Constant-Rank Codes

Constant-Rank Codes Maximilien Gadouleau and Zhiyuan Y an Department of Elec trical a nd Computer Engine ering Lehigh Uni versity , P A 1 8015, USA E-mails: { mag c, yan } @lehigh.edu Abstract — Constant-dimension codes ha ve r ecently rec eived attention d ue to their significance to err or control in noncoherent random networ k coding. In this paper , we show th at constant- rank codes ar e closely related to constant-dimension codes and we study the p roperties of constant-rank codes. W e first introduce a relation between v ectors in GF( q m ) n and subspaces of GF( q ) m or GF( q ) n , and u se it to establish a relation between constant- rank codes and constant-d imension codes. W e then deriv e bound s on th e maximum cardin ality of constant-rank codes with given rank weight and minimu m rank di stance. Fi nally , we in vestigate the asymptotic behavio r of the maximal cardin ality of constant- rank codes with giv en rank weight and minimu m rank di stance. I . I N T R O D U C T I O N While random network co ding [1] has proved to be a powerful tool fo r dissemina ting infor mation in networks, it is hig hly su sceptible to error s. Thus, e rror control f or ra ndom network co ding is critical and has received g rowing a ttention recently . Er ror control sche mes p ropo sed for random network coding assume two typ es o f tr ansmission models: some (see, e.g., [2], [3]) depend o n the under lying network topo logy or the p articular linear network coding operation s perfor med at various network no des; other s [4 ], [5] assume that th e transmitter and r eceiv er have no k nowledge o f such ch annel transfer ch aracteristics. The contrast is similar to that be tween coheren t an d no ncoher ent commu nication systems. Error contro l for nonco herent ra ndom ne twork cod ing is first co nsidered in [ 4]. Motiv ated by the proper ty th at r andom network coding is vector-space p reserving , [4] defines an operator channel that capture s the essence of the no ncoher ent transmission mo del. Hence , codes d efined in finite field Gra ss- mannians [6], referr ed to as constan t-dimension cod es, play a significant role in erro r control for no ncohere nt r andom net- work codin g. In [ 4], a Singleto n bou nd for co nstant-dim ension codes and a family o f code s that are n early Singleton-bo und achieving ar e p roposed . Despite the asym ptotic op timality of the Singleton bou nd and the cod es design ed in [4 ], the maximal cardin ality of a constant-dim ension code with finite dimension and min imum distance r emains unkn own, and it is not clear how an op timal code th at achieves th e ma ximal cardinality can b e constru cted. It is difficult to an swer the above questio ns based on con stant-dimen sion codes dire ctly since th e set of all subspaces o f th e am bient space lacks a natural g roup structur e [5]. The class of nearly Singleto n boun d achieving con stant- dimension codes in [4] are related to rank metr ic codes. The relev ance of rank metric codes to no ncoher ent rando m netw ork coding is f urther estab lished in [5]. In additio n to n etwork coding, r ank metric codes [7 ]–[9] have b een r eceiving steady attention in the literature due to th eir applications in storage systems [9], public- key cr yptosystems [10], and space-time coding [11]. The pion eering works in [ 7]–[9 ] h av e estab lished many importan t p roperties of rank metric codes. Independen tly in [ 7]–[9 ], a Singleton bound (u p to som e variations) o n the minimum rank d istance of co des was established, and a class of codes that ach iev e the boun d with equality was constru cted. W e refer to cod es that attain the Singleton b ound as maximu m rank d istance (MRD) codes, and the class o f MRD codes propo sed in [8] as Gabidulin co des hen ceforth. In this paper, we in vestigate the pro perties o f co nstant-rank codes, wh ich are the coun terparts in rank m etric c odes of constant (Hamming ) weight codes [12]. W e first introduce a relation between vectors in GF( q m ) n and subspaces of GF( q ) m or GF( q ) n , an d u se it to establish a relation b etween constant-ra nk codes and constant-dim ension codes. W e also derive a lower bo und on the maximum card inality of constant- rank cod es which d epends on the maxim um cardin ality of constant-d imension c odes. W e then d erive bou nds on the maximum cardin ality o f constant-r ank co des with g iv en ran k and m inimum r ank d istance. Fin ally , we chara cterize the asymptotic behavior of the maxim al card inality of constant- rank codes with given ran k and minimu m rank distance, and co mpare it with asympto tic behavior of the maximal cardinality of constan t-dimension codes. The rest of the paper is organized as follows. Section I I briefly revie ws some imp ortant concepts in o rder to keep this p aper self-con tained. In Sectio n II I, we establish a re- lation between constant- dimension and constant-ran k cod es. In Section IV, we derive b ound s on the max imum card inality of constant-rank codes with a given minim um ran k distance . Finally , Sectio n V in vestigates the asymptotic b ehavior of the maximum ca rdinality of constant- rank codes. I I . P R E L I M I N A R I E S A. Ran k metric codes a nd elementary linear subspaces Consider a vector x of length n over GF ( q m ) . The field GF( q m ) may be viewed as an m -dimension al vector space over GF( q ) . The rank weight o f x , denoted as rk( x ) , is defined to be the ma ximum number of coordinates of x that are linearly indepen dent over GF( q ) [8 ]. For any basis B m of GF( q m ) over GF ( q ) , each coo rdinate of x can be exp anded to an m - dimensiona l colu mn vector over GF( q ) with respect to B m . The rank weight of x is hen ce th e r ank o f the m × n matrix over GF( q ) o btained by expand ing all the coordin ates o f x . For all x , y ∈ GF( q m ) n , it is easily verified that d R ( x , y ) def = rk( x − y ) is a metric over GF ( q m ) n , referr ed to as the rank metric hen ceforth [8]. The minimum rank distance o f a co de C , denoted as d R , is simply the m inimum ran k distance over all po ssible pairs of distinct cod ew ords. It is shown in [ 7]–[9 ] that the minim um ran k d istance o f a block code of leng th n and card inality M over GF( q m ) satisfies d R ≤ n − log q m M + 1 . In this paper, we refer to this boun d as the Sin gleton bound for rank metric codes and co des that attain the equality as maximum rank d istance (MRD) codes. W e re fer to the subc lass of linear MRD codes introdu ced