A large family of maximum scattered linear sets of \mathrm{PG}(1,q^n) and their associated MRD codes
Giovanni Longobardi, Giuseppe Marino, Rocco Trombetti, Yue Zhou
aa r X i v : . [ m a t h . C O ] F e b A LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETSOF
PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES
G. LONGOBARDI , GIUSEPPE MARINO , ROCCO TROMBETTI , AND YUE ZHOU Abstract.
The concept of linear set in projective spaces over finite fieldswas introduced by Lunardon [16] and it plays central roles in the study ofblocking sets, semifields, rank-distance codes and etc. A linear set with thelargest possible cardinality and rank is called maximum scattered. Despitetwo decades of study, there are only a limited number of maximum scatteredlinear sets of a line PG(1 , q n ). In this paper, we provide a large family of newmaximum scattered linear sets over PG(1 , q n ) for any even n ≥ q .In particular, the relevant family contains at least j q t +18 rt k , if t j q t +14 rt ( q +1) k , if t ≡ , inequivalent members for given q = p r and n = 2 t >
8, where p = char( F q ).This is a great improvement of previous results: for given q and n >
8, the num-ber of inequivalent maximum scattered linear sets of PG(1 , q n ) in all classesknown so far, is smaller than q . Moreover, we show that there are a largenumber of new maximum rank-distance codes arising from the constructedlinear sets. AMS subject classification:
Keywords: linearized polynomial, rank distance code, linear set1.
Introduction
Let V be a vector space over F q n of dimension r and Ω = PG( V, q n ) = PG( r − , q n ). A set of points L U of Ω is called an F q - linear set of rank k if it consists ofthe points defined by the non-zero elements of an F q -subspace U of V of dimension k , that is, L U = (cid:8) h u i F qn : u ∈ U \ { } (cid:9) . The term linear was introduced by Lunardon [16] who considered a special kindof blocking sets. In the pasting two decades after this work, linear sets have beenintensively investigated and applied to construct and characterize various objectsin finite geometry, such as blocking sets, two-intersection sets, translation spreadsof the Cayley generalized Hexagon, translation ovoids of polar spaces, semifieldsand rank-metric codes. We refer to [1, 13, 21, 22, 23] and the references therein.The most interesting linear sets are those satisfying certain extremal properties.Firstly, it is clear that | L U | ≤ q k − q − . When the equality is achieved, L is called scattered . A scattered linear set L U of Ω with largest possible rank k is called a maximum scattered linear set . In [5], it is proved that the largest possible rank is k = rn/ r is even, and ( rn − n ) / ≤ k ≤ rn/ r is odd. In particular, when r = 2, i.e., L U is a maximum scattered linear set over a projective line, its rank k equals n .For a given linear set L U of rank n of a projective line, by a suitable collineationof PG(1 , q n ), we may always assume that the point h (0 , i F qn is not in L U . Thismeans U = U f := { ( x, f ( x )) : x ∈ F q n } , for a q -polynomial f ( x ) over F q n ; i.e, an element of the set L n,q [ x ] = ( n − X i =0 c i x q i : c i ∈ F q n ) . Since polynomials in this set define F q -linear maps of F q n seen as F q -vector space,they are also known in the literature as linearized polynomials . Given a q -polynomial f , we use L f to denote the linear set defined by U f . It is not difficult to show that L f is scattered if and only if for any z, y ∈ F ∗ q n the condition f ( z ) z = f ( y ) y implies that z and y are F q -linearly dependent. Hence, a q -polynomials satisfyingthis condition is usually called a scattered polynomial (over F q n ). The condition for a q -polynomial f ( x ) to be scattered can be rephrased by saying that if f ( γx ) = γf ( x ),for x and γ ∈ F q n with x = 0, then γ ∈ F q .Scattered polynomials are also strongly related to a topic in network codingtheory: the rank-distance codes. More precisely, a rank-distance code (or RD codefor short) C is a subset of the set of m × n matrices F m × nq over F q endowed withthe rank distance d ( A, B ) = rk( A − B )for any A, B ∈ F m × nq . The minimum distance of an RD code C , |C| ≥
2, is definedas d ( C ) = min M,N ∈C M = N d ( M, N ) . A rank-distance code of F m × nq with minimum distance d has parameters ( m, n, q ; d ).If C is an F q -linear subspace of F m × nq , then C is called F q - linear RD code and its dimension dim F q C is defined to be the dimension of C as a subspace over F q . The Singleton-like bound [11] for an ( m, n, q ; d ) RD-code C is |C| ≤ q max { m,n } (min { m,n }− d +1) . If C attains this size, then C is a called Maximum Rank-Distance code , MRD code for short. In this paper we will consider only the case in which the codewords aresquare matrices, i.e. m = n . Note that if n = d , then an MRD code C consists of q n invertible endomorphisms of F q n ; such C is called spread set of End F q ( F q n ). Inparticular if C is F q -linear, it is called a semifield spread set of End F q ( F q n ), see [12].Two F q -linear codes C and C ′ are called equivalent if there exist A, B ∈ GL( n, q )and a field automorphism σ of F q such that C ′ = { AC σ B : C ∈ C} . The aforementioned link lies in the fact that rank-distance codes can be describedby means of q -polynomials over F q n , considered modulo x q n − x . After fixing anordered F q -basis { b , b , . . . , b n } for F q n it is possible to give a bijection Φ which LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES3 No. Families C f L f (i) Pseudo-regulus ϕ ( n ) / ≈ ( q − r , ∤ n, ( q +1)( q − r , | n, / ( q − r , ∤ n, ( q +1)( q − r , | n, (iii) Longobardi-Zanella ϕ ( n ) / ≤ ϕ ( n ) / Table 1.
Numbers of inequivalent C f and PΓL(2 , q n )-inequivalent L f , where f is a scattered polynomial in (i), (ii) or (iii), q = p r with p = char( F q ) and ϕ denote Euler’s totient function.associates for each matrix M ∈ F n × nq a unique q -polynomial f M ∈ L n,q [ x ]. Moreprecisely, put b = ( b , b , . . . , b n ) ∈ F nq n , then Φ( M ) = f M where for each u =( u , u , . . . , u n ) ∈ F nq we have f M ( b · u ) = b · u M .Given a scattered polynomial f , the set of q -polynomials C f = { ax + bf ( x ) : a, b ∈ F q n } defines a linear MRD code of minimum distance n − F q . For recent surveyson MRD codes and their relations with linear sets, we refer to [22, 24].Up to now, there are only three families of maximum scattered linear sets inPG(1 , q n ) for infinitely many n . We list the corresponding scattered polynomialsover F q n below:(i) x q s , where 1 ≤ s ≤ n − s, n ) = 1; see [5].(ii) δx q s + x q n − s , where n ≥ N q n /q ( δ ) / ∈ { , } , gcd( s, n ) = 1 and N q n /q : x ∈ F q n x qn − q − ∈ F q , is the norm function of F q n over F q ; see [17, 23].(iii) ψ ( k ) ( x ), where ψ ( x ) = (cid:16) x q + x q t − − x q t +1 + x q t − (cid:17) , q odd, n = 2 t and- t is even and gcd( k, t ) = 1, or- t is odd, gcd( k, t ) = 1, and q ≡ n ∈ { , } , there are other families of scattered polynomials over F q n ; see[3, 8, 9, 20, 25]. According to the asymptotic classification results of them obtainedin [2, 4], maximum scattered linear sets in PG(1 , q n ) seem rare.Two linear sets L U and L U ′ of PG(1 , q n ) are said to be PΓL- equivalent (or projectively equivalent ) if there exists ϕ ∈ PΓL(2 , q n ) such that L ϕU = L U ′ . Fortwo given q -polynomials, it is well-known that C f is equivalent to C g if and only if U f and U g are on the same ΓL(2 , q n )-orbit, which further implies that L f and L g are PΓL-equivalent. However, the converse statement is not true in general. Forinstance, if U f = { ( x, x q ) : x ∈ F q n } and U g = (cid:8) ( x, x q s ) : x ∈ F q n (cid:9) with s = 1 andgcd( s, n ) = 1, then U f and U g are not ΓL(2 , q n )-equivalent, but obviously L f = (cid:8) h (1 , x q − ) i F qn : x ∈ F ∗ q n (cid:9) = L g . For more results on the equivalence problems, werefer to [7, 10].In Table 1, we list the numbers of inequivalent C f and PΓL(2 , q n )-inequivalent L f , for a scattered polynomial f in each one of the three known families. The proofof what is stated in Table 1 will be provided in Section 2.In this paper, we present a new family of maximum scattered linear sets inPG(1 , q n ) where q = p r , p is an odd prime and n = 2 t ≥
6; see Theorem 3.1. In
G. LONGOBARDI, G. MARINO, R. TROMBETTI, AND Y. ZHOU particular, when t >
4, this new family provides q t +12 , if t q t +1 q +1 , if t ≡ . inequivalent number of F q n -MRD codes (Corollary 4.2) and at least j q t +18 rt k , if t j q t +14 rt ( q +1) k , if t ≡ , q n )-inequivalent maximum linear sets (Theorem 5.1). Therefore, the num-ber of maximum scattered linear sets in PG(1 , q n ) (and hence of F q n -MRD codes)grows exponentially with respect to n .The remaining part of this paper is organized as follows. In Section 2, we intro-duce more results on the equivalence of maximum scattered linear sets in PG(1 , q n )and the associated MRD codes, and explain Table 1 in details. In Section 3, weexhibit a family of scattered polynomials f over F q n and provide its proof. Theequivalence between the MRD codes C f associated to the members of this family arecompletely determined in Section 4, in which we also study the MRD codes derivedfrom the adjoint maps of our scattered polynomials. Based on these results, thePΓL-equivalence of the associated maximum linear sets are investigated in Section5. 2. Preliminaries
Equivalence of MRD codes and linear sets.
