned o v er F q . (c) W e prov e that W is absolutely irreducible for each k satisfying our constraints, and thus so is V . F urthermore, W is ﬁxed by the shift ψ that sends each m i to m i +1 (mod 5) and raises the co eﬃcients to the p ow er q . This shows that V is F q -rational (i.e., ﬁxed by the F rob enius map ϕ q ). First, we consider the following set of equations { h 1 ( m ) , h 2 ( m ) , h 3 ( m ) , h 4 ( m ) } , where 20 h 1 ( m ) := 2 m q 4 + q +1 + k q 4 + q 3 m q +1 + k q 3 + q m q 4 +1 + 2 k q 4 + q 3 + q m + 2 k q 3 + q m + k q m + m q 4 + q + 2 k q 4 + q 3 m q + 2 k q +1 m q 4 + k m q 4 + 2 k q 3 m q 4 + k q 4 + q 3 + q +1 + 2 k q 4 + q 3 +1 + k q 4 + q 3 + k q 3 + 2; h 2 ( m ) := 2 m q 3 + q 2 + q + k q 3 +1 m q 2 + q + m q 3 + q + 2 k m q + 2 k q 3 + q 2 m q + k q 2 m q + k q +1 m q 3 + q 2 + 2 k q 3 + q 2 +1 m q 2 + 2 k q 3 +1 m q 2 + k q 3 m q 2 +2 k q +1 m q 3 + k q 3 + q 2 + q +1 + 2 k q 2 + q +1 + k q +1 + k + 2; h 3 ( m ) := 2 q 2 + q +1 + k q 4 + q 2 m q +1 + m q 2 +1 + 2 k q 2 + q m + k q m + 2 k q 4 m + k q 4 +1 m q 2 + q + 2 k q 4 + q 2 +1 m q + 2 k q 4 + q 2 m q + k q 2 m q + 2 k q 4 +1 m q 2 + k q 4 + q 2 + q +1 + 2 k q 4 + q +1 + k q 4 +1 + k q 4 + 2; h 4 ( m ) := ( k q 4 + q 3 + q 2 + 2) m 2 q +2 + (2 k q 4 + q 3 + q 2 + q + 2 k q 4 + q 3 + q 2 + k q 3 + q + k q + 2 k q 4 + q 3 + 1) m 2 q +1 +( k q 3 + q 2 +2 q + 2 k q 3 +2 q + k q 4 + q 3 + q + 2 k q ) m 2 + (2 k q 4 + q 3 + q 2 +1 + k q 4 +1 + 2 k q 4 + q 3 + q 2 +2 k q 4 + q 2 + k q 2 + 1) m q +2 + (2 k q 4 + q 3 + q 2 + q +1 + k q 3 + q 2 + q +1 + k q 4 + q 2 + q +1 + 2 k q 2 + q +1 +2 k q 4 + q 3 + q +1 + 2 k q 4 + q +1 + 2 k q +1 + 2 k q 4 + q 3 + q 2 +1 + k q 4 + q 3 +1 + 2 k q 4 +1 + k + k q 4 + q 3 + q 2 + q + 2 k q 3 + q 2 + q + k q 2 + q + 2 k q + k q 4 + q 3 + q 2 + k q 3 + q 2 + 2 k q 2 + k q 4 + q 3 + 2 k q 3 + k q 4 + 1) m q +1 + (2 k q 4 + q 3 + q 2 +2 q +1 + 2 k q 4 + q 3 + 2 k q 2 +2 q +1 + k q 4 + q 3 +2 q +1 + k 2 q +1 + k q 2 + q +1 + 2 k q 4 + q 3 + q +1 + 2 k q 3 + q +1 + k q +1 + k q 4 + q 3 +1 + 2 k + 2 k q 3 + q 2 + q + 2 k q 4 + q 3 + q +2 k q 3 + q + 2 k q 4 + q 3 + 1) m + ( k q 4 + q 3 + q 2 +1 + 2 k q 4 +1 + k q 4 + q 2 + 2 k q 2 ) m 2 q +(2 k q 4 + q 3 + q 2 + q +2 + k q 4 + q +2 + k q 4 + q 3 + q 2 +2 + 2 k q 4 +2 + k q 4 + q 2 + q +1 + k q 2 + q +1 + k q 4 + q +1 +2 k q 4 + q 3 + q 2 +1 + 2 k q 3 + q 2 +1 + k q 4 + q 2 +1 + k q 4 +1 + 2 k q 4 + q 3 + q 2 + k q 2 + q 3 + 2 k q 4 + 1) m q + k q 4 + q 2 +2 q +2 + 2 k q 4 +2 q +2 + 2 k q 4 + q 2 + q +2 + k q 4 + q 3 + q +2 + 2 k q 4 + q 3 +2 + k q 4 +2 + k q 4 + q 3 + q 2 + q +1 + 2 k q 4 + q 3 + q +1 + k q 4 + q +1 + 2 k q +1 + k q 3 +1 + 2 k q 4 +1 + k q 4 + q 3 + 2 k q 3 . A direct computation using MAGMA conﬁrms that if m is a common ro ot of the four preceding p olynomials, then the rank of the matrix M is less than 4. This can b e v