ind epende ntly in [7]– [9] as Gabidulin co des. W e den ote the number of vectors of rank r ( 0 ≤ r ≤ min { m, n } ) in GF( q m ) n as N r ( q m , n ) =  n r  α ( m, r ) [8], where α ( m, 0 ) def = 1 and α ( m, r ) def = Q r − 1 i =0 ( q m − q i ) fo r r ≥ 1 . The  n r  term is often ref erred to as a Gau ssian polyno mial [13] , defin ed as  n r  def = α ( n, r ) /α ( r, r ) . The volume of a ball with r ank rad ius r in GF( q m ) n is d enoted as V r ( q m , n ) = P r i =0 N i ( q m , n ) . For all q , 1 ≤ d ≤ r ≤ n ≤ m , the num ber of codew ords of rank r in an ( n, n − d + 1 , d ) linear MRD co de over GF( q m ) is given b y [8 ] M d,r def =  n r  r X j = d ( − 1) r − j  r j  q ( r − j 2 )  q m ( j − d +1) − 1  . (1) An elementa ry linear subspace (ELS) [14] is define d to be a linear subspace V ⊆ GF( q m ) n for which there exists a basis o f vectors in GF( q ) n . W e denote the set of all ELS’ s of GF( q m ) n with dimensio n v as E v ( q m , n ) . It c an be easily shown that | E v ( q m , n ) | =  n v  for all m . An ELS has properties similar to tho se for a set of coor dinates [14]. In par ticular , any vector belongin g to an E LS with dimension r has ran k no more than r ; conversely , any vector x ∈ GF( q m ) n with rank r belong s to a u nique ELS in E r ( q m , n ) . B. Constan t-dimension codes A co nstant-dimen sion cod e [4] of length n and con stant- dimension r over GF( q ) is defined to be a nonempty subset of E r ( q , n ) . For all U , V ∈ E r ( q , n ) , it is easily verified that d S ( U , V ) def = dim( U + V ) − dim( U ∩ V ) = 2 dim( U + V ) − 2 r (2) is a metr ic over E r ( q , n ) , refe rred to as the sub space metric hencefo rth [4]. The sub space distance b etween U an d V thus satisfies d S ( U , V ) = 2r k( X T | Y T ) − 2 r , where X and Y are generato r matrices of U and V , r espectively . The minimum subsp ace distance of a co nstant-dim ension code Ω ⊆ E r ( q , n ) , denoted as d S , is the minimu m subspac e distance ov er all possible pairs of distinct subspaces. W e say Ω is an ( n, d S , r ) constan t-dimension co de over GF ( q ) and we denote the m aximum cardin ality of an ( n, 2 d, r ) constant-d imension c ode over GF( q ) a s A S ( q , n, 2 d, r ) . Since A S ( q , n, 2 d, r ) = A S ( q , n, 2 d, n − r ) [15 ], only the case wher e 2 r ≤ n needs to be co nsidered. Also, since A S ( q , n, 2 , r ) =  n r  and A S ( q , n, 2 d, r ) = 1 fo r d > r , we shall assume 2 ≤ d ≤ r h enceforth . Upp er and lower boun ds on A S ( q , n, 2 d, r ) were derived in [4] , [ 15], [ 16]. In p articular, for all q , 2 r ≤ n , and 2 ≤ d ≤ r , q ( n − r )( r − d +1) ≤ A S ( q , n, 2 d, r ) ≤ α ( n, r − d + 1) α ( r , r − d + 1) . (3) C. Pr eliminary graph-theo r etic r esults W e revie w some results in g raph th eory given in [ 17]. T wo adjacent vertices u, v in a gr aph are denoted as u ∼ v . Definition 1: Let G and H b e two graphs. A mapping f from V ( G ) to V ( H ) is a homom orphism if for all u , v ∈ V ( G ) , u ∼ v ⇒ f ( u ) ∼ f ( v ) . Definition 2: Let G be a graph and φ a bijection from V ( G ) to itself. φ is called an autom orphism of G if for all u, v ∈ V ( G ) , u ∼ v ⇔ φ ( u ) ∼ φ ( v ) . Definition 3: W e say th at the g raph G is vertex tr ansitiv e if fo r all u, v ∈ V ( G ) , th