The following result concern-ing the equivalence of MRD codes associated with scattered polynomials is provedin [23].
Theorem 2.1.
Let f and g be two scattered polynomials over F q n , respectively.The MRD-codes C f and C g are equivalent if and only if U f and U g are ΓL(2 , q n ) -equivalent. For this paper, we only need the necessary and sufficient conditions in Theorem2.1 to interpret the equivalence problem in Section 4. However, in general, theequivalence problem for MRD codes could be more complicated. We refer to thesurveys [24, 22] for its precise definition and related results. See [6] for the hardnessof testing the equivalence between rank-distance codes. The left idealizer and the right idealizer of any given rank-distance code C are invariant under equivalence.These two concepts were introduced in [14], and in [18] in different names. If arank-distance code C is given as a subset of L n,q [ x ], then its left idealizer and rightidealizer are defined as I L ( C ) = { ϕ ∈ L n,q [ x ] : ϕ ◦ f ∈ C for all f ∈ C} , and I R ( C ) = { ϕ ∈ L n,q [ x ] : f ◦ ϕ ∈ C for all f ∈ C} , respectively. When C is an MRD-code, it is well known that all nonzero elements in I L ( C ) and I R ( C ) are invertible and each of the idealizers must be a subfield of F q n .In particular, if C is an F q n -subspace of L n,q [ x ], then I L ( C ) is isomorphic to F q n and C is said to be an F q n -MRD code . Obviously, for every MRD code C f associated LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES5 with a scattered polynomial f over F q n , its left idealizer I L ( C f ) is isomorphic to F q n .For a q -polynomial f ( x ) = P n − i =0 a i x q i over F q n , the adjoint map ˆ f of it withrespect to the bilinear form h x, y i = Tr q n /q ( xy ) isˆ f ( x ) = a x + n − X n =1 a q n − i i x q n − i . For a given scattered polynomial f over F q n , its adjoint ˆ f is a scattered polyno-mial over F q n as well, and U f and U ˆ f (and hence C f and C ˆ f ) are not necessarilyequivalent. However, they define exactly the same linear set of PG(1 , q n ); see [1, 7].To investigate the PΓL-equivalence among linear sets of a line, we need thefollowing result proved in [7]. Lemma 2.2.
Let f ( x ) = P n − i =0 α i x q i and g ( x ) = P n − i =0 β i x q i be two q -polynomialsover F q n such that L f = L g . Then α = β , and α k α q k n − k = β k β q k n − k for k = 1 , , · · · , n − , and α α qk − α q k n − k + α k α qn − α q k n − k +1 = β β qk − β q k n − k + β k β qn − β q k n − k +1 , for k = 2 , , · · · , n − . Details of Table 1.
In the following we provide details on the estimatesstated in Table 1. The number of inequivalent MRD codes defined in (i) comesfrom the well-known results on the equivalence of Delsarte-Gabidulin codes andtheir generalizations; see [19, Theorem 4.4] for instance. Moreover all monomials f ( x ) = x q s with gcd( s, n ) = 1, determine the same linear set in PG(1 , q n ); in fact, L f := n h (1 , x q s − ) i F qn : x ∈ F ∗ q n o ;the so-called linear set of pseudo-regulus type.For the equivalence of MRD codes defined by (ii), we need the following resultwhich is a special case of [19, Theorem 4.4]. Proposition 2.3.
For θ, η ∈ F q n such that N q n /q ( θ ) , N q n /q ( η ) / ∈ { , } , with ≤ s, t ≤ n − satisfying gcd( s, n ) = 1 , let f ( x ) = ηx q s + x q n − s and g ( x ) = θx q t + x q n − t .Then C f and C g are equivalent if and only if s = t and θ = η τ z q s − for some τ ∈ Aut( F q n ) and z ∈ F ∗ q n . By Hilbert’s Theorem 90, if m | n , { x ∈ F ∗ q n : N q n /q m ( x ) = 1 } = { y q m − : y ∈ F ∗ q n } . As gcd( q s − , q n −
1) = q gcd(2 s,n ) − ( q − , | n,q − , ∤ n, if N q n /q gcd(2 s,n ) ( θ ) = N q n /q gcd(2 s,n ) ( η ), then we can always find z ∈ F ∗ q n such that θ = ηz q s − . Hence, under the maps η ηz q s − for z ∈ F ∗ q n , the elements in F ∗ q n are partitioned into q gcd(2 s,n ) − θ and θ are in the same G. LONGOBARDI, G. MARINO, R. TROMBETTI, AND Y. ZHOU orbit under the maps η η τ z q s − for z ∈ F ∗ q n and τ ∈ Aut( F q n ) if and only if N q n /q gcd(2 s,n ) ( θ ) = (cid:0) N q n /q gcd(2 s,n ) ( θ ) (cid:1) τ ′ for some τ ′ ∈ Aut( F q gcd(2 s,n ) ).Note that for most choices of θ satisfying N q n /q ( θ ) / ∈ { , } , N q n /q gcd( s,n ) ( θ ) isnot in any proper subfield of F q gcd( s,n ) . Therefore, by Proposition 2.3, the numberof inequivalent MRD codes from family (ii) is approximately and less than q − r , ∤ n, q − − ( q +1)2 r , | n, where q = p r and p = char( F q ). This also provides an upper bound on the number ofPΓL-inequivalent linear sets associated with scattered polynomials in (ii). Actually,the precise value of this number could be smaller; see [9, Section 3] for n = 6 , q and n , there are exactly ϕ ( n ) / ϕ ( n ) / Construction
In this section, our goal is to prove the following main result.
Theorem 3.1.
