aa r X i v : . [ m a t h . C O ] N ov Subspace codes in PG(2n − , q) Antonio CossidenteDipartimento di Matematica, Informatica ed EconomiaUniversit`a della BasilicataContrada Macchia RomanaI-85100 [email protected] PaveseDipartimento di Matematica, Informatica ed EconomiaUniversit`a della BasilicataContrada Macchia RomanaI-85100 [email protected] roposed Running Head: Subspace codes in PG(2n − , q) Corresponding Author:
Antonio CossidenteDipartimento di Matematica, Informatica ed EconomiaUniversit`a della BasilicataContrada Macchia RomanaI-85100 [email protected] 2 bstract
An ( r, M, δ ; k ) q constant–dimension subspace code, δ >
1, is acollection C of ( k − r − , q )such that every ( k − δ )–dimensional projective subspace of PG( r − , q )is contained in at most a member of C . Constant–dimension subspacecodes gained recently lot of interest due to the work by Koetter andKschischang [18], where they presented an application of such codesfor error-correction in random network coding. Here a (2 n, M, n ) q constant–dimension subspace code is constructed, for every n ≥
4. Thesize of our codes is considerably larger than all known constructions sofar, whenever n >
4. When n = 4 a further improvement is provided byconstructing an (8 , M,
4; 4) q constant–dimension subspace code, with M = q + q ( q + 1) ( q + q + 1) + 1. KEYWORDS: hyperbolic quadric; subspace code; Segre variety; rank dis-tance codes.
AMS MSC:
Let V be an r –dimensional vector space over GF( q ), q any prime power.The set S ( V ) of all subspaces of V , or subspaces of the projective spacePG( V ), forms a metric space with respect to the subspace distance definedby d s ( U, U ′ ) = dim( U + U ′ ) − dim( U ∩ U ′ ). In the context of subspacecoding theory, the main problem asks for the determination of the largersize of codes in the space ( S ( V ) , d s ) ( subspace codes ) with given minimumdistance and of course the classification of the corresponding optimal codes.Codes in the projective space and codes in the Grassmannian over a finitefield referred to as subspace codes and constant–dimension codes (CDCs),respectively, have been proposed for error control in random linear networkcoding, see [18]. An ( r, M, d ; k ) q constant–dimension subspace code is a set C of k –subspaces of V , where |C| = M and minimum subspace distance d s ( C ) = min { d s ( U, U ′ ) | U, U ′ ∈ C , U = U ′ } = d . The maximum size of an( r, M, d ; k ) q constant–dimension subspace code is denoted by A q ( r, d ; k ).For general results on bounds and constructions of subspaces codes, see[16]. More recent constructions and results can be found in [5], [6], [7], [9],[11], [14], [25]. For a geometric approach to subspace codes see also [3], wherea connection between certain subspace codes and particular combinatorialstructures is highlighted.From a combinatorial point of view an ( r, M, δ ; k ) q constant–dimensionsubspace code, δ >
1, is a collection C of ( k − r − , q ) such that every ( k − δ )–dimensional projectivesubspace of PG( r − , q ) is contained in at most a member of C .The set M m × n ( q ) of m × n matrices over the finite field GF( q ) formsa metric space with respect to the rank distance defined by d r ( A, B ) = rk ( A − B ). The maximum size of a code of minimum distance d , 1 ≤ d ≤ min { m, n } , in ( M m × n ( q ) , d r ) is q n ( m − d +1) for m ≤ n and q m ( n − d +1) for m ≥ n . A code A ⊂ M m × n ( q ) attaining this bound is said to be a q –ary( m, n, k ) maximum rank distance code ( MRD ), where k = m − d +1 for m ≤ n and k = n − d + 1 for m ≥ n . A rank code A is called GF( q )–linear if A is a subspace of M m × n ( q ). Rank metric codes were introduced by Delsarte[4] and rediscovered in [8] and [21]. Recently, these codes have found