ere exists an automo rphism φ of G such that φ ( u ) = v . An indepen dent set of a graph G is a subset of V ( G ) with no ad jacent vertices. Th e ind ependen ce num ber α ( G ) of G is the max imum cardinality of an in depend ent set of G . I f H is a vertex transitive graph and if th ere is a ho momor phism from G to H , then [17] α ( G ) ≥ α ( H ) | G | | H | . (4) I I I . C O N S TA N T - R A N K A N D C O N S TA N T - D I M E N S I O N C O D E S A. Defin itions and technical r esults Definition 4: A constant-ran k code of length n and constant-ra nk r over GF( q m ) is a non empty subset of GF( q m ) n such that all elements have ran k weigh t r . W e denote a co nstant-ran k code with leng th n , min imum rank distance d , and constant-r ank r as an ( n, d, r ) constant- rank code over GF( q m ) . W e define the term A R ( q m , n, d, r ) to be the maximum cardina lity of an ( n, d, r ) constant-ran k code over GF( q m ) . If C is an ( n, d, r ) constan t-rank cod e over GF( q m ) , then the code obtain ed by transposing all the expansion m atrices of co dew ords in C fo rms an ( m, d, r ) constant-ra nk code over GF( q n ) with the same car dinality . Therefo re A R ( q m , n, d, r ) = A R ( q n , m, d, r ) , and hencefo rth we assum e n ≤ m witho ut loss of g enerality . W e now define two families of graphs which ar e instrumen - tal in ou r analysis of co nstant-rank cod es. Definition 5: The bilin ear forms graph R q ( m, n, d ) has as vertices all the vectors in GF( q m ) n and two vertices x and y are adjacent if and o nly if d R ( x , y ) < d . Th e constant-rank graph K q ( m, n, d, r ) is th e subgr aph of R q ( m, n, d ) ind uced by the vectors in GF( q m ) n with r ank r . The orders of the bilinear forms a nd constant-r ank gr aphs are thus giv en by | R q ( m, n, d ) | = q mn and | K q ( m, n, d, r ) | = N r ( q m , n ) . A n indepen dent set of R q ( m, n, d ) corresp onds to a code with minimum rank distance ≥ d . Du e to th e existence of MRD codes fo r all par ameter values, we hav e α ( R q ( m, n, d )) = q m ( n − d +1) . Similarly , an ind epende nt set of K q ( m, n, d, r ) corr esponds to a constant-r ank co de with minimum ran k distance ≥ d , and hence α ( K q ( m, n, d, r )) = A R ( q m , n, d, r ) . Lemma 1: The bilinear forms graph R q ( m, n, d ) is vertex transitiv e for all q , m , n , and d . T he con stant-rank gr aph K q ( m, m, d, m ) is vertex transitive fo r all q , m , a nd d . Pr oof: Let u , v ∈ GF( q m ) n . For all x ∈ GF( q m ) n , define φ ( x ) = x + v − u . It is easily shown that φ is a graph automo rphism of R q ( m, n, d ) satisfying φ ( u ) = v . By Definition 3, R q ( m, n, d ) is hence vertex transitive. Let u , v ∈ GF( q m ) m have rank m , and denote their expansions with r espect to a basis B m of GF( q m ) over GF( q ) as U and V , respectively . For a ll x ∈ GF( q m ) m with r ank m , define φ ( x ) = y such that Y = XU − 1 V , wh ere X , Y are the expansions of x an d y with re spect to B m , respectiv ely . W e have φ ( u ) = v , rk( φ ( x )) = m , and f or all x , z ∈ GF( q m ) m , d R ( φ ( x ) , φ ( z )) = rk( XU − 1 V − ZU − 1 V ) = rk( X − Z ) = d R ( x , z ) . By Definition 2, φ is an autom orph ism which takes u to v and hence K q ( m, m, d, m ) is vertex