eriﬁed b y noting that the p olynomials h 1 , h 2 , and h 3 are linear in m q 4 , m q 3 , and m q 2 , resp ectively . One can therefore solv e for these terms and substitute the resulting expressions in to the conditions det( M ) = 0 and det( M 2 , 3 , 4 , 5 1 , 2 , 3 , 4 ) = 0. After making these substitutions, the tw o resulting p olynomials in m and m q are found to b e divisible by h 4 . This accomplishes T ask (a). W e now establish the existence of an element m ∈ F q 5 that is a common zero of the p olyno- mials h 1 , h 2 , h 3 , and h 4 . T o this end, we employ an approach based on algebraic v arieties. W e represent an element m ∈ F q 5 using a normal basis { ξ , ξ q , . . . , ξ q 4 } , such that m = P 4 i =0 m i ξ q i with co ordinates m i ∈ F q . The four equations h j ( m 0 , m 1 , m 2 , m 3 , m 4 ) = 0 for j = 1 , 2 , 3 , 4 deﬁne a v ariet y V ⊆ A 5 ( F q 5 ) ov er F q 5 . Let W = ϕ ( V ), where ϕ is deﬁned in (11). The F rob enius automorphism Φ q : x 7→ x q acts on an element m by mapping it to P 4 i =0 m i ξ q i +1 , whic h induces a cyclic p erm utation on its co ordinates. Let ψ be the corresponding action on the p olynomial ring, which raises the co eﬃcients of p olynomials to the q -th p ow er and cyclically p erm utes the v ariables m i . T o prov e that the v ariet y V is deﬁned o ver F q , we m ust sho w that the deﬁning ideal of W = ϕ ( V ) is inv ariant under the action of ψ . 21 Let e f i = h i ( ϕ ( m 0 , m 1 , m 2 , m 3 , m 4 )). T o ac hiev e our goal, it is suﬃcien t to v erify the following conditions: • ψ ( f h 1 ) = f h 3 • ψ ( f h 3 ) = f h 2 • The p olynomials ψ ( f h 2 ) and ψ ( f h 4 ) are multiples of f h 4 mo dulo the ideal ⟨ f h 1 , f h 2 , f h 3 ⟩ . The ab o v e conditions can b e easily veriﬁed by using MAGMA. The next step is to pro ve that W is absolutely irreducible. W e b egin b y noting that the equations f h 1 = 0 , f h 2 = 0 , and f h 3 = 0 are of degree one in the v ariables m 2 , m 3 , and m 4 . This linearity allo ws us to eliminate these v ariables, thereb y establishing a birational equiv alence b et w een W and a plane curv e deﬁned ov er F q 5 in the v ariables m 0 and m 1 . Note that f h 4 ( m 0 , m 1 ) can b e written as P i,j ≤ 2 a i,j m i 0 m j 1 , where a 2 , 2 := k q 4 + q 3 + q 2 + 2; a 2 , 1 := 2 k q 4 + q 3 + q 2 + q + 2 k q 4 + q 3 + q 2 + k q 3 + q + k q + 2 k q 4 + q 3 + 1; a 2 , 0 := k q 3 + q 2 +2 q + 2 k q 3 +2 q + k q 4 + q 3 + q + 2 k q ; a 1 , 2 := 2 k q 4 + q 3 + q 2 +1 + k q 4 +1 + 2 k q 4 + q 3 + q 2 + 2 k q 4 + q 2 + k q 2 + 1; a 1 , 1 := 2 k q 4 + q 3 + q 2 + q +1 + k q 3 + q 2 + q +1 + k q 4 + q 2 + q +1 + 2 k q 2 + q +1 + 2 k q 4 + q 3 + q +1 + 2 k q 4 + q +1 +2 k q +1 + 2 k q 4 + q 3 + q 2 +1 + k q 4 + q 3 +1 + 2 k q 4 +1 + k + k q 4 + q 3 + q 2 + q + 2 k q 3 + q 2 + q + k q 2 + q + 2 k q + k q 4 + q 3 + q 2 + k q 3 + q 