a newapplication in the construction of error-correcting codes for random networkcoding [24].A constant–rank code (CRC) of constant rank r in M m × n ( q ) is a non–empty subset of M m × n ( q ) such that all elements have rank r . We denote aconstant–rank code with length n , minimum rank distance d , and constant–rank r by ( m, n, d, r ). The term A ( m, n, d, r ) denotes the maximum cardinal-ity of an ( m, n, d, r ) constant–rank code in M m × n ( q ). From [9, Proposition8] we have that A ( m, n, d, r ) ≤ (cid:2) nr (cid:3) q Q r − di =0 ( q m − q i ) and if this upper boundis attained the CRC is said to be optimal. Here (cid:2) nr (cid:3) q := ( q n − · ... · ( q n − r +1 − q r − · ... · ( q − .In this paper we will construct a (2 n, M, n ) q constant–dimension sub-space code, for every n ≥
4. The size of our codes is considerably larger thanall known constructions so far whenever n > n = 4, by exploring in more details the geometry of the hyperbolicquadric Q + (7 , q ), a further improvement is provided by constructing an(8 , M,
4; 4) q constant–dimension subspace code, with M = q + q ( q +1) ( q + q + 1) + 1. An (8 , M,
4; 4) q constant–dimension subspace code withthe same size has also been constructed in [5] with a completely differenttechnique. We do not know if the two constructions are equivalent butcertainly both codes contain a lifted MRD code.In the sequel θ n,q := (cid:2) n +11 (cid:3) q = q n + . . . + q + 1 .4 The geometric setting
The
Segre map may be defined as the map σ : PG( n − , q ) × PG( n − , q ) → PG( n − , q ) , taking a pair of points x = ( x , . . . x n ), y = ( y , . . . y n ) of PG( n − , q ) totheir product ( x y , x y , . . . , x n y n ) (the x i y j are taken in lexicographicalorder). The image of the Segre map is an algebraic variety called the Segrevariety and denoted by S n − ,n − . The Segre variety S n − ,n − has two rulingsof projective ( n − R and R , such that twosubspaces in the same ruling are disjoint, and each point of S n − ,n − iscontained in exactly one member of each ruling. Also, a member of R meetsan element of R in exactly one point. From [13, Theorem 25.5.14] certainlinear sections of dimension n ( n + 1) / − S n − ,n − are Veronese varieties[13, § Let (
V, k ) be a non–degenerate formed space with associated polar space P where V is a ( d + 1)–dimensional vector space over GF( q e ) and k is asesquilinear (quadratic) form. The vector space V can be considered as an( e ( d +1))–dimensional vector space V ′ over GF( q ) via the inclusion GF( q ) ⊂ GF( q e ). Composition of k with the trace map T : z ∈ GF( q e ) P ei =1 z q i ∈ GF( q ) provides a new form k ′ on V ′ and so we obtain a new formed space( V ′ , k ′ ). If our new formed space ( V ′ , k ′ ) is non–degenerate, then it has anassociated polar space P ′ . The isomorphism types and various conditionsare presented in [17], [10]. Now each point in PG( d, q e ) corresponds toa 1–dimensional vector space in V , which in turn corresponds to an e –dimensional vector space in V ′ , that is an ( e − e ( d + 1) − , q ). Extending this map from points of PG( d, q e )to subspaces of PG( d, q e ), we obtain an injective map from subspaces ofPG( d, q e ) to certain subspaces of PG( e ( d + 1) − , q ): φ : PG( d, q e ) → PG( e ( d + 1) − , q ) . The map φ is called the GF( q )– linear representation of PG( d, q e ).A partial t –spread of a projective space P is a collection S of mutuallydisjoint t –dimensional projective subspaces of P . A partial t –spread of P is5aid to be a t –spread if each point of P is contained in an element of P . Thepartial t –spread S of P is said to be maximal , if there is no partial t –spread S ′ of P containing S as a proper subset.The set D = { φ ( P ) | P ∈ PG( d, q e ) } is an example of ( e − e ( d + 1) − , q ), called a Desarguesian spread (see [23], Section 25).The incidence structure whose points are the elements of D and whose linesare the (2 e − e ( d + 1) − , q ) joiningtwo distinct elements of D , is isomorphic to PG( d, q e ). One immediateconsequence of the definitions is that the image of the pointset of the originalpolar space P is contained in the new polar space P ′ (but is not necessarilyequal to it). PG(2 n − , q ) A Hermitian variety H of PG( n − , q ), is the set of absolute points forsome Hermitian form defined on the underlying vector space. The variety H is called degenerate if the corresponding Hermitian form is degenerate;else, it is called non–degenerate . Let H ( n − , q ) be the non–degenerateHermitian variety of PG( n − , q ), n ≥ H ( n − , q ) has thefollowing number of points: ( q n − q n − + 1) q − . The generators of H ( n − , q ) are ( n − / H ( n − , q ) is equal to( q + 1)( q + 1) · . . . · ( q n − + 1) . For further details on Hermitian varieties we refer to [22].Let H and H be the two distinct Hermitian varieties of PG( n − , q )having the following homogeneous equations f : X X q n +22 + . . . + X n X qn + X q X n +22 + . . . + X q n X n = 0 ,f : X X q n +22 + . . . + X n X qn + ω q − ( X q X n +22 + . . . + X q n X n ) = 0 , respectively, where ω is a primitive element of GF( q ). Then the Hermitianpencil F defined by H and H is the set of all Hermitian varieties withequations af + bf = 0, as a and b vary over the subfield GF( q ), not bothzero. Note that there are q + 1 distinct Hermitian varieties in the pencil6 , none of which is degenerate. The set X = H ∩ H is called the baselocus of F . Since the Hermitian varieties of a pencil cover all the points ofPG( n − , q ), a counting argument shows that |X | = ( q n − + 1)( q n − q − F intersect precisely in X . In particular X is a variety defined by the following equation: X X q n +22 + . . . + X n X qn = 0 . Straightforward computations show that X contains the following two ( n − / X = . . . = X n = 0 , Σ ′ : X n +22 = . . . = X n = 0 . Also, through a point P of Σ (resp. Σ ′ ) there pass θ n − ,q lines entirelycontained in X and these lines are contained in a generator of H ( n − , q )meeting Σ (resp. Σ ′ ) exactly in P .Let Π r − be a ( r − ≤ r ≤ ( n − /
2, and let Π ⊥ r − be the polar space of Π r − with respect to the unitarypolarity of H (or, equivalently, H ). The intersection of Π ⊥ r − and Σ ′ isa (( n − / − r )–dimensional projective space, say Π ′ ( n − / − r . Note that h Π r − , Π ′ ( n − / − r i is a generator of H contained in X . In particular, onecan see that the above construction produces ( n − / X r =1 (cid:20) n r (cid:21) q distinct generators of H lying on X and these are all the generators incommon between two Hermitian varieties belonging to the pencil F exceptΣ and Σ ′ .A hyperbolic quadric Q + (2 n − , q ) of PG(2 n − , q ), is the set of singularpoints for some non–degenerate quadratic form of hyperbolic type definedon the underlying vector space. The hyperbolic quadric Q + (2 n − , q ) hasthe following number of points:( q n − q n − + 1) q − . Q + (2 n − , q ) are ( n − Q + (2 n − , q ) is equal to2( q + 1)( q + 1) · . . . · ( q n − + 1) . The set of all generators of the hyperbolic quadric Q + (2 n − , q ) is dividedin two distinct subsets of the same size, called systems of generators anddenoted by M and M , respectively. Let A and A ′ two distinct generatorsof Q + (2 n − , q ). Then their possible intersections are projective spaces ofdimension (cid:26) , , , . . . , n − A, A ′ ∈ M i , i = 1 , − , , , . . . , n − A ∈ M i , A ′ ∈ M j , i, j ∈ { , } , i = j if n is odd or (cid:26) , , , . . . , n − A ∈ M i , A ′ ∈ M j , i, j ∈ { , } , i = j − , , , . . . , n − A, A ′ ∈ M i , i = 1 , n is even. For further details on hyperbolic quadrics we refer to [13].From [17], if n ≥ φ ( H ( n − , q )) is a hyperbolic quadric Q + (2 n − , q ) of PG(2 n − , q ). In particular, points of the Hermitian va-riety are mapped, under the GF( q )–linear representation map, to mutuallydisjoint lines contained in the corresponding hyperbolic quadric and cov-ering all the points of the quadric. Now, let φ ( H i ) = Q i , i = 1 ,