transitive. It is worth noting that K q ( m, n, d, r ) is not vertex transitive in g eneral. B. Constan t-dimension and co nstant-rank codes In [4], con stant-dimen sion codes were co nstructed from rank d istance codes as follows. L et C be a code with len gth n over GF ( q m ) . For any c ∈ C , consider its expansion C with respect to th e basis B m of GF( q m ) over GF( q ) , an d c onstruct I ( C ) = ( I m | C ) ∈ GF( q ) m × m + n . Then I ( C ) def = { I ( C ) | c ∈ C } is a co nstant-dim ension c ode in E m ( q , m + n ) . This relation between rank co des and co nstant-dimen sion code s was also co mmented in grap h-theor etic terms in [18 ]. W e introdu ce a relation b etween vectors in GF( q m ) n and subspaces of GF( q ) m or GF( q ) n . For any x ∈ GF( q m ) n with r ank r , con sider the ma trix X ∈ GF( q ) m × n obtained by expanding all the coord inates o f x with respect to a basis B m of GF( q m ) over GF( q ) . Th e column span of X , denoted as S ( x ) , is an r - dimension al subspace of GF ( q ) m , which corresp onds to the sub space o f GF( q m ) spann ed by the coordin ates of x . The row sp an of X , denoted as T ( x ) , is an r -d imensional sub space o f GF( q ) n , which co rrespon ds to th e unique EL S V ∈ E r ( q m , n ) such that x ∈ V . Lemma 2: For all S ∈ E r ( q , m ) an d T ∈ E r ( q , n ) , there exists x ∈ GF( q m ) n with rank r such that S ( x ) = S and T ( x ) = T . Pr oof: Conside r the gen erator matrices G ∈ GF( q ) r × m and H ∈ GF( q ) r × n of S an d T , respectively . Let X = G T H and x be the vector wh ose expansion with respect to B m is giv en by X . Th en S ( x ) = S and T ( x ) = T . By Lemma 2, the fu nctions S and T are surjective. They are not injec ti ve, h owe ver . For all V ∈ E r ( q m , n ) , th ere exist exactly α ( m, r ) vector s x ∈ V with r ank r [14 ], h ence for all T ∈ E r ( q , n ) there exist exactly α ( m, r ) vectors x such that T ( x ) = T . By tr ansposition, it follows th at the re exist exactly α ( n, r ) vector s x such that S ( x ) = S for all S ∈ E r ( q , m ) . For any C ⊆ GF ( q m ) n , de fine S ( C ) def = { S ( c ) | c ∈ C } and T ( C ) def = { T ( c ) | c ∈ C } . W e obtain the following lemma. Lemma 3: For all C ⊆ GF( q m ) n , we h av e | S ( C ) | ≤ | C | ≤ α ( n, r ) | S ( C ) | and | T ( C ) | ≤ | C | ≤ α ( m, r ) | T ( C ) | . Pr oposition 1: For any constant-d imension cod e Γ ⊆ E r ( q , m ) , there exists a constant-ran k code C with length n and constant-r ank r over GF( q m ) such that r ≤ n ≤ m and S ( C ) = Γ . The cardinality o f C satisfies | Γ | ≤ | C | ≤ α ( n, r ) | Γ | . On the oth er hand, for any constant- dimension code ∆ ⊆ E r ( q , n ) , ther e exists a con stant-rank code D with length n and con stant-rank r over GF( q m ) such that r ≤ n ≤ m and T ( D ) = ∆ . Th e car dinality of D satisfies | ∆ | ≤ | D | ≤ α ( m, r ) | ∆ | . Pr oof: By Lemm a 2, fo r any U ∈ Γ there exists c U ∈ GF( q m ) n with rank r such that S ( c U ) = U . Therefore , the code C = { c U |U ∈ Γ } satisfies S ( C ) = Γ . C is a c onstant- rank code with length n an d co nstant-ran k r over GF( q m ) , and by Lemma 3, | C | satisfies | Γ | ≤ | C | ≤ α ( n, r ) | Γ | . The proof f or ∆ ⊆ E r ( q , n ) is similar and h ence om itted. Proposition 1 shows