Let n = 2 t , t ≥ and let q be an odd prime power. For each h ∈ F q n \ F q t such that h q t +1 = − , the F q -linearized polynomial (1) ψ h,t ( x ) = x q + x q t − − h − q t +1 x q t +1 + h − q t − x q t − ∈ F q n [ x ] is scattered. First we note that if t = 3 in Theorem 3.1, then the scattered polynomials ψ h, ( x )are exactly those constructed by Bartoli, Zanella and Zullo in [3].Furthermore, if we allowed h ∈ F q t , since − h q t +1 = h , then h ∈ F q . Thenwe may distinguish two cases:(a) q ≡ h ∈ F q and ψ h,t ( x ) becomes x q + x q t − − x q t +1 + x q t − . This polynomial was proven to be scattered for each t ≥ q ≡ h ∈ F q and h q = − h ; hence, t must be evenand ψ h,t ( x ) becomes x q + x q t − + x q t +1 − x q t − . This polynomial was proven in turn to be scattered in [15].Now we note that polynomials described in (1) can be rewritten in the followingfashion:(2) ψ h,t ( x ) = L ( x ) + M ( x ) , where L ( x ) = x q − h − q t +1 x q t +1 and M ( x ) = x q t − + h − q t − x q t − .It is straightforward to see that L ( x ) and M ( x ) are F q t -semilinear maps of F q n with companion automorphisms x x q and x x q t − , respectively. Moreover, wehave that(3) ker L = { x ∈ F q n : x − h q t − − q t x q t = 0 } LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES7 and similarly(4) ker M = { x ∈ F q n : x + h q t +1 − q t x q t = 0 } . In addition, since h q t +1 = −
1, we have L ( x ) q t = ( x q − h − q t +1 x q t +1 ) q t = x q t +1 − h q t − q x q = − h q t − q ( x q − h − q t +1 x q t +1 ) = − h q t − q L ( x )and similarly, we may prove that M ( x ) q t = h q t − q t − M ( x ). Hence, we obtain that(5) im L = { z ∈ F q t : z q t + h q t − q z = 0 } and(6) im M = { z ∈ F q t : z q t − h q t − q t − z = 0 } . Clearly, the sets in (3), (4), (5) and (6) are 1-dimensional F q t -subspaces of F q n . Proposition 3.2.
Let n = 2 t , t ≥ and let h be in F q t such that h q t +1 = − .Then h q +1 = 1 and h q t − = − h .Proof. First, as q is odd, gcd( q + 1 , q t −
1) = 2 if t is odd, andgcd( q + 1 , q t + 1) = ( , t ≡ q + 1 , t ≡ . Assume on the contrary that h q +1 = 1. Together with h q t +1 = − t is odd, then h = 1 which contradicts h q t +1 = −
1. If 4 | t , then h = − h q +1 = − | t but 4 ∤ t , then h q t +1 = 1 contradicting h q t +1 = − h q t +1 = − h q +1 = 1, we finally derive h q t − = − h directly. (cid:3) Proposition 3.3.
Let n = 2 t with t ≥ . The finite field F q n , seen as F q t -vectorspace, is both the direct sum of ker L and ker M , and of im L and im M .Proof. Since ker L and ker M are 1-dimensional F q t -subspaces of F q n ; it is enoughto prove that ker L ∩ ker M = { } . In this regard, let u ∈ ker L ∩ ker M . By(3) and (4), we get h q t − = − h q t +1 , i.e. ( h q t − ) q t +1 = − h q t +1 . This implies that h q t − = − h , and since h q t +1 = − F q n , seen as F q t -vector space, can be also written as im L ⊕ im M . (cid:3) Consider now the following F q t -linear maps of F q n (7) R ( x ) = x q t + h q t − − q x and T ( x ) = x q t + h q − q t − x. It is straightforward to see that dim F qt ker R = dim F qt ker T = 1; moreover, ker T = h q t − − q ker R . Lemma 3.4.
Let ρ, τ ∈ F ∗ q n , n = 2 t and t ≥ such that ρ ∈ ker R and τ ∈ ker T .Then(i) { , ρ } and { , τ } are F q t -bases of F q n . G. LONGOBARDI, G. MARINO, R. TROMBETTI, AND Y. ZHOU (ii) if τ = h q t − − q ρ and an element γ ∈ F q n has components ( λ , µ ) in the F q t -basis { , ρ } , then the components of γ in { , τ } are (8) (cid:16) λ + µ ρ (cid:16) − h q t − − q (cid:17) , µ (cid:17) . Proof. ( i ) It is enough showing that ρ and τ are not in F q t . We will show that ρ F q t . A similar argument can be applied to τ as well. Suppose that ρ ∈ F q t ,then ρ q t − = 1. Then, by hypothesis,1 = ρ q t − = − h q t − − q . Hence h q t − = − h which, by Proposition 3.2, is not the case.( ii ) Let γ ∈ F q n and suppose that γ = λ + µ ρ with λ , µ ∈ F q t . Also, denote by λ and µ the components of γ in the F q t -basis { , τ } . Of course, we have(9) λ + µ τ = λ + µ ρ Raising (9) to the q t -th power, and taking into account that ρ ∈ ker R and τ ∈ ker T ,we get the following linear system in the unknowns λ and µ ( λ + µ τ = λ + µ ρλ − µ h q − q t − τ = λ − µ h q t − − q ρ. Clearly, this linear system has a unique solution; i.e., λ = λ + µ ρ (cid:16) − h q t − − q (cid:17) and µ = µ . Hence, the assertion follows. (cid:3)
Proposition 3.5.
For any nonzero vectors u ∈ ker L , v ∈ ker M and any a ∈ F q n ,the following statements are equivalent:(i) a ∈ ker R ;(ii) av ∈ ker L ;(iii) aM ( u ) ∈ im L .Proof. Clearly, if a is zero the statement is trivially verified. Suppose that a ∈ F ∗ q n .Let ρ be a nonzero vector in ker R which means ker R = h ρ i q t .( i ) ⇒ ( ii ). Let a ∈ h ρ i q t , then there exists λ ∈ F q t such that a = λρ . Then L ( av ) = λ q L ( ρv ) = λ q (cid:16) ( ρv ) q − h − q t +1 ( ρv ) q t +1 (cid:17) . Since ρ ∈ ker R and v ∈ ker M , by (4) and (7), we get( λρv ) q (cid:16) − h − q t +2 +1+ q t − q (cid:17) . Moreover, since h q t +1 = −
1, the latter expression is equal to 0; hence, av ∈ ker L .( ii ) ⇒ ( iii ) Let v ∈ ker M . Since av ∈ ker L ,0 = L ( av ) = ( av ) q − h − q t +1 ( av ) q t +1 = v q (cid:16) a q + h − q t +2 a q t +1 (cid:17) , this is(10) a + h q t − − q t +1 a q t = 0 . We will prove that aM ( u ) ∈ im L . So, putting z = M ( u ), by (5), this is equivalentto prove that ( az ) q t + h q t − q ( az ) = 0 . LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES9 By Proposition 3.3, since u ∈ ker L , z = M ( u ) = 0. Also, since h q t +1 = −
1, by (6),we have ( az ) q t + h q t − q ( az ) = zh q t − q · (cid:16) a q t h q t − − q t +1 + a (cid:17) and by (10) the last expression equals 0, proving the result.( iii ) ⇒ ( i ) As before, by Proposition 3.3, z = M ( u ) is a nonzero element of im M .Since az ∈ im L , by (5) and (6), we obtain0 = ( az ) q t + h q t − q ( az ) = z (cid:16) a q t h q t − q t − + h q t − q a (cid:17) = h q t − q t − z (cid:16) a q t + h q t − − q a (cid:17) , which implies a q t + h q t − − q a = 0. Then, by (7), a ∈ ker R . Finally, since ker R is a1-dimensional F q t -subspace of F q n , a = λρ for some λ ∈ F q t . (cid:3) Similarly, we have the following result.
Proposition 3.6.