2. Thenthe hyperbolic quadrics Q , Q generate a pencil of PG(2 n − , q ), say F ′ ,containing other q − Q i , 3 ≤ i ≤ q + 1,none of which is degenerate. It turns out that the base locus of F ′ , say X ′ ,consists of the ( q n − + 1)( q n − q − φ ( X ). In particular X ′ contains two distin-guished generators, say S and S ′ , corresponding to Σ and Σ ′ , respectively,that are disjoint. Hence S and S ′ belong to the same system of generators,say M i of Q i , 1 ≤ i ≤ q + 1. Finally, if we denote by G the set of genera-tors meeting non–trivially both S and S ′ and belonging to each hyperbolicquadric of the pencil F ′ , we have that |G| = ( n − / X r =1 (cid:20) n r (cid:21) q . The construction
Let M n × n ( q ) be the vector space of all n × n matrices over the finite fieldGF( q ). Let PG( n − , q ) be the ( n − q ) equipped with homogeneous projective coordinates ( X , . . . , X n ).With the identification a i +1 ,j = a in + j , 0 ≤ i ≤ ( n − ≤ j ≤ n , wemay associate, up to a non-zero scalar factor, to a matrix A = ( a i,j ) ∈M n × n ( q ) a unique point P = ( a , . . . , a n ) ∈ PG( n − , q ), and viceversa.In this setting the Segre variety S n − ,n − can be represented by all n × n matrices of rank 1. Let G be the subgroup of PGL( n , q ) fixing S n − ,n − ,then | G | = 2 | P GL ( n, q ) | . In this context the subspace of all symmetricmatrices of M n × n ( q ) is represented by the ( n ( n + 1) / − n − , q ) defined by the following equations: X in + j = X ( j − n + i +1 , ≤ i ≤ n − , i + 2 ≤ j ≤ n. In particular Γ meets the Segre variety S n − ,n − in a Veronese variety V . Thesubgroup of G fixing V leaves invariant a ( n ( n − / − ′ , which corresponds to the subspace of all skew–symmetricmatrices of M n × n ( q ). In particular, Γ ′ is either contained in or disjoint toΓ according as q is even or odd, respectively. In any case Γ ′ is disjoint from S n − ,n − .In PG( n − , q n ) consider a q -order subgeometry PG( n − , q ). Let C ∈ PGL( n, q ) be a Singer cycle of PG( n − , q ), then h C i is a Singer cyclicgroup of order θ n − ,q = ( q n − / ( q − h C i partitions thepoints of PG( n − , q n ) into n hyperplanes and the remaining orbits are q -order subgeometries, see [2]. In particular h C i fixes n points in generalpositions and each of the n fixed hyperplanes contains n − q )–linear representation of PG( n − , q n ), a point ofPG( n − , q n ) becomes a PG( n, q ) that is member of a Desarguesian spreadof a PG( n − , q ). In particular points of a PG( n − , q ) become maximalspaces of a ruling of a Segre variety S n − ,n − of PG( n − , q ), see [20]. Itfollows that PG( n − , q ) is partitioned into n ( n − n − P denotesthe above partition of PG( n − , q ), then there exists a subgroup J of G of order 2 θ n − ,q fixing P . The group J is generated by the projectivitiesof PGL( n , q ) induced by ¯ ι, I ⊗ ¯ C, ¯ C ⊗ I ∈ GL( n , q ). Here ⊗ denotes the9ronecker product and C is induced by ¯ C ∈ GL( n, q ).