that constant-dim ension c odes can be viewed as a special class of c onstant-ran k codes. Althoug h the rank m etric is n ot dir ectly related to the su bspace metr ic in general, the maximal cardinalities of constant-dimension codes and co nstant-ran k cod es are r elated. Pr oposition 2: For all q and 1 ≤ r < d ≤ n ≤ m , A R ( q m , n, d, r ) ≥ min { A S ( q , n, 2( d − r ) , r ) , A S ( q , m, 2 r , r ) } . (5) Pr oof: Let Γ b e an optimal ( m, 2 r , r ) constant-dimen sion code over GF( q ) and ∆ b e an optimal ( n, 2 d, r ) co nstant- dimension co de ov er GF( q ) . Denote th eir card inalities as µ = A S ( q , m, 2 r , r ) and ν = A S ( q , n, 2 d, r ) and the g ener- ator matrices of their com ponen t sub spaces a s { X i } µ − 1 i =0 and { Y j } ν − 1 j =0 , respectively . By (2), for all 0 ≤ i < j ≤ ν − 1 , 2rk( Y T i | Y T j ) − 2 r ≥ 2 d , and hence r k( Y T i | Y T j ) ≥ d + r . For all 0 ≤ i ≤ µ − 1 , define b i = ( β i, 0 , β i, 1 , . . . , β i,r − 1 ) ∈ GF( q m ) r such that the expansion of β i,l with respect to a basis B m of GF( q m ) is given by the l -th row o f X i . For a ll 0 ≤ i < j ≤ ν − 1 , the matrix ( X T i | X T j ) h as full rank by (2) and h ence the elemen ts { β i, 0 , . . . , β i,r − 1 , β j, 0 , . . . , β j,r − 1 } are linearly independ ent. W e thus define the basis γ i,j = { β i, 0 , . . . , β i,r − 1 , β j, 0 , . . . , β j,r − 1 , γ 2 r , . . . , γ m − 1 } of GF( q m ) over GF( q ) . W e define the cod e C ⊆ GF( q m ) n such that c i = b i Y T i for 0 ≤ i ≤ min { µ, ν } − 1 . Expandin g c i and c j with respec t to th e basis γ i,j , we ob tain rk( c i ) = rk  Y T i | 0  = r and d R ( c i , c j ) = rk  Y T i | − Y T j | 0  = rk( Y T i | Y T j ) ≥ d + r . Therefo re, C is an ( n, d + r , r ) constant- rank code over GF( q m ) with card inality min { µ, ν } . Cor ollary 1: For all q and m , A R ( q m , n, 2 r , r ) ≥ A S ( q , n, 2 r , r ) for n ≤ m (6) A R ( q m , m, d, r ) ≥ A S ( q , m, 2 r , r ) for r < d. (7) Therefo re, a lower bound on A S is a lso a lower bound o n A R for r < d . W e may use the lower bo und on A S in ( 3). I V . B O U N D S O N C O N S TA N T - R A N K C O D E S W e derive b ound s on the maximu m ca rdinality of con stant- rank codes. W e first obser ve th at A R ( q m , n, d, r ) is a non- decreasing fu nction of m and n , an d a no n-incre asing function of d . W e also remark that the b ound s on A R ( q m , n, d, r ) derived in Section III-B for 2 r ≤ n can b e easily adapted for 2 r > n by applyin g them to n − r instead. Finally , since A R ( q m , n, 1 , r ) = N r ( q m , n ) an d A R ( q m , n, d, r ) = 1 for d > 2 r , we shall assume 2 ≤ d ≤ 2 r hencefo rth. By considering the Singleton bound for rank metric codes or MRD codes, we obtain a lower b ound and some upper b ounds on A R ( q m , n, d, r ) . Pr oposition 3: For all q and 1 ≤ r, d ≤ n ≤ m , A R ( q m , n, d, r ) ≥ M d,r for r ≥ d (8) A R ( q m , n, d, r ) ≤ q m ( n − d +1) − X j ∈ J a A R ( q m , n, d, j ) (9) A R ( q m , n, d, r ) ≤ q m ( n − d +1) − X i ∈ I r M d,i (10) A R ( q m , n, d, r ) ≤ q m ( n − d +1) − 1 for r ≥ d, (11) where I r def = { i : 0 ≤ i ≤ n, | i − r | ≥ d } and J a def = I r ∩ { a + k d : k ∈ Z } for 0 ≤ a < d . Pr oof: The codewords o f ran k r in an ( n, n − d + 1 , d ) linear