2 + 2 k q 2 + k q 4 + q 3 + 2 k q 3 + k q 4 + 1; a 1 , 0 := 2 k q 4 + q 3 + q 2 +2 q +1 + 2 k q 4 + q 3 + 2 k q 2 +2 q +1 + k q 4 + q 3 +2 q +1 + k 2 q +1 + k q 2 + q +1 +2 k q 4 + q 3 + q +1 + 2 k q 3 + q +1 + k q +1 + k q 4 + q 3 +1 + 2 k + 2 k q 3 + q 2 + q + 2 k q 4 + q 3 + q +2 k q 3 + q + 2 k q 4 + q 3 + 1; a 0 , 2 := k q 4 + q 3 + q 2 +1 + 2 k q 4 +1 + k q 4 + q 2 + 2 k q 2 ; a 0 , 1 := 2 k q 4 + q 3 + q 2 + q +2 + k q 4 + q +2 + k q 4 + q 3 + q 2 +2 + 2 k q 4 +2 + k q 4 + q 2 + q +1 + k q 2 + q +1 + k q 4 + q +1 +2 k q 4 + q 3 + q 2 +1 + 2 k q 3 + q 2 +1 + k q 4 + q 2 +1 + k q 4 +1 + 2 k q 4 + q 3 + q 2 + k q 2 + q 3 + 2 k q 4 + 1; a 0 , 0 := k q 4 + q 2 +2 q +2 + 2 k q 4 +2 q +2 + 2 k q 4 + q 2 + q +2 + k q 4 + q 3 + q +2 + 2 k q 4 + q 3 +2 + k q 4 +2 + k q 4 + q 3 + q 2 + q +1 + 2 k q 4 + q 3 + q +1 + k q 4 + q +1 + 2 k q +1 + k q 3 +1 + 2 k q 4 +1 + k q 4 + q 3 + 2 k q 3 . Supp ose that ( m 1 − λ ) is a factor of e f 4 ( m 0 , m 1 ) for some λ ∈ F q . Then f h 4 ( m 0 , λ ) must be the zero p olynomial in m 0 . By insp ecting the co eﬃcien t of m 2 0 , we deduce that either λ = k q or λ = k q 3 + q 2 + q + 2 k q 3 + q + k q 4 + q 3 + 2 k q 4 + q 3 + q 2 + 2 . Note that, since N q 5 /q ( k ) = 1, the denominator abov e v anishes only if k = 1. • In the former case, w e substitute m 1 = k q in to e f 4 ( m 0 , m 1 ). Setting the co eﬃcien t of m 0 to zero yields ( k q 3 + q + k q 4 + q + k q + 2 k q 4 + q 3 + 1)( k q 2 + q + 2)( k + 2) = 0 . The last tw o factors yield k = 1, so w e are left with the condition k q 3 + q + k q 4 + q + k q + 2 k q 4 + q 3 + 1 = 0. It can b e v eriﬁed that no k ∈ F q 5 satisfying N q 5 /q ( k ) = 1, k  = 1, u 1 ( k ) u 2 ( k ) = 0, and t ( k ) = 0 also satisﬁes this last condition. 22 • In the latter case, we again substitute the expression for m 1 in to f h 4 ( m 0 , m 1 ) and consider the co eﬃcien t of m 0 . Setting this co eﬃcient to zero implies that either k = 1 or k q 4 + k 1+2 q +2 q 2 +3 q 3 +3 q 4 + k 1+2 q +2 q 2 +3 q 3 +2 q 4 + k 1+2 q +2 q 2 +2 q 3 + q 4 + k 1+2 q + q 2 +3 q 3 +2 q 4 + 2 k q 3 + 2 k 1+2 q + q 2 +3 q 3 + q 4 + 2 k 1+2 q + q 2 +2 q 3 +2 q 4 + 2 k 1+2 q + q 2 +2 q 3 + q 4 + k 1+2 q + q 2 +2 q 3 + 2 k 1+2 q + q 2 + q 3 + q 4 + 2 k 1+2 q + q 2 + q 3 + 2 k 1+2 q + q 3 + k 1+2 q + k 1+ q +2 q 2 +3 q 3 +2 q 4 + 2 k 1+ q +2 q 2 +2 q 3 +2 q 4 + 2 k 1+ q + q 2 +3 q 3 +3 q 4 + k 1+ q + q 2 +3 q 3 +2 q 4 + 2 k 1+ q + q 2 +2 q 3 +3 q 4 + 2 k 1+ q + q 2 +2 q 3 +2 q 4 + k 1+ q + q 2 + q 3 +2 q 4 + k 1+ q + q 2 + q 3 + q 4 + 2 k 1+ q +2 q 3 +2 q 4 + k 1+ q +2 q 3 + q 4 + k 1+ q + q 3 +2 q 4 + 2 k 1+ q + q 4 + k 1+2 q 3 +3 q 4 + 2 k 1+2 q 3 +2 q 4 + 2 k 1+ q 3 +2 q 4 + k 1+ q 3 + q 4 + 2 k 2 q +2 q 2 +3 q 3 +2 q 4 + 2 k 2 q +2 q 2 +3 q 3 + q 4 + 2 k 2 q +2 q 2 +2 q 3 + 2 k 2 q + q 2 +3 q 3 + q 4 + k 2 q + q 2 +2 q 3 + q 4 + 2 k 2 q + q 2 +2 q 3 + k 