For any nonzero vectors u ∈ ker L , v ∈ ker M and any b ∈ F q n ,the following statements are equivalent:(i) b ∈ ker T ;(ii) b u ∈ ker M ;(iii) bL ( v ) ∈ im M .Proof. As before, we may suppose that b ∈ F ∗ q n , otherwise the statement is easilyverified. Let τ be a nonzero vector in ker T which means ker T = h τ i q t .( i ) ⇒ ( ii ). Let b ∈ h τ i q t , then there exists λ ∈ F q t such that b = λτ . Consider M ( bu ) = M ( λτ u ). Then M ( bu ) = λ q t − M ( τ u ) = λ q t − (cid:16) ( τ u ) q t − + h − q t − ( τ u ) q t − (cid:17) . Since τ ∈ ker T , u ∈ ker L , by (3) and (7), we get( λτ u ) q t − (cid:16) − h − q t − + q t − q t − (cid:17) . Moreover, since h q t +1 = −
1, the latter expression is equal to 0; hence, bu ∈ ker M .( ii ) ⇒ ( iii ) Let u ∈ ker L . Since bu ∈ ker M ,0 = M ( bu ) = ( bu ) q t − + h − q t − ( bu ) q t − = u q t − ( b q t − + h − q t − b q t − ) , this is(11) b q t + h q − q t − b = 0 . We will prove that bL ( v ) ∈ im M . So, putting z = L ( v ), by (6), this is equivalentto prove that ( bz ) q t − h q t − q t − ( bz ) = 0 . By Proposition 3.3, since v ∈ ker M then z = L ( v ) ∈ im L and L ( v ) = 0. Since h q t +1 = −
1, by (5) we get( bz ) q t − h q t − q t − ( bz ) = − zh q t − q ( b q t + h q − q t − b )which equals 0 taking (11) into account, proving the result.( iii ) ⇒ ( i ) As before, by Proposition 3.3, z = L ( v ) is a nonzero element of im L .Since bz ∈ im M , by (5) and (6), we obtain0 = ( bz ) q t − h q t − q t − ( bz ) = − zh q t − q ( b q t + h q − q t − b ) , this is b q t + h q − q t − b = 0. By (7), then b ∈ ker T . Since ker T is a 1-dimensional F q t -subspace of F q n , b = λτ for some λ ∈ F q t . (cid:3) We are now in the position to prove our main result of this section.
Proof of Theorem 3.1.
Let ψ ( x ) := ψ h,t ( x ), we want to prove that for each x ∈ F ∗ q n and for each γ ∈ F q n such that(12) ψ ( γx ) = γψ ( x )we get γ ∈ F q . Recall that ψ ( x ) = L ( x ) + M ( x )as in (2). Also, by Proposition 3.3, any x ∈ F q n can be uniquely written as x = x + x , where x ∈ ker L and x ∈ ker M . Similarly, by Lemma 3.4, if γ ∈ F q n there are exactly two elements λ , µ ∈ F q t and two elements λ , µ ∈ F q t such that λ + µ ρ = γ = λ + µ τ where ρ ∈ ker R and τ = h q t − − q ρ . It is easy to check that τ ∈ ker T . Putting a = µ ρ and b = µ τ , which imply a ∈ ker R and b ∈ ker T , Condition (12) may bere-written as follows L (( λ + a )( x + x ))+ M (( λ + b )( x + x )) == ( λ + b ) L ( x + x ) + ( λ + a ) M ( x + x ) . (13)Also, since x ∈ ker L , x ∈ ker M , L ( x ) and M ( x ) are F q t -semilinear maps and by( ii ) of Propositions 3.5 and ( ii ) of Proposition 3.6, Equation (13) is equivalent to L ( λ x ) + L ( ax ) + M ( λ x ) + M ( bx ) == λ L ( x ) + bL ( x ) + λ M ( x ) + aM ( x ) . and hence λ q L ( x ) + L ( ax ) − λ L ( x ) − aM ( x ) == bL ( x ) + λ M ( x ) − λ q t − M ( x ) − M ( bx ) . (14)Now, since the image spaces of the maps L ( x ) and M ( x ) are F q t -spaces, taking( iii ) of Proposition 3.5 and ( iii ) of Proposition 3.6 into account, the expressionson left and right hand sides of (14) belong to im L and im M , respectively. ByProposition 3.3, both sides of (14) must be equal to zero an hence we obtain thefollowing system ( L ( ax ) − aM ( x ) = ( λ − λ q ) L ( x ) bL ( x ) − M ( bx ) = ( λ q t − − λ ) M ( x ) . Raising to the q -th power the second equation, we get(15) ( L ( ax ) − aM ( x ) = ( λ − λ q ) L ( x ) b q L ( x ) q − M ( bx ) q = ( λ − λ q ) M ( x ) q . Since a = µ ρ , b = µ τ and τ = h q t − − q ρ , from Lemma 3.4 it follows that µ = µ and b = h q t − − q a .If a = 0, we have µ = 0 and hence γ = λ = λ ∈ F q t . Also, from (15), if λ = λ q , then L ( x ) = M ( x ) = 0. By Proposition 3.3, x = x = x = 0, acontradiction. Then λ = λ = λ q , which gives λ ∈ F q , i.e. γ ∈ F q and ψ ( x ) is ascattered polynomial. LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES11 In the remainder of the proof, we are going to show that a = 0, i.e. γ ∈ F q t \ F q t leads to contradictions. Depending on the value of x and x , we separate the proofinto three cases. Case 1. x = 0. The system in (15) is reduced to ( ( λ − λ q ) L ( x ) = 0 b q L ( x ) q − M ( bx ) q = 0 . Since x ∈ ker M , by (7), from second equation we get b q − = h q − (cid:0) x (1 + h q − q t − ) (cid:1) q − . Then, there exists λ ∈ F q ∗ such that b = λ · h (cid:0) x (1 + h q − q t − ) (cid:1) q +1 . Since b ∈ ker T , then b q t + h q − q t − b = 0 and we get h q t − (cid:0) x (1 + h q − q t − ) (cid:1) q t ( q +1) + h q − q t − (cid:0) x (1 + h q − q t − ) (cid:1) q +1 = 0 , whence, since x ∈ ker M , h q − q t − h q − (1 + h q t − − q ) ! q +1 = − h q − q t − − q t . This is equivalent to h q t − − (1 + h q − q t − ) h q − (1 + h q t − − q ) ! q +1 = − , whence we have 1 q +1 = −
1, a contradiction.
Case 2. x = 0. The system in (15) is reduced to ( L ( ax ) − aM ( x ) = 0( λ − λ q ) M ( x ) q = 0 . By the first equation, taking into account that a ∈ ker R , x ∈ ker L and h q t +1 = −
1, we obtain a q − = ( x q ) q t − − · h − q t − h q t +2 − = (cid:16) x q (1 + h q t +2 − ) (cid:17) q t − − . Then there exists λ ∈ F q ∗ such that a = λ (cid:16) x q (1 + h q t +2 − ) (cid:17) ν , where ν = ( q t − − / ( q − a ∈ ker R , then (cid:16) x q t +1 (1 + h q − q t ) (cid:17) ν + h q t − − q (cid:16) x q (1 + h q t +2 − ) (cid:17) ν = 0 . Moreover, since x ∈ ker L , then h q t +1 − (1 + h q − q t )1 + h q t − q ! ν = − h q t − − q . The last expression is equivalent to h q t − q (1 + h q − q t ) h q − q (1 + h q t − q ) ! ν = − , whence 1 ν = −
1, leading to a contradiction.