MRD code ov er GF ( q m ) fo rm an ( n, d, r ) constant-rank code. Thus, A R ( q m , n, d, r ) ≥ M d,r for r ≥ d . Let C be an ( n, n − d + 1 , d ) linear MR D code over GF ( q m ) , and den ote its co dewords with r anks belo nging to I r as C ′ . For 0 ≤ j ≤ n , let C j be optim al ( n, d, j ) con stant-rank cod es and define C ′′ def = S j ∈ J a C j . The Singleton bound on the codes C r ∪ C ′ and C r ∪ C ′′ yields (1 0) and (9), r espectively . Finally , the Singleton bou nd on C ∪ { 0 } , where C is an ( n, d, r ) ( r ≥ d ) co nstant-rank code over GF( q m ) , yield s (11). Pr oposition 4: For all q and 1 ≤ r, d ≤ n ≤ m , A R ( q m , n, d, r ) ≥ N r ( q m , n ) q m ( − d +1) (12) A R ( q m , m, d, m ) ≤ A R ( q m − 1 , m − 1 , d, m − 1) · q m − 1 ( q m − 1) fo r d < m (13) A R ( q m , n, d, r ) ≤ A R ( q m , n − 1 , d, r ) · q n − 1 q n − r − 1 for r < n. (14) Pr oof: Since K q ( m, n, d, r ) is a subg raph of R q ( m, n, d ) , the in clusion map is a trivial hom omorp hism from K q ( m, n, d, r ) to R q ( m, n, d ) . B y Lemma 1, R q ( m, n, d ) is vertex transitive. W e h ence ap ply (4) to these grap hs, wh ich yields (1 2). Let B m − 1 and B m be b ases sets over GF ( q ) of GF( q m − 1 ) and GF( q m ) , respectiv ely . For all x ∈ GF( q m − 1 ) m − 1 with rank m − 1 , define g ( x ) = y ∈ GF( q m ) m such that Y =  X 0 0 1  ∈ GF( q ) m × m , (15) where X and Y are the expansion s of x and y with resp ect to B m − 1 and B m , respec ti vely . By ( 15), for all x , z ∈ GF( q m − 1 ) m − 1 with rank m − 1 , we have rk( g ( x )) = r k( x ) + 1 = m and r k( g ( x ) − g ( z )) = rk( x − z ) . Therefore g is a h omom orphism fr om K q ( m − 1 , m − 1 , d, m − 1 ) to K q ( m, m, d, m ) . Applying ( 4) to these g raphs, and noticing that α ( m, m ) = q m − 1 ( q m − 1) α ( m − 1 , m − 1 ) , we obtain (13). W e now p rove (14). No te that any vector x ∈ GF( q m ) n with r ank r belongs to  n − r 1  ELS’ s of dimen sion n − 1 . Indeed , such ELS’ s are of the form E ( x ) ⊕ N , where N ∈ E n − r − 1 ( q m , n − r ) . Let C be an optimal ( n, d, r ) constant- rank c ode over GF( q m ) . For all c ∈ C an d all V ∈ E n − 1 ( q m , n ) , we define f ( V , c ) = 1 if c ∈ V and f ( V , c ) = 0 otherwise. For all c , P V ∈ E n − 1 ( q m ,n ) f ( V , c ) =  n − r 1  , and f or all V , P c ∈ C f ( V , c ) = | C ∩ V | . Summing over all po ssible pairs, we o btain X V ∈ E n − 1 ( q m ,n ) X c ∈ C f ( V , c ) = X c ∈ C X V ∈ E n − 1 ( q m ,n ) f ( V , c ) = X c ∈ C  n − r 1  =  n − r 1  A R ( q m , n, d, r ) . Hence there exists U ∈ E n − 1 ( q m , n ) such that | C ∩ U | = P c ∈ C f ( U , c ) ≥ [ n − r 1 ] [ n 1 ] A R ( q m , n, d, r ) . The r estriction of C ∩ U to the E LS U [14] is an ( n − 1 , d, r ) constant- rank code over GF( q m ) , an d hence its cardinality satisfies q n − r − 1 q n − 1 A R ( q m , n, d, r ) ≤ | C ∩ U | ≤ A R ( q m , n − 1 , d, r ) . Eq. (12) is the counter part in r ank metric codes of the Bassalygo-Elias bo und [ 19], while (14) is analo gous to a well- known result by Johnson [20]. No te that (1 2) can be trivial for d app roachin g 2 r . Pr oposition 5: For all q and 1 ≤ r ≤ n ≤ m , A R ( q m , n, r , r ) =  n r  ( q m − 1) . (16) Pr oof: First, by (8), we o btain A R ( q m , n, r , r ) ≥  n r  ( q m − 1) . Seco