2 q + q 2 + q 3 + k 2 q +2 q 3 + 2 k 2 q + q 3 + 2 k q +2 q 2 +3 q 3 + q 4 + k q +2 q 2 +2 q 3 + q 4 + k q + q 2 +3 q 3 +2 q 4 + k q + q 2 +3 q 3 + q 4 + k q + q 2 +2 q 3 +2 q 4 + 2 k q + q 2 +2 q 3 + q 4 + k q + q 2 +2 q 3 + 2 k q + q 2 + q 3 + q 4 + k q + q 2 + q 3 + 2 k q +2 q 3 + q 4 + 2 k q +2 q 3 + k q + q 3 + q 4 + 2 k q + q 3 + 2 k q + 2 k q 2 +3 q 3 +2 q 4 + k q 2 +2 q 3 +2 q 4 + k q 2 +2 q 3 + q 4 + 2 k q 2 + q 3 + q 4 + k 2 q 3 + q 4 + 2 k q 3 +2 q 4 = 0 . As in the previous case, it can b e v eriﬁed that no k ∈ F q 5 satisfying N q 5 /q ( k ) = 1 and k  = 1, along with u 1 ( k ) u 2 ( k ) = 0 and t ( k ) = 0, satisﬁes this ﬁnal equation. Since a similar argument holds for factors of the form ( m 0 − λ ), where λ ∈ F q , w e conclude that f h 4 ( m 0 , m 1 ) has no linear (degree-one) factors. Therefore, if f h 4 ( m 0 , m 1 ) is not absolutely irreducible, it must split in to tw o quadratic (degree-tw o) factors. F urthermore, both quadratic factors m ust b e linear in eac h v ariable, m 0 and m 1 . Otherwise, a factor dep ending on only one v ariable w ould hav e to exist, which contradicts the previous conclusion. Th us, the only p ossibilit y is that the tw o factors are j 1 ( m 0 , m 1 ) := ( m 1 + 2 k q ) m 0 + Am 1 + B , j 2 ( m 0 , m 1 ) := (( k q 4 + q 3 + q 2 + 2) m 1 + 2 k q 3 + q 2 + q + k q 3 + q + 2 k q 4 + q 3 + 1) m 0 + C m 1 + D , for some A, B , C , D ∈ F q . The p olynomial J ( m 0 , m 1 ) := f h 4 ( m 0 , m 1 ) − j 1 ( m 0 , m 1 ) j 2 ( m 0 , m 1 ) must v anish identically . By setting the co eﬃcien ts of m 0 and m 0 m 2 1 in J ( m 0 , m 1 ) to zero, we get: − k q D = 2 k 1+2 q + q 2 + q 3 + q 4 + 2 k 1+2 q + q 2 + k 1+2 q + q 3 + q 4 + k 1+2 q + k 1+ q + q 2 + 2 k 1+ q + q 3 + q 4 + 2 k 1+ q + q 3 + k 1+ q + k 1+ q 3 + q 4 + 2 k + 2 k q + q 2 + q 3 + 2 k q + q 3 + q 4 + 2 k q + q 3 + 2 k q 3 + q 4 + 1 + ( k q 3 + q 2 + q + 2 + k q 4 + q 3 + 2 k q 3 + q ) B , C = (2 k q 4 + q 3 + q 2 + 1) A + 2 k q 4 + q 3 + q 2 +1 + k q 4 +1 + 2 k q 4 + q 3 + q 2 + 2 k q 4 + q 2 + k q 2 + 1 . F rom the co eﬃcien ts of m 2 1 and m 0 m 1 , we also obtain the conditions: ( A + 1)( k q 4 + q 3 + q 2 +1 + 2 k q 4 +1 + k q 4 + q 3 + q 2 A + k q 4 + q 2 + 2 k q 2 + 2 A ) = 0 , and 23 ( k q + q 2 + q 3 + q 4 + 2 k q + q 2 + q 3 + k q + q 3 + 2 k q + 2 k q 3 + q 4 + 1) B + ( k 2 q + q 2 + q 3 + q 4 + 2 k 2 q + q 2 + q 3 + k 2 q + q 3 + 2 k 2 q + 2 k q + q 3 + q 4 + k q ) A + 2 k 1+2 q + q 2 + q 3 + 2 k 1+2 q + q 2 + q 4 + 2 k 1+2 q + q 2 + k 1+ q + q 2 + q 3 + q 4 + 2 k 1+ q + q 2 + k 1+ q + q 3 + k 1+ q + q 4 + k 1+ q + 2 k 1+ q 3 + q 4 + 2 k + k 2 q + q 2 + q 3 + k 2 q + q 2 + q 4 + k 2 q + q 2 + 2 k q + q 2 + q 3 + q 4 + k q + q 2 + 2 k q + q 3 + 2 k q + q 4 + 2 k q + k q 3 + q 4 + 2 = 0 . The equation inv olving A yields t w o p ossibilities. Either A = − 1 or A = − k q 4 + q 3 + q 2 +1 + 2 k q 4 +1 + k q 4 + q 2 + 2 k q 2 k q 4 + q 3 + q 2 + 2 . It can b e veriﬁed that b oth cases lead to a contradiction when substituted into the other co ef- ﬁcien t equations, given the conditions on k (namely that N q 5 /q ( k ) = 1, k  = 1, u 1 ( k ) u 2 ( k ) = 0, and t ( k ) = 0). This shows that the curve V is F q -rational and absolutely irreducible. An upp er b ound on its gen us can b e determined by examining the plane curv e deﬁned b y f h 4 ( m 0 , m 1 ) = 0. This curve has degree four and at least t w o singular p oints (its points at inﬁnit y), so its geometric gen us is at most one. Since the genus is a birational inv ariant (i.e., it is preserved by such transformations), and b oth V and W are birationally equiv alen t to this plane curve, we conclude that the genus of b oth V and W is at most one. Therefore, by the celebrated Hasse-W eil Theorem [19], the curve V has at least one aﬃne F q -rational p oin t (recalling that q = 3 2 h +1 and h > 1). This implies that Z k is not scattered, which concludes the pro of. W e can now prov e our main result. Theorem 3.12. L et q = 3 2 h +1 , with h > 1 . The subsp ac e U 1 := { ( x, x q + x q 4 ) : T r q 5 /q ( x ) = 0 } ≤ F 2 q 5 is maximal ly sc atter e d. Pr o of. This directly follows from Propositions 3.6, 3.7, 3.8, 3.9, and 3.11. 4 Self-dualit y of the co de C δ Let C b e an [ n, k ] q m /q rank-metric code. Its dual co de C ⊥ is the [ n, n − k ] q m /q co de deﬁned by the standard inner pro duct on F n q m . It is a well-kno wn geometric prop ert y (see [4, 6]) that if U is a q -system asso ciated with C , then the q -system associated with the dual co de C ⊥ is exactly the Delsarte dual ¯ U of U . In our sp eciﬁc case, U δ is a 4-dimensional F q -subspace of F 2 q 5 . The asso ciated co de C δ is a [4 , 2] q 5 /q co de, and its dual C ⊥ δ is also a [4 , 2] q 5 /q co de. Computing the Delsarte dual ¯ U δ reduces directly to determining the generators of C ⊥ δ . Theorem 4.1. L et q b e an o dd prime p ower, δ ∈ F q 5 with N q 5 /q ( δ ) = 1 . L et C δ b e the [4 , 2] q 5 /q c o de asso ciate d with the q -system U δ := { ( x, x q + δ x q 4 ) : T r q 5 /q ( x ) = 0 } ⊆ F 2 q 5 . Then C ⊥ δ is e quivalent to C δ . Ge ometric al ly, this me ans U δ is Delsarte self-dual. 24 In order to prov e Theorem 4.1, we need a few auxiliary results. Fix an F q -basis x = ( x 1 , x 2 , x 3 , x 4 ) of k er(T r q 5 /q ) and write x ( q i ) = ( x q i 1 , . . . , x q i 4 ) for i ∈ { 1 , . . . , 4 } . The co de asso ciated with U δ is C δ = ⟨ x, x ( q ) + δ x ( q 4 ) ⟩ F q 5 ⊆ F 4 q 5 . F or c = ( c 1 , c 2 , c 3 , c 4 ) ∈ F 4 q 5 , set σ i ( c ) := P 4 j =1 x q i j c j . Then c ∈ C ⊥ δ if and only if σ 0 ( c ) = 0 and σ 1 ( c ) + δ σ 4 ( c ) = 0 . (12) Since x j ∈ ker(T r q 5 /q ), we hav e x j q 4 = − x j − x j q − x q 2 j − x j q 3 , giving σ 4 ( c ) = − σ 