Case 3. x , x = 0. Recall that a ∈ ker R , b = h q t − − q a , λ = λ + (1 − h q t − − q ) a , x ∈ ker L and x ∈ ker M . Then, by (15), a turns out to be a nonzero solution ofthe following linear system(16) x q (1 + h q t − q ) a q − (cid:16) M ( x ) + (1 − h q t − − q ) L ( x ) (cid:17) a = ( λ − λ q ) L ( x ) h q t − q L ( x ) q a q + (cid:16) x q t (1 + h q t − − q ) − (1 − h q t − − q ) M ( x ) q (cid:17) a = ( λ − λ q ) M ( x ) q . By x ∈ ker L and x ∈ ker M , we obtain the following two equations which willbe frequently used later, L ( x ) = x q (1 + h − q t +2 ) ,M ( x ) = x q t − (1 + h − q t − ) . - Case 3.1 First of all, suppose that λ ∈ F q , then System (16) becomes(17) x q (1 + h q t − q ) a q − (cid:16) M ( x ) + (1 − h q t − − q ) L ( x ) (cid:17) a = 0 h q t − q L ( x ) q a q + (cid:16) x q t (1 + h q t − − q ) − (1 − h q t − − q ) M ( x ) q (cid:17) a = 0 . and since a is a nonzero solution then x q (1 + h q t − q ) (cid:16) x q t (1 + h q t − − q ) − (1 − h q t − − q ) M ( x ) q (cid:17) == − h q t − q L ( x ) q (cid:16) M ( x ) + (1 − h q t − − q ) L ( x ) (cid:17) . (18)Since L ( x ) = 0 = M ( x ), from (17) we get M ( x ) q (cid:16) x q (1 + h q t − q ) a q − M ( x ) a (cid:17) = L ( x ) (cid:16) h q t − q L ( x ) q a q + x q t (1 + h q t − − q ) a (cid:17) whence (cid:16) x q M ( x ) q (1 + h q t − q ) − h q t − q L ( x ) L ( x ) q (cid:17) a q == (cid:16) M ( x ) M ( x ) q + x q t (1 + h q t − − q ) L ( x ) (cid:17) a. (19)Next we want to show that the coefficient of a q in (19) cannot be 0. By way ofcontradiction, suppose that(20) x q (cid:16) h q t − q (cid:17) M ( x ) q = h q t − q L ( x ) L ( x ) q , LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES13 from (18) it follows(21) x q x q t (cid:16) h q t − q (cid:17) (cid:16) h q t − − q (cid:17) = − h q t − q L ( x ) q M ( x ) . Since x ∈ ker L and x ∈ ker M , this is equivalent to(22) x q t − − = − ( x q ) q t − − (1 + h q − q t − )(1 + h q t − − )(1 + h − q t +2 )(1 + h q t − − q t − ) . This formula is equivalent to(23) x q t − − = − x q · h − q t +2 (1 + h q − q t − ) h q t − ! q t − − . Since d = gcd (2 t, t −
2) = gcd(4 , t −
2) = , if t odd2 , if t ≡ , if t ≡ x q t − − = − t ≥ t F q n . Thus, for t ≡ x = ωx q · h − q t +2 (1 + h q − q t − ) h q t − , for some ω ∈ F ∗ q n satisfying ω q t − − = −
1. By substituting this expression in (20),since x ∈ ker L , we get(24) ω q +1 = − , and hence ω ∈ F q .If t is odd, since ω q t − = − ω , then ω q = − ω and from (24), we get ω = ±
1. If t ≡ ω = ω q t − = − ω . In both cases we get a contradiction.Then, by (19), we get a q − = M ( x ) M ( x ) q − h q t − q t +1 x L ( x )(1 + h q t − − q ) x q (1 + h q t − q ) M ( x ) q − h q t − q L ( x ) L ( x ) q == h q − · h q t − − q h q t − q · x M ( x ) − x L ( x ) x q M ( x ) q − x q L ( x ) q == h (1 + h q t − − q )( x M ( x ) − x L ( x )) ! q − , (25)whence a = λ · h (1 + h q t − − q )( x M ( x ) − x L ( x ))for some λ ∈ F ∗ q . Since a ∈ ker R , then h q t (cid:0) (1 + h q t − − q )( x M ( x ) − x L ( x )) (cid:1) q t + h q t − − q · h (1 + h q t − − q )( x M ( x ) − x L ( x )) = 0 . Recalling that x ∈ ker L and x ∈ ker M , we get x M ( x ) − x L ( x ) = x q t − +11 (cid:16) h − q t − (cid:17) − x q +12 (cid:16) h − q t +2 (cid:17) . This implies that − h q t (1 + h q − q t − ) + h q t − − q · h h q t − − q = 0 , which means h q t +1 = 1, a contradiction. Then λ may not belong to F q .- Case 3.2
Let λ / ∈ F q and let a be a nonzero solution of System (16). If thissystem admits more than one solution, then each 2 × a, a q ) ∈ F q n . Bycomputing the ratio a q − of its components, we get a q − = (cid:12)(cid:12)(cid:12)(cid:12) L ( x ) − M ( x ) M ( x ) q x q t (1 + h q t − − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q (1 + h q t − q ) L ( x ) h q t − q L ( x ) q M ( x ) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) == M ( x ) M ( x ) q − h q t − q t +1 x L ( x )(1 + h q t − − q ) x q (1 + h q t − q ) M ( x ) q − h q t − q L ( x ) L ( x ) q . This is again Equation (25). Repeating the arguments as in
Case 3.1 we get acontradiction. (cid:3) A new family of MRD codes
Let ψ h,t be defined as in Theorem 3.1 and(26) C h,t = { ax + bψ h,t ( x ) : a, b ∈ F q n } . By a result in [23], C h,t is an F q n -linear MRD code.The following result is about the equivalence among C h,t ’s for different h and theautomorphism group of C h,t . Theorem 4.1.
Let n = 2 t with t > . For each h, k ∈ F q n satisfying h q t +1 = k q t +1 = − , the following hold(a) if t , then C h,t and C k,t are equivalent if and only if h = ± k ;(b) if t ≡ , then C h,t and C k,t are equivalent if and only if h = ℓk where ℓ q +1 = 1 .Furthermore, the automorphism group Aut( C h,t ) of C h,t is isomorphic to the multi-plicative group of F q .Proof. Let U h = { ( x, ψ h,t ( x )) : x ∈ F q n } and U k = { ( x, ψ k,t ( x )) : x ∈ F q n } . ByTheorem 2.1, we only have to consider the ΓL(2 , q n )-equivalence U h and U k .Without loss of generality we may always suppose that ρ ∈ Aut( F q n ) is theidentity. Hence, we only have to consider the existence of invertible matrix (cid:18) a bc d (cid:19) over F q n such that for each x ∈ F q n there exists y ∈ F q n satisfying (cid:18) a bc d (cid:19) (cid:18) xψ h,t ( x ) (cid:19) = (cid:18) yψ k,t ( y ) (cid:19) . LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES15 This is equivalent to(27) cx + dψ h,t ( x ) = ψ k,t ( ax + bψ h,t ( x )) , for all x ∈ F q n . The right-hand-side of (27) is (cid:16) b q + k − q t +1 b q t +1 h q t +1 − q (cid:17) x q + (cid:16) b q t − h q t − − q t − + b q t − k − q t − (cid:17) x q t − ++ (cid:16) b q + b q t − − h q t +1 − q t k − q t +1 b q t +1 − k − q t − h q t − − q t b q t − (cid:17) x q t ++ (cid:16) b q t − + k − q t − h q t − − q t − b q t − (cid:17) x q t − − (cid:16) h q − q t +2 b q + k − q t +1 b q t +1 (cid:17) x q t +2 ++ (cid:16) b q h q − − b q t − h q t − − − k − q t +1 b q t +1 + k − q t − b q t − (cid:17) x + ψ k,t ( ax ) . As t >
4, it is easy to see that the coefficients of x q , x q t − , x q t , x q t +2 and x q t − in the right-hand-side of (27) must be 0. Depending on whether the value of b equals 0 or not, we separate the proof into two cases. Case 1. b = 0. By the coefficient of x q (or equivalently, by the coefficient of x q t +2 ), we get(28) b q t +1 − q = − k − q − h q + q . Similarly, by the coefficient of x q t − (or equivalently, by the coefficient of x q t − ),we get(29) b q t +1 − q = − k − q + q h q − . By (28) and (29), we obtain h q +1 = k q +1 . Let ℓ = h/k . Then ℓ q +1 = 1. By the assumption that h q t +1 = k q t +1 = −
1, wehave ℓ q t +1 = 1.If t ℓ = ±