¯ ι = A A . . . A n A A . . . A n ... ... . . . ... A n A n . . . A nn , where A ij are ( n × n )-matrices defined as follows: A ij = ( a rs ) , a rs = (cid:26) i, j ) = ( r, s )0 ( i, j ) = ( r, s ) . The projectivity ι induced by ¯ ι is either an involutory homology having Γas axis and Γ ′ as center, if q is odd, or an involutory elation having Γ as axisand Γ ′ as center, if q is even. Also, notice that the projectivity of J inducedby ¯ C ⊗ ¯ C has order θ n − ,q and fixes V .From [15], ¯ C is conjugate in GL( n, q n ) to the a diagonal matrix DD = diag ( ω, ω q , . . . ω q n − ) , for some primitive element ω of GF( q n ). In other words, there exists amatrix E ∈ GL( n, q n ) with E − ¯ CE = D . Let ˆ J be the group generated bythe projectivities of PGL( n , q n ) induced by ¯ ι, I ⊗ D, D ⊗ I ∈ GL( n , q n ).Since( E ⊗ E ) − ( I ⊗ ¯ C )( E ⊗ E ) = I ⊗ D, ( E ⊗ E ) − ( ¯ C ⊗ I )( E ⊗ E ) = D ⊗ I, and ( E ⊗ E ) − ¯ ι ( E ⊗ E ) = ¯ ι, it turns out that the group ˆ J fixes the q –order subgeometry Π of PG( n − , q n ) whose points are as follows:( α , . . . , α n , α qn , α q , . . . , α qn − , α q n − , α q n , . . . , α q n − , . . . , α q n − , . . . , α q n − ) , where α i ∈ GF( q n ), 1 ≤ i ≤ n , Q ni =1 α i = 0, and the Segre variety ˆ S n − ,n − corresponding to S n − ,n − contained in it. Furthermore the group ˆ J fixesthe following n ( n − X = h U ( a − n + a i , ≤ a ≤ n, X k = h U ( a − k ) n + a , U ( n − k ) n + a ( n +1) i , k ≤ a ≤ n, ≤ a ≤ k − , ≤ k ≤ n, U i denotes the point with coordinates (0 , . . . , , , , . . . , i − th position. The projectivity of ˆ J induced by D ⊗ D has order θ n − ,q and fixes the Veronese variety ˆ V = ˆΓ ∩ ˆ S n − ,n − . In particular X is contained in ˆΓ and the involution ι fixes X pointwise and interchanges X k with X n − k +2 , 2 ≤ k ≤ n . Then the involution ι fixes the ( n − n − Y = hX k i , 2 ≤ k ≤ n . It follows that thecenter of ι , ˆΓ ′ , must be contained in ˆ Y . We have proved the following result. Proposition 3.1.
There exists an ( n − n − –dimensional projective space Y that is disjoint from S n − ,n − and contains Γ ′ . We denote by A the set consisting of q n ( n − / matrices correspondingto the points of Γ ′ (together with the zero matrix). Since Y is disjointfrom the Segre variety S n − ,n − , the set M , consisting of the q n − n matricescorresponding to the points of Y (together with the zero matrix), form alinear ( n, n, n −
1) MRD code.Let A be a n × n matrix over GF( q ), and let I n be the n × n identitymatrix. The rows of the n × n matrix ( I n | A ) can be viewed as pointsin general position of an ( n − n − , q ). This subspace is denoted by L ( A ). From [24], a q -ary ( n, n, n − q -ary (2 n, q n − n , n ) constant–dimension subspace code. Aconstant–dimension code such that all its codewords are lifted codewordsof an MRD code is called a lifted MRD code . Let L = { L ( A ) | A ∈ M} be the constant–dimension code obtained by lifting the ( n, n, n −
1) MRDcode contructed above. Then L consists of ( n − n − L are disjoint from the special ( n − S = h U n +1 , . . . , U n i and therefore every ( n − L is disjoint from S . Moreover, from [14, Lemma 6], every ( n − n − , q ) disjoint from S is covered by a member of L exactly once.From [8] it is known that a linear ( n, n, n −
1) MRD code contains an( n, n, , r ) CRC of size (cid:20) nr (cid:21) q r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q q ( r − j )( q n ( j − − . Let C r denotes the ( n, n, , r ) CRC contained in Y . Let A be an elementof C r , 2 ≤ r ≤ ( n − n × n matrix ( A | I n ) canbe viewed as points in general position of an ( n − n − , q ). This subspace is denoted by L ′ ( A ). The subspace11 ′ ( A ) is disjoint from the special ( n − S ′ = h U , . . . , U n i and meets S in a ( n − r − n − L ′ ( A )meets S in at least a point and is disjoint from S ′ . Let L r = { L ′ ( A ) | A ∈ C r } be the constant–dimension code obtained by lifting the ( n, n, , r ) CRC codes C r , 2 ≤ r ≤ ( n −
2) constructed above. If A ∈ C r , A ∈ C r , then L ′ ( A )meets L ′ ( A ) in at most in ( n − Proposition 3.2.