0 ( c ) − σ 1 ( c ) − σ 2 ( c ) − σ 3 ( c ) . (13) Consider the 4 × 4 Mo ore matrix M 4 ( x ) =     x 1 x 2 x 3 x 4 x 1 q x 2 q x 3 q x 4 q x q 2 1 x q 2 2 x q 2 3 x q 2 4 x 1 q 3 x 2 q 3 x 3 q 3 x 4 q 3     . Since x 1 , . . . , x 4 are F q -indep enden t in F q 5 , the matrix M 3 ( x ) has rank 3 and M 4 ( x ) is non- singular. Set s := det( M 4 ( x ))  = 0 and deﬁne y = ( y 1 , y 2 , y 3 , y 4 ) as the vector of cofactors of the last row of M 4 ( x ): y j := ( − 1) j det( M 3 ( x 1 , . . . , b x j , . . . , x 4 )) , j = 1 , 2 , 3 , 4 , where ( x 1 , . . . , b x j , . . . , x 4 ) denotes the vector without the j -th component. By the standard prop erties of cofactor expansions, σ i ( y ) = 0 for i = 0 , 1 , 2 (expansion along a wrong ro w produces a matrix with tw o identical rows) and σ 3 ( y ) = s (expansion along the correct ro w). T ogether with (13): σ 0 = σ 1 = σ 2 = 0 , σ 3 = s, σ 4 = − s. (14) Lemma 4.2. Setting σ ik := P j x q i j y q k j = σ i ( y ( q k ) ) , one has σ ik = σ ( i − k ) ( y ) q k . Pr o of. σ ik =  P j x q i − k j y j  q k . Com bined with (14), the complete table of σ i ( y ( q k ) ) is: k σ 0 σ 1 σ 4 σ 1 + δ σ 4 0 0 0 − s − δ s 1 − s q 0 s q δ s q 2 s q 2 − s q 2 0 − s q 2 3 0 s q 3 0 s q 3 4 0 0 0 0 25 The row k = 4 shows that y ( q 4 ) ∈ C ⊥ δ . F or the second generator, set c 2 = y + α y ( q 3 ) , with α ∈ F q 5 . The condition σ 0 ( c 2 ) = 0 + α · 0 = 0 is automatic, while condition (12) gives α s q 3 + δ ( − s ) = 0 = ⇒ α = δ s 1 − q 3 . The tw o generators are F q 5 -indep enden t since they inv olv e distinct F rob enius shifts of y , so C ⊥ δ = ⟨ y ( q 4 ) , y + δ s 1 − q 3 y ( q 3 ) ⟩ F q 5 . (15) Prop osition 4.3. s = det( M 4 ( x )) ∈ F × q . Pr o of. Applying the q -th p o wer to the determinant shifts e ac h row by one F robenius: s q = det( M 4 ( x ( q ) )) = det      x 1 q x 2 q x 3 q x 4 q x q 2 1 x q 2 2 x q 2 3 x q 2 4 x 1 q 3 x 2 q 3 x 3 q 3 x 4 q 3 x 1 q 4 x 2 q 4 x 3 q 4 x 4 q 4      . Substituting x j q 4 = − ( x j + x j q + x q 2 j + x j q 3 ) in the last row and expanding by multilinearit y , all terms with a rep eated ro w v anish, leaving s q = − det     x 1 q · · · x 4 q x q 2 1 · · · x q 2 4 x 1 q 3 · · · x 4 q 3 x 1 · · · x 4     = ( − 1)( − 1) 3 s = s, where three row swaps bring the last row to the top, restoring the standard Moore order. Since s ∈ F × q , we hav e s q 3 = s , hence α = δ s 1 − q 3 = δ , and (15) simpliﬁes to C ⊥ δ = ⟨ y ( q 4 ) , y + δ y ( q 3 ) ⟩ F q 5 . (16) Prop osition 4.4. L et y = ( y 1 , y 2 , y 3 , y 4 ) . The c omp onents of y form an F q -b asis of ker(T r q 5 /q ) . Pr o of. Set v = P 4 k =0 y ( q k ) , so that v j = T r q 5 /q ( y j ). Then σ i ( v ) = 4 X k =0 σ ik = s q ( i − 3) − s q ( i − 4) = s − s = 0 for all i = 1 , . . . , 4, where the last equality uses s ∈ F q . Since the 4 × 4 Mo ore matrix M 