1; if t ≡ ℓ q +1 = 1,which implies ℓ ∈ F q . Case 2. b = 0. If (27) holds, then c = 0 and(30) d = a q = a q t − dh − q t +1 = k − q t +1 a q t +1 dh − q t − = k − q t − a q t − . The first equation in (30) implies that a ∈ F q gcd(2 t,t − which means a ∈ F q gcd( t − , .Let ℓ = h/k . The last two equations in (30) become(31) dℓ − q t +1 = a q = dℓ − q t − . Thus ℓ q t − − q t +1 = 1. This means ℓ ∈ F q gcd( t − , t ) = F q gcd( t − , . By the assumptionthat h q t +1 = k q t +1 = −
1, we have ℓ q t +1 = 1. If gcd( t − , ∈ { , } , then ℓ = ± t − ,
4) = 4, then we obtain ℓ q +1 = 1.Next, let us handle the case t ≡ a q − q = ℓ − q t +1 = ℓ − q = ℓ (1 − q )( q + q +1) = ℓ q − q = ℓ q +1 . For a given ℓ ∈ F q , we can always find a ∈ F q satisfying the above equation.Moreover, it is routine to verify that such a satisfies (30) provided that h = ℓk with ℓ q +1 = 1; note here that by (30) d depends on a . This complete the proof ofequivalence between C h,t and C k,t . Finally, we determine the automorphism group of C h,t . Depending on whether b = 0 or not, there are two types of elements in it.When b = 0, we only have to let k = h in (30). From there we derive that d = a q = a q t − = a q t +1 . Therefore a ∈ F ∗ q gcd( t, .When b = 0, by letting k = h , we see that the coefficient of x q t in the right-hand-side of (27) is b q + b q t − − h − q t b q t +1 − h − q t b q t − which must be 0. Plugging (28) into it and taking into account that h q t = − /h ,we get b q (1 + h q − q t ) + b q t − (1 + h − q t − ) = 0 . Raising it to the q -th power and plugging (28) again, we obtain b q (1 − h q + q ) − b q ( h q − − h q ) = 0 , which means b q ( h − − h q ) = (cid:16) b q ( h − − h q ) (cid:17) q . By Proposition 3.2, h − = h q . Thus b = − δh − q t − − h q = δh q t − + h q for some δ ∈ F q . Substitute it into (28), δ q t h q t − + h q t +1 − h q + q t − δh q t − + h q = 0 , which means δ q t + δ = 0 . When t is even, it means 2 δ = 0. Thus b = 0.When t is odd, it implies δ q = − δ . Plugging this value back into (27), we seethat the coefficients of x q , x q t − , x q t , x q t +2 and x q t − on the right-hand-side of(27) are all 0. Hence, the coefficient of x on the left-hand-side of (27) is c which iscompletely determined by b , and a can take any value in F q when b = 0. Moreover,the value of d is determined by a which is independent of the value of b . One maycheck directly by computation that they form a cyclic group of order q −
1, oruse the fact that the nonzero elements of the right idealizer of an MRD code formthe multiplicative group of a finite field to conclude the proof of the automorphismgroup of C h,t . (cid:3) Theorem 4.1 shows that our construction provides a big family of inequivalentMRD codes.
Corollary 4.2.
For a given q and t > , the total number N of inequivalent MRDcodes C h,t is N = q t +12 , if t q t +1 q +1 , if t ≡ . Proof.
First, there are totally q t + 1 elements in F q t satisfying h q t +1 = −
1. ByTheorem 4.1, we get the results immediately. (cid:3)
LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES17 As f f ◦ ϕ defines an automorphism of C h,t for each nonzero element ϕ ∈ I R ( C h,t ), the following result is a direct consequence of Theorem 4.1. Corollary 4.3.
Let C h,t be defined as in (26) , with t > . Then I R ( C h,t ) ∼ = F q . Hence, we can prove the following result.
Theorem 4.4.
Let n = 2 t with t > and let q be an odd prime power. The familyof F q n -MRD codes of minimum distance n − C h,t = { ax + bψ h,t ( x ) : a, b ∈ F q n } , where ψ h,t ( x ) = x q + x q t − − h − q t +1 x q t +1 + h − q t − x q t − ∈ F q n [ x ] and h is anyelement of F q n such that h q t +1 = − , is new.Proof. As q can be any odd prime power and n can be any even integer larger than 6,by comparing Corollary 4.2 with the numbers of known inequivalent constructionsof F q n -MRD codes of minimum distance n − (cid:3) Finally, let us investigate the adjoint of ψ h,t and the associated MRD code. Theorem 4.5.
The F q n -MRD ˆ C h,t = { ax + b ˆ ψ h,t ( x ) : a, b ∈ F q n } , where ˆ ψ h,t isthe adjoint map of ψ h,t is equivalent to C h,t .Proof. Consider the polynomial g ( x ) := h ˆ ψ h,t ( x/h ) = x q − x q t − + h − q t +1 x q t +1 + h − q t − x q t − , we investigate the equivalence between C g and C h,t .To prove that C g is equivalent to C h,t , as in the proof of Theorem 4.1, we onlyhave to consider the existence of invertible matrix (cid:18) a bc d (cid:19) over F q n such that foreach x ∈ F q n there exists y ∈ F q n satisfying (cid:18) a bc d (cid:19) (cid:18) xg ( x ) (cid:19) = (cid:18) yψ h,t ( y ) (cid:19) . This is equivalent to(32) cx + dg ( x ) = ψ h,t ( ax + bg ( x )) , for all x ∈ F q n .Let a = d = 0. We only have to check whether there exist nonzero b and c suchthat cx = ψ h,t ( bg ( x )) . The right-hand-side of it is (cid:16) b q − h − q b q t +1 (cid:17) x q + (cid:16) b q t − h q t − − q t − − b q t − h − q t − (cid:17) x q t − ++ (cid:16) − b q + b q t − − h − q t b q t +1 + h − q t b q t − (cid:17) x q t ++ (cid:16) − b q t − + h − q t − b q t − (cid:17) x q t − + (cid:16) h q − q t +2 b q − h − q t +1 b q t +1 (cid:17) x q t +2 ++ (cid:16) b q h q − + b q t − h q t − − + h − q t +1 b q t +1 + h − q t − b q t − (cid:17) x. Choose z ∈ F q and let b = (cid:16) zhh q − (cid:17) q t − . Since z q t − = 1, we get( b q ) q t − = z q t − h q t − ( h q +1 − h q t +2 + q t − z q t − h q t + q = h q − , which means b q t +1 = h q − b q .It is easy to verify that the coefficients of the terms of degree q , q t − , q t − and q t +2 in ψ h,t ( bg ( x )) all equal 0.The computation of the coefficient of x q t is a bit more complicated. It equals − b q + b q t − − h − q t b q t +1 + h − q t b q t − == − b q + b q t − + h b q t +1 − h b q t − == − b q + b q t − + h q +1 b q − h q t − +1 b q t − == b q ( h q +1 − − b q t − ( h − q − − q t − == b q ( h q +1 − − b q t − (cid:16) h q +1 − (cid:17) q t − h − q t − . It equals 0 if and only if b q h ( h q +1 −
1) = (cid:18) b q h ( h q +1 − (cid:19) q t − , which holds because b q h ( h q +1 −
1) = z ∈ F q ⊆ F q t − ∩ F q t .Therefore, we have shown that there exist c and b such that cx = ψ h,t ( bg ( x )),which means that C g and C k,t are equivalent. (cid:3) Equivalence of the associated linear sets
Let ψ h,t be the scattered polynomial over F q n , n = 2 t , defined in Theorem 3.1.Let(33) L h,t := (cid:8) h ( x, ψ h,t ( x )) i F qn : x ∈ F ∗ q n (cid:9) , which is a maximum scattered linear set of PG(1 , q n ).In this part, we consider the PΓL-equivalence between L h,t and L k,t . Our mainresult shows that there is a large number of inequivalent maximum scattered linearsets associated with our family of scattered linear sets. Theorem 5.1.