The set S n − i =1 L i is a (2 n, M, n ) q constant–dimensionsubspace code, where M = q n − n + n − X r =2 (cid:20) nr (cid:21) q r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q q ( r − j )( q n ( j − − . Now, we introduce the non–degenerate hyperbolic quadric Q of PG(2 n − , q ) having the following equation: X X n + X X n − + . . . + X n X n +1 = 0 . The subspaces S and S ′ are maximals of Q that belong to the same or todifferent systems, according as n is even or odd, respectively. Let M be thesystem of maximals of Q containing S and let D ( X ) and I ( X ) denote the setof maximals in M disjoint from X or meeting non–trivially X , respectively.Let A be a skew–symmetric matrix in M n × n ( q ), then it is not difficult tosee that L ( A ) (resp. L ′ ( A )) is a maximal of Q disjoint from S (resp. S ′ ).Since the number of maximals of Q disjoint from S equals q n ( n − / [26, p.175 Ex. 11.4], we have that each such a maximal is of the form L ( A ), forsome A ∈ A . n even Assume that n is even. In this case we have that M = D ( S ) ∪ ( D ( S ′ ) ∩ I ( S )) ∪ ( I ( S ) ∩ I ( S ′ ))and | D ( S ) | = q n ( n − . On the other hand, a maximal L ′ ( A ) in D ( S ′ ) is disjoint from S if andonly if A is a skew–symmetric matrix of rank n . From [19], the number of12kew–symmetric matrices of rank n is equal to q n ( n − ( q n − − q n − − · . . . · ( q −
1) = q n ( n − n − Y i =0 ( q i +1 − . Therefore, we have that | D ( S ′ ) ∩ I ( S ) | = q n ( n − − q n ( n − n − Y i =0 ( q i +1 − | I ( S ) ∩ I ( S ′ ) | = |M | − q n ( n − + q n ( n − n − Y i =0 ( q i +1 − . Notice that both D ( S ) and D ( S ′ ) ∩ I ( S ) are contained in S n − i =1 L i , whereas I ( S ) ∩ I ( S ′ ) is disjoint from S n − i =1 L i . Then it turns out that ( S n − i =1 L i ) ∪ ( I ( S ) ∩ I ( S ′ )) is a set of ( n − n − q n − n + n − X r =2 (cid:20) nr (cid:21) q r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q q ( r − j )( q n ( j − − n − Y i =1 ( q i + 1) − q n ( n − + q n ( n − n − Y i =0 ( q i +1 − . In this case, from Section (2.3), there exists a pencil F ′ comprising q hyperbolic quadrics Q i , 2 ≤ i ≤ q + 1 of PG(2 n − , q ) distinct from Q . Let I i ( X ) denote the set of maximals in M i meeting non–trivially X , 2 ≤ i ≤ ( q + 1) and let G = T q +1 i =2 ( I i ( S ) ∩ I i ( S ′ )) ∩ ( I ( S ) ∩ I ( S ′ )). Then, from Section(2.3), we have that | q +1 [ i =2 ( I i ( S ) ∩ I i ( S ′ )) | = q ( | I ( S ) ∩ I ( S ′ ) | − |G| ) = q |M | − q n ( n − + q n ( n − n − Y i =0 ( q i +1 − − n − X r =1 (cid:20) n r (cid:21) q .
13t follows that ( S n − i =1 L i ) ∪ ( S q +1 i =2 ( I i ( S ) ∩ I i ( S ′ ))) ∪ ( I ( S ) ∩ I ( S ′ )) is a set of( n − n − q n − n − q + 1) q n ( n − + n − X r =2 (cid:20) nr (cid:21) q r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q q ( r − j )( q n ( j − − q + 1) n − Y i =1 ( q i + 1) + q n ( n − n − Y i =0 ( q i +1 − − q n − X r =1 (cid:20) n r (cid:21) q . The set G contains a subset D consisting of θ ( n − / ,q generators belong-ing to each hyperbolic quadric of the pencil F ′ such that every element in D meets S in a line and S ′ in an ( n − D S = { A ∩ S | A ∈ D} is a Desarguesian line–spread of S . In other words D S = { φ ( P ) | P ∈ Σ } . On the other hand, the set D S ′ = { A ∩ S ′ | A ∈ D} is a set of ( n − n − ℓ ∈ D S there exists a unique element in D S ′ , say A ℓ , such that h ℓ, A ℓ i is in D , andviceversa. Furthermore, if ℓ ∈ D S and B ∈ D S ′ \ { A ℓ } , then h ℓ, B i is an( n − F ′ in a cone having as vertex A ℓ ∩ B and as base a Q + (3 , q ) contain-ing ℓ . Notice that such a cone meets a generator