4 ( x ) is non-singular, v must b e the null vector. Hence y j ∈ ker(T r q 5 /q ) for every j . Since det( M 4 ( y ))  = 0 (which follows from standard cofactor matrix prop erties), the elements y 1 , y 2 , y 3 , y 4 are F q - linearly indep enden t, forming a basis of ker(T r q 5 /q ). W e are now ready to prov e Theorem 4.1. Pr o of of The or em 4.1. By (16), the dual co de C ⊥ δ is generated by the rows of the matrix H = y ( q 4 ) y + δ y ( q 3 ) ! . 26 Let u = y ( q 4 ) . Notice that ker(T r q 5 /q ) is closed under the F rob enius map z 7→ z q , since T r q 5 /q ( z q ) = T r q 5 /q ( z ) q = 0. Because the map z 7→ z q 4 is an F q -automorphism of F q 5 that preserv es this kernel, the comp onents of u also form an F q -basis of ker(T r q 5 /q ). Note that u ( q ) = y ( q 5 ) = y and u ( q 4 ) = y ( q 8 ) = y ( q 3 ) . Th us, the generator matrix H can b e rewritten as H =  u u ( q ) + δ u ( q 4 )  . Recall that the generator matrix of the original co de C δ is G =  x x ( q ) + δ x ( q 4 )  . Since b oth x = ( x 1 , x 2 , x 3 , x 4 ) and u = ( u 1 , u 2 , u 3 , u 4 ) are F q -bases of the same 4-dimensional F q -v ector space ker(T r q 5 /q ), there exists an inv ertible matrix A ∈ GL(4 , q ) such that u = xA. Because A has en tries in F q , the F rob enius map acts comp onent-wise and leav es A inv ariant. Therefore, u ( q ) = x ( q ) A and u ( q 4 ) = x ( q 4 ) A . Substituting this into H , we obtain H =  xA x ( q ) A + δ x ( q 4 ) A  =  x x ( q ) + δ x ( q 4 )  A = GA. Since t wo rank-metric co des are equiv alen t if they hav e t wo generator matrices diﬀering b y right m ultiplication by a matrix in GL( n, q ), the co des C ⊥ δ and C δ are equiv alen t. 5 Concluding remarks and op en problems In this paper, we presen ted the ﬁrst inﬁnite family of non-extendable F q m -linear MRD codes that do not reach the maxim um p ossible length (i.e., n < m ). Geometrically , these correspond to scattered subspaces that are not maxim um scattered, but are maximal with resp ect to inclusion. Our results rely heavily on the partial classiﬁcation of maximum length MRD co des in F 2 q 5 (corresp onding to maxim um scattered subspaces) obtained in [13]. Consequently , if the co des corresp onding to families (C3) and (C4) from Theorem 3.3 are indeed non-existent—as conjec- tured and supp orted by computational evidence—then our construction holds for any v alue of q and any δ satisfying the conditions in Proposition 3.6. W e conclude with some op en problems arising from this co ding theory p ersp ectiv e: (i) Are the MRD co des corresp onding to families (C3) and (C4) in Theorem 3.3 non-existent for any q ? (ii) Exp erimen tal results suggest that the co des asso ciated with the q -systems U δ , N q m /q ( δ ) = 1, whenev er they are MRD, are also non-extendable, at least when m is o dd. Is this