Let p be an odd prime number and let r, t be positive integers with t > and q = p r . The total number M of inequivalent maximum scattered linearsets L h,t of PG(1 , q n ) , n = 2 t , satisfies M ≥ j q t +18 rt k , if t j q t +14 rt ( q +1) k , if t ≡ . To prove Theorem 5.1, we first restrict to the equivalence of linear sets underPGL(2 , q n ) and consider two cases which will be handled in Lemma 5.2 and Lemma5.3, respectively. Then we will consider the PΓL(2 , q n )-equivalence and present theproof of Theorem 5.1. LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES19 Let f ( x ) = P n − i =0 α i x q i and g ( x ) = P n − i =0 β i x q i be two scattered polynomialsover F q n with α = β = 0. The associated linear sets L f and L g are PGL(2 , q n )-equivalent if and only if there exists an invertible matrix (cid:18) a bc d (cid:19) over F q n suchthat(34) (cid:26) f ( x ) x : x ∈ F ∗ q n (cid:27) = (cid:26) cx + dg ( x ) ax + bg ( x ) : x ∈ F ∗ q n (cid:27) . Depending on whether the value of b equals 0 or not, we may consider the equiva-lence in two cases.If b = 0, then we can assume that a = 1, and (34) becomes ( n − X i =0 α i x q i − : x ∈ F ∗ q n ) = ( c + d n − X i =0 β i x q i − : x ∈ F ∗ q n ) . As α = β = 0, by Lemma 3.6 in [7], c must be 0. Hence(35) ( n − X i =1 α i x q i − : x ∈ F ∗ q n ) = ( d n − X i =1 β i x q i − : x ∈ F ∗ q n ) . Lemma 5.2.
Suppose f = ψ h,t , g = ψ k,t with t > . If there exists d such that (35) holds, then ( h/k ) q +1 = 1 which means U h is ΓL(2 , q n ) -equivalent to U k .Proof. By Lemma 2.2, α j α q j n − j = d q j +1 β k β q j n − j for j = 1 , , · · · , n −
1. Plugging the coefficients of ψ h,t and ψ k,t , we get( h − q t − ) q = d q +1 ( k − q t − ) q and ( − h − q t +1 ) q t − = d q t − +1 ( − k − q t +1 ) q t − . By setting ℓ = h/k , we get(36) ( ℓ q − = d q +1 ,ℓ q t − − = d q t − +1 . From (36), we derive d ( q +1)( q t − + q t − + ··· +1) = ℓ q t − − = d q t − +1 , which means d q ( q t − + ··· +1) = 1. Hence d q t − − = d q t − + ··· +1) · q − = 1. It followsthat ℓ q t − − = d q t − +1 = d q +1 = ℓ q − , whence ℓ q t − = ℓ .As h q t +1 = k q t +1 = − ℓ q t = 1 /ℓ . By ℓ q t − = ℓ , we get ℓ q +1 = 1. By Theorem4.1, U h is ΓL(2 , q n )-equivalent to U k . (cid:3) Next we consider the case b = 0. Without loss of generality, we assume that b = 1. Then cx + dg ( x ) ax + g ( x ) = ( c − da ) x + d ( ax + g ( x )) ax + g ( x ) = ¯ cxax + g ( x ) + d, for any x ∈ F ∗ q n , where ¯ c = c − da = 0. Noting that ax + g ( x ) = 0 must have nononzero solution, ¯ cxax + g ( x ) + d = ¯ c ¯ g ( y ) y + d, where ¯ g ( y ) = P n − i =0 γ i y q i is the inverse of the map x ax + g ( x ). Furthermore, (34) becomes ( n − X i =1 α i x q i − : x ∈ F ∗ q n ) = ( ¯ c n − X i =1 γ i x q i − + d + ¯ cγ : x ∈ F ∗ q n ) . By Lemma 2.2,(37) d + ¯ cγ = 0 . Thus(38) ( c n − X i =1 α i x q i − : x ∈ F ∗ q n ) = ( n − X i =1 γ i x q i − : x ∈ F ∗ q n ) . Lemma 5.3.
Let f = ψ h,t , g = ψ k,t with t > . Suppose that there exist a , c and d such that (38) holds.(a) If t is even, then a must be .(b) If t is odd and a = 0 , then f = g .In particular, when a = 0 , γ = d = 0 .Proof. By the second identity in Lemma 2.2, we know that(39) γ j γ t − j = 0 , for j ∈ { , , · · · , t − } \ { , t − , t + 1 , t − } ,(40) γ γ q t − = (cid:18) c (cid:19) q +1 τ q , and(41) γ t − γ q t − t +1 = (cid:18) c (cid:19) q t − θ q t − , where τ = h − q t − and θ = − h − q t +1 = h q .By the third identity in Lemma 2.2, we have(42) γ γ qj − γ q j t − j + γ j γ q t − γ q j t − j +1 = 0 , for j ∈ { , , · · · , n − } . Letting j = 2, we obtain γ γ q γ q t − + γ γ q t − γ q t − = 0 . As γ γ t − = 0 and γ γ t − = 0, we derive γ = γ t − = 0. Similarly, by letting j = t −
1, we have γ γ qt − γ q t − t +1 + γ t − γ q t − γ q t − t +2 = 0 . Since γ t − γ t +2 = 0 and γ , γ t +1 , γ t − , γ t +1 = 0, from the above equation we deduce γ t − = γ t +2 = 0 . Moreover, by (39), (42) and replacing j − j in (42),(43) γ j = 0 ⇒ γ t − j = γ t − j +1 = γ t − j − = 0 , for j ∈ { , , · · · , t − } \ { , t − , t + 1 , t − } .Now, we will use the fact that ¯ g ( ax + g ( x )) = x , namely,(44) ¯ g ( ax + x q + x q t − + ux q t +1 + vx q t − ) = x, LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES21 for all x ∈ F q n , where u = − k − q t +1 and v = k − q t − . The coefficient of x q j in theleft-hand-side of (44) is(45) a q j γ j + γ j − + γ j + t +1 + u q j + t − γ j + t − + v q j +1 γ j +1 . By letting j = 0 ,
1, 2 t − t + 1, t − t in (45) and comparing it with theright-hand-side of (44), we get aγ + γ t − + γ t +1 + u q t − γ t − + v q γ = 1 , (46) a q γ + γ = 0 , (47) a q t − γ t − + vγ = 0 , (48) a q t +1 γ t +1 + uγ = 0 , (49) a q t − γ t − + γ = 0 , (50) γ t − + γ + u q t − γ t − + v q t +1 γ t +1 = 0 . (51)Here we have used the result that γ t = γ = γ t − = γ t +2 = γ t − = 0.It is clear that if a = 0, then γ must be 0. By (37), d = 0.Assume that a = 0. By (47), (48), (49) and (50) into (46) and (51), we see that γ = 0 is completely determined by a , and − γ a q t − + 1 a q + u q t − va q t − + v q t +1 ua q t +1 ! = 0 , respectively. Recall that u = − k − q t +1 , v = k − q t − and k q t = − /k , from theabove equation we deduce 1 a q t − + 1 a q + k a q t − + k a q t +1 = 0 . Therefore(52) 1 a q t − + 1 a q = − k (cid:18) a q t − + 1 a q (cid:19) q t . Our goal of the next step is to prove(53) 1 a q t = − k q − q t − a , always holds.Let j = 2 , t + 2 , t − , t − γ + γ t +3 + u q t +1 γ t +1 + v q γ = 0 , (54) γ t +1 + γ + u q γ + v q t +3 γ t +3 = 0 , (55) γ t − + γ t − + u q t − γ t − + v q t − γ t − = 0 , (56) γ t − + γ t − + u q t − γ t − + v q t − γ t − = 0 . (57)Depending on the value of γ , γ t +3 , γ t − and γ t − , we separate the proof of(53) into four different cases. Case (i) . γ = γ t +3 = 0. By (47), (49) and (54), − ua q t +1 γ = γ t +1 = − u q γ = u q a q γ , which means a qt = − u − q t − a = − k q − q t − a . Case (ii) . γ t − = γ t − = 0. By a similar computation of (56) as in Case (i),we get (53) again. Case (iii) . γ = 0 and γ t − = 0. By (43), γ t − = γ t − = γ t +3 = γ t +4 = 0.Now, (54) and (55) become γ + k − q − q γ t +1 + k q − q γ = 0 ,γ + γ t +1 + k q + q γ = 0 . Canceling γ , we get(1 − k q + q ) γ t +1 = − (1 − k q + q ) k q + q γ . By Proposition 3.2, k q +1 = 1. Hence, γ t +1 = − k q + q γ . By plugging (47) and(49) into it, we derive (53). Case (iv) . γ t +3 = 0 and γ t − = 0. By (43), γ t − = γ t − = γ = γ = 0. As inCase (iii), by canceling γ t +3 using (54) and (55), we obtain (53) again.By (43), we have covered all possible cases. Therefore, (53) is proved.Now we are ready to prove (a) and (b). Our strategy is to give a precise expres-sion for a , which is strong enough to prove (a). Then we further use (40) to getmore restrictions on the value of h which leads to (b).Plugging (53) in (52), we have1 a q t − (cid:16) − k q t − (cid:17) + 1 a q (cid:16) − k q (cid:17) = 0 , that is (cid:18) a q (cid:16) − k q (cid:17)(cid:19) q t − − k − − q t − a q (cid:16) − k q (cid:17) = 0 . By k q t − = − k − q t − , we have (cid:18) k − a q (cid:16) − k q (cid:17)(cid:19) q t − + k − a q (cid:16) − k q (cid:17) = 0 . Therefore,(58) a = k − q t − (cid:16) − k q + q t − (cid:17) η, where η satisfies η q t − + η = 0. Since 0 = ( η q t − + η ) q t +2 = η + ( η q t − ) q = η − η q ,we get η ∈ F q . Moreover, η = − η q t − = − η q = − η q , t ≡ , − η = 0 , t ≡ , − η q , t ≡ , − η q , t ≡ . Hence(59) η q − = , t ≡ , , t ≡ , , t ≡ , − , t ≡ . LARGE FAMILY OF MAXIMUM SCATTERED LINEAR SETS OF PG(1 , q n ) AND THEIR ASSOCIATED MRD CODES23 Substitute a in (53) by (58), − k q − q t − k − q t − (1 − k q t +1 + q t − ) η q t = k − q t − (1 − k q + q t − ) η, which equals k q (1 − k − q − q t − ) η q t = k − q t − (1 − k q + q t − ) η. It implies(60) η q t = − η. Together with η q t − + η = 0, we deduce η q = η . It contradicts (59) when t ≡ t is even, a must be 0 and (a) is proved.Finally, let us plugging (47) and (48) into (40), we have(61) (¯ cγ ) q +1 = τ q v q a q +1 = ℓ q − a q +1 , where ℓ = h/k which means ℓ q t = 1 /ℓ .Similarly, by (50) and (49) into (41)(62) (¯ cγ ) q t − +1 = θ q t − u q t − a q t − +1 = ℓ q t − − a q t − +1 . Raising (61) to its q t − +12 -th power and (62) to its q +12 -th power and canceling(¯ cγ ) ( q +1)( q t − +1) and a ( q +1)( q t − +1) , we obtain ℓ q t − − = 1 . Again, by ℓ q t +1 = 1, we have ℓ q +1 = 1 which means ℓ gcd( q +1 ,q t +1) = 1. Therefore,since t is odd, we get ℓ = 1, and (b) is proved. (cid:3) Now we are ready to prove Theorem 5.1.
Proof of Theorem 5.1.
Suppose that f = ψ h,t , g = ψ k,t , and L f is PΓL(2 , q n )-equivalent to L g . Then there exists an invertible matrix (cid:18) a bc d (cid:19) over F q n and σ ∈ Aut( F q n ) such that (cid:26) f ( x ) x : x ∈ F ∗ q n (cid:27) = (cid:26) cx σ + d ( g ( x )) σ ax σ + b ( g ( x )) σ : x ∈ F ∗ q n (cid:27) = (cid:26) cx + d ¯ g ( x ) ax + b ¯ g ( x ) : x ∈ F ∗ q n (cid:27) , where ¯ g ( x ) = ψ k σ ,t ( x ).For a given h satisfying h q t +1 = −
1, let ξ h denote the number of k for which U k is ΓL(2 , q n )-equivalent to U h . By Theorem 4.1, the value of ξ h is independent of h and ξ h = ( , t q + 1 , t ≡ . Depending on the value of b , we separate the remainder part of the proof intotwo cases: Case (a). b = 0. By Lemma 5.2, ( h/k σ ) q +1 = 1 which means that U h isΓL(2 , q n )-equivalent to U k σ . Thus there are at most nrξ h choices of k for which L k,t is equivalent to L h,t . Case (b). b = 0. Without loss of generality, we assume b = 1 and f = ¯ g . ByLemma 5.3, d = a = 0. Thus (cid:26) f ( x ) x : x ∈ F ∗ q n (cid:27) = (cid:26) cx ¯ g ( x ) : x ∈ F ∗ q n (cid:27) . Suppose that there is another ˜ g ( x ) = ψ ˜ k,t ( x ) for certain ˜ k ∈ F q t satisfying ˜ k q t +1 = − (cid:26) f ( x ) x : x ∈ F ∗ q n (cid:27) = (cid:26) ˜ cx ˜ g ( x ) : x ∈ F ∗ q n (cid:27) for some ˜ c ∈ F q t . Then (cid:26) ¯ g ( x ) x : x ∈ F ∗ q n (cid:27) = (cid:26) c ˜ c · ˜ g ( x ) x : x ∈ F ∗ q n (cid:27) . By Lemma 5.2, (˜ k/k σ ) q +1 = 1 which means that U k σ is ΓL(2 , q n )-equivalent to U ˜ k . Hence, for the case b = 0, there are at most nrξ ˜ k choices of k for which L k,t isequivalent to L h,t .Combining Case (a) and Case (b), for a given L h,t , there are at most 2 nrξ h =4 rtξ h choices of k for which L k,t is equivalent to L h,t . (cid:3) Remark 5.4.
By Theorem 5.1, Family (33) contains much more inequivalent ele-ments compared with the known constructions for infinitely many n listed in Table1. Therefore, this family must be new. Remark 5.5.
For t = 3 and 4, we do not have any result about the equivalenceproblems among the linear sets defined by different ψ h,t ’s. The equivalence forthe associated MRD codes are also not completely known. One of the reasons isthat the comparing of coefficients of equations, such as (4.1), becomes much moreinvolved. We leave them as open questions. Acknowledgments
The research that led to the present paper was partially supported by a grantof the group GNSAGA of INdAM. Yue Zhou is partially supported by NaturalScience Foundation of Hunan Province (No. 2019JJ30030) and Training Programfor Excellent Young Innovators of Changsha (No. kq1905052).
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Email address : [email protected] Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universit`a degliStudi di Napoli “Federico II”, Via Vicinale Cupa Cintia, 80126 Napoli, Italy,
Email address : [email protected], [email protected] College of Liberal Arts and Sciences, National University of Defense Technology,410073 Changsha, China
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