of a hyperbolic quadricof the pencil F ′ in at most an ( n − D ′ be the set of ( n − h ℓ, B i ,where ℓ ∈ D S and B ∈ D S ′ \ { A ℓ } . Then D ′ is disjoint from D . Also |D ′ | = θ ( n − / ,q ( θ ( n − / ,q − S n − i =1 L i ) ∪ ( S q +1 i =2 ( I i ( S ) ∩ I i ( S ′ ))) ∪ ( I ( S ) ∩ I ( S ′ )) ∪ D ′ ∪ { S } is a set of( n − n − Theorem 3.3. If n is even, there exists a (2 n, M, n ) q constant–dimensionsubspace code, where M = q n − n − q + 1) q n ( n − + n − X r =2 (cid:20) nr (cid:21) q r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q q ( r − j )( q n ( j − − q +1) n − Y i =1 ( q i + 1) + q n ( n − n − Y i =0 ( q i +1 − − q n − X r =1 (cid:20) n r (cid:21) q + (cid:20) n (cid:21) q (cid:20) n (cid:21) q − ! +1 . .2 n odd Assume that n is odd. In this case M = ( D ( S ′ ) ∩ I ( S )) ∪ ( I ( S ) ∩ I ( S ′ ))and | D ( S ) | = 0 , | D ( S ′ ) ∩ I ( S ) | = | D ( S ′ ) | = q n ( n − . On the other hand, a maximal L ′ ( A ) in D ( S ′ ) is not in S n − i =1 L i if and onlyif A is a skew–symmetric matrix of rank n −
1, i.e., L ′ ( A ) meets S in a point.From [19], the number of skew–symmetric matrices of rank n − q ( n − n − ( q n − q n − − · . . . · ( q −
1) = q ( n − n − n − Y i =1 ( q i +1 − . Therefore, if we denote by I the subset of D ( S ′ ) consisting of maximalintersecting S in exactly a point, we have that |I| = q ( n − n − n − Y i =1 ( q i +1 − | I ( S ) ∩ I ( S ′ ) | = |M | − q n ( n − . Notice that { L ( A ) | A ∈ A} ⊆ L . Then, if L ′ = L \ { L ( A ) | A ∈ A} , then L ′ ∪ ( S n − i =2 L i ) ∪ I ∪ ( I ( S ) ∩ I ( S ′ )) ∪ { S } is a set of ( n − n − q n − n + n − X r =2 (cid:20) nr (cid:21) r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q ( r − j )( q n ( j − − n − Y i =1 ( q i + 1) − q n ( n − + q ( n − n − n − Y i =1 ( q i +1 −
1) + 1 . From [1, Theorem 4.6] a partial 1–spread of PG( n − , q ), n ≥ y := q n − + q n − + · · · + q + 1 and actually examples of this sizeexist. Arguing as in the last part of the previous paragraph we prove thefollowing result. 15 heorem 3.4. If n is odd, there exists a (2 n, M, n ) q constant–dimensionsubspace code, where M = q n − n + n − X r =2 (cid:20) nr (cid:21) r X j =2 ( − ( r − j ) (cid:20) rj (cid:21) q ( r − j )( q n ( j − − n − Y i =1 ( q i + 1) − q n ( n − + q ( n − n − n − Y i =1 ( q i +1 −
1) + y ( y −
1) + 1 . PG(7 , q ) In this section we will improve, in the case n = 4, the result established inTheorem 3.3 by considering some more suitable projective 3–spaces (solids).In this case S and S ′ are generators of Q + (7 , q ) belonging to the samesystem. Here, D consists of q + 1 generators belonging to each hyperbolicquadric of the pencil F ′ such that every element in D meets S and S ′ in aprojective line. It follows that D S = { A ∩ S | A ∈ D} and D S ′ = { A ∩ S ′ | A ∈ D} are both Desarguesian line–spreads of S and S ′ , respectively.In other words D S = { φ ( P ) | P ∈ Σ } and D S ′ = { φ ( P ) | P ∈ Σ ′ } . Let r ′ bea line of S ′ . Then, r ′⊥ (here ⊥ denotes the orthogonal polarity of PG(7 , q )induced by Q + (7 , q )) meets S in a line r . If r ′ belongs to D S ′ , then r belongsto D S . Assume that r ′ does not belong to D S ′ . Of course, r ′ meets q +1 lines l ′ , . . . , l ′ q +1 of D S ′ and r meets q + 1 lines l , . . . , l q +1 of D S . The subgroup ofthe orthogonal group PGO + (8 , q ) fixing Q + (7 , q ) and stabilizing both S and S ′ (but that does not interchange them) is isomorphic to PGL(4 , q ) (whichin turn is isomorphic to a subgroup of index two of PGO + (6 , q )). Under theKlein correspondence between lines of S and points of the Klein quadric