true? Unfortunately , in these cases, we cannot use an y partial classiﬁcation of scattered subspaces in F 2 q m . (iii) A general strategy for constructing non-extendable MRD co des could b e as follows: (a) Start with an F q m -linear code C f asso ciated with U f := { ( x, f ( x )) : x ∈ F q m } , whic h is not MRD. 27 (b) Puncture the co de by restricting its geometric domain to a suitable ( n − 1)-dimensional F q -subspace H ⊂ F q m , creating a candidate MRD co de C f ,H of length n − 1. This approac h raises a k ey question: F or a giv en f and H , can w e determine all the p ossible co des C g that can act as one-column extensions of C f ,H ? Solving this would provide a p o w erful to ol to prov e non-extendability by ruling out known families of extended co des. (iv) The discussion so far has fo cused exclusively on co des of dimension 2 (scattered subspaces in F 2 q m ). Un til very recently , no examples were known of non-extendable MRD co des with higher dimension ( k ≥ 3 or h > 1). How ev er, using MAGMA, w e found a sp oradic example of a non-extendable [6 , 3 , 4] 2 7 / 2 MRD co de (geometrically , a maximally 2-scattered subspace of dimension 6 in F 3 2 7 ). This isolated case suggests that such optimal, non-extendable co des ma y exist more generally . Remark 5.1. A sp or adic non-extendable [6 , 3 , 4] 2 7 / 2 MRD c o de c an b e describ e d explicitly by its asso ciate d q -system: U = { ( x, x 2 , x 4 + ξ 3 x 64 ) + µ ( ξ 6 , ξ 27 , ξ 34 ) : x ∈ S, µ ∈ F 2 } , wher e S := ⟨ 1 , ξ , ξ 2 , ξ 3 , ξ 4 ⟩ F 2 and ξ is a primitive element of F 2 7 . It c an b e pr oven using MA GMA that U is 2 -sc atter e d but not c ontaine d in any lar ger 2 -sc atter e d subsp ac e, henc e the asso ciate d MRD c o de c annot b e extende d. (v) Are there structural diﬀerences b et w een extendable and non-extendable MRD co des in terms of their generalized rank weigh ts? Geometrically , this translates to in vestigating dif- ferences in the in tersection n umbers of maximally scattered v ersus non-maximally scattered subspaces with F q m -subspaces. Ac kno wledgemen ts The authors thank the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA—INdAM) which supp orted the research. Declarations Conﬂicts of interest. The authors hav e no conﬂicts of interest to declare that are relev ant to the conten t of this article. References [1] G. N. Alfarano, M. Borello, A. Neri, and A. Ra v agnani. Linear cutting blo c king sets and minimal co des in the rank metric. J. Combin. The ory Ser. A , 192:105658, 2022. [2] D. Bartoli, B. Csa jb´ ok, G. Marino, and R. T rombetti. Ev asiv e subspaces. J. Combin. Designs , 29(8):533–551, 2021. [3] A. Blokhuis and M. Lavrau w. 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An infinite family of non-extendable MRD codes

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