K ,the lines of D S are mapped to a 3–dimensional elliptic quadric E embeddedin K and the lines l , . . . , l q +1 are mapped to a conic section C of E , see [12].Also, there exists a subgroup H ′ of the orthogonal group PGO + (6 , q ) fixing K , isomorphic to C q +1 × PGL(2 , q ), stabilizing E and permuting in a singleorbit the remaining points of K . It follows that there exists a group H inthe orthogonal group PGO + (8 , q ) corresponding to H ′ , stabilizing Q + (7 , q )and fixing both S , S ′ , their line–spreads D ( S ), D ( S ′ ) and permuting in asingle orbit the remaining lines of S (respectively S ′ ). In this setting the line r corresponds, under the Klein correspondence, to a point P ∈ C ⊥ K (here ⊥ K denotes the orthogonal polarity of PG(5 , q ) induced by K ). Let H ′ P bethe stabilizer of P in H ′ . Then | H ′ P | = | PGL(2 , q ) | . On the other hand, H ′C , the stabilizer of C in H ′ , is contained in H ′ P and contains a subgroup16somorphic to PGL(2 , q ). It follows that H ′ P = H ′C ≃ PGL(2 , q ). The group H ′C has q ( q − / q − q . Each of them together with C givesrise to an elliptic quadric of K on C and these are all the elliptic quadrics of K on C . Let E ′ be one of the above orbits of H ′C of size q − q disjoint from E . Let L E ′ be the set of lines of S corresponding to E ′ . Let Y denotes thesolid generated by r ′ and a line of L E ′ and consider the orbit Y H of Y underthe action of the group H . Since the lines in L E ′ are mutually disjoint,then two distinct solids in Y H containing r ′ have in common exactly theline r ′ . Let l be a line of L E ′ . Under the Klein correspondence, the line l corresponds to a point P ′ ∈ E ′ . Notice that P ′⊥ K meets E in a conic, say C ′ , that is necessarily disjoint from C . Assume on the contrary that thereexists a point in common between C and C ′ , say Q . Then the line P ′ Q isentirely contained in K . Also, P ′ Q ⊂ E ′ = h P ′ , Ci ∩ K , contradicting the factthat E ′ is a 3–dimensional elliptic quadric (and so does not contain lines).Now, we claim that the solid h P, C ′ i meets K in a 3–dimensional ellipticquadric. Indeed, otherwise, there would be a line entirely contained in K and passing through P . But such a line would contain a point of C ′ , thatclearly is a contradiction, since P ∈ C ⊥ K and C ′ is disjoint from C . It followsthat if H l denotes the stabilizer of l in H , then r H l contain q − q mutuallydisjoint lines. Therefore r ′ H l contain q − q mutually disjoint lines and twosolids in Y H containing l have in common exactly the line l . Then Y H is aset of solids mutually intersecting in at most a line. The set Y H contains( q − q )( q + q )( q + 1) = q − q solids. Notice that none of the solids in Y H is a generator of Q + (7 , q ) or of a quadric of the pencil F ′ . Finally, assumethat a solid T in Y H generated by a line l ∈ S and a line r ∈ S ′ contains aplane π that is entirely contained in Q + (7 , q ) or in a quadric of the pencil F ′ . Then, π would meet l ′ in a point U and hence T would meet S ′ in a linethrough U that is not the case. We have proved the following result. Theorem 4.1.
There exists an (8 , M,
4; 4) q constant–dimension subspacecode, where M = q + q ( q + 1) ( q + q + 1) + 1 . Corollary 4.2. A q (8 ,
4; 4) ≥ q + q ( q + 1) ( q + q + 1) + 1 . Remark 4.3.
The result of Theorem 4.1 was obtained with different tech-niques in [5], where the authors, among other interesting results, proved that q + q ( q + 1) ( q + q + 1) + 1 is also the maximum size of an (8 , M,
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