nd, app lying (14) successively n − r times leads to A R ( q m , n, r , r ) ≤  n r  A R ( q m , r, r , r ) . By (1 1), we obtain A R ( q m , n, r , r ) ≤  n r  ( q m − 1) . Equality in (1 6) is thus ac hieved by the codewords of ran k r in an ( n, n − r + 1 , r ) linear MRD c ode. Pr oposition 6: For all q and 0 ≤ r < d ≤ n ≤ m , A R ( q m , n, d, r ) ≤  n r  . (17) Pr oof: Consider a co de C with min imum rank distance d and c onstant-ran k r < d . If | C | >  n r  = | E r ( q m , n ) | , then there exist two codewords in C belonging to the same ELS V ∈ E r ( q m , n ) . Their distance is hence at most equal to r , wh ich contrad icts the m inimum distance of C . Therefore, | C | ≤  n r  . Cor ollary 2: For all q , m , and n , A R ( q m , n, 2 , 1) =  n 1  . Pr oof: First, by Proposition 6, we obtain A R ( q m , n, 2 , 1) ≤  n 1  . Seco nd, by Corollary 1, we obtain A R ( q m , n, 2 , 1) ≥ A S ( q , n, 2 , 1) . W e now prove that A S ( q , n, 2 , 1) =  n 1  . For any U , V ∈ E 1 ( q , n ) , U 6 = V , we have dim( U ∩ V ) = 0 and hence d S ( U , V ) = 2 . Theref ore, E 1 ( q , n ) is a constant-dim ension co de with minimu m subspace d istance 2 and A S ( q , n, 2 , 1) =  n 1  . V . A S Y M P T OT I C R E S U LT S In this section, we study the asymptotic behavior of A R ( q m , n, d R , r ) . In order to comp are it to the asymp - totic b ehavior of A S ( q , m, d S , r ) , we use a set o f nor- malized parameters different f rom those introduced in [4]: ν = n m , ρ = r m , δ R = d R m , and δ S = d S 2 m . By d ef- inition, 0 ≤ ρ, δ R ≤ ν , and since we assume n ≤ m , ν ≤ 1 . W e consider the asymptotic rates defined as a R ( ν, δ R , ρ ) def = lim m →∞ sup h log q m 2 A R ( q m , n, d R , r ) i and a S ( δ S , ρ ) def = lim m →∞ sup h log q m 2 A S ( q , m, d S , r ) i . Adapting the re sults in [5] u sing the par ameters defined above, we obtain a S ( δ S , ρ ) = min { (1 − ρ )( ρ − δ S ) , ρ (1 − ρ − δ S ) } for 0 ≤ δ S ≤ min { ρ, 1 − ρ } and a S ( δ S , ρ ) = 0 oth erwise. W e now in vestigate ho w the A R ( q m , n, d, r ) term b ehaves as the parameter s ten d to infinity . W ithout loss of ge nerality , we only co nsider the case where 0 ≤ δ R ≤ 2 ρ , sin ce a R ( ν, δ R , ρ ) = 0 for δ R > 2 ρ . Pr oposition 7: Sup pose ν ≤ 1 . For 0 ≤ δ R ≤ ρ , a R ( ν, δ R , ρ ) = ρ (1 + ν − ρ ) − δ R . ( 18) For ρ ≤ δ R ≤ min { 2 ρ, ν } , max { 0 , ρ (1 + ν − ρ ) − δ R } ≤ a R ( ν, δ R , ρ ) ≤ ρ ( ν − δ R ) . (19) Suppose ν > 1 . For 0 ≤ δ R ≤ ρ , a R ( ν, δ R , ρ ) = ρ (1 + ν − ρ ) − ν δ R . For ρ ≤ δ R ≤ min { 2 ρ, 1 } , max { 0 , ρ (1 + ν − ρ ) − ν δ R } ≤ a R ( ν, δ R , ρ ) ≤ ρ (1 − δ R ) . Pr oof: W e g iv e the p roof fo r ν ≤ 1 , and the proo f for ν > 1 is similar and hen ce o mitted. W e first d erive a lower bound on a R ( ν, δ R , ρ ) f or all ρ . Using the com- binatorial boun ds in [14] , (12) yields A R ( q m , n, d R , r ) > q r ( m + n − r ) − σ ( q )+ m ( − d R +1) , where σ ( q ) < 2 for q ≥ 2 . Th is asymptotically becomes a R ( ν, δ R , ρ ) ≥ ρ (1 + ν − ρ ) − δ R for 0 ≤ δ R ≤ min { 2 ρ, ν } . W e now derive an up per boun d on a R ( ν, δ R , ρ ) . First, suppose r ≥ d R . App lying ( 14), we easily obtain A R ( q m , n, d R , r ) ≤  n r  A R ( q m , r, d R , r ) . Combining with ( 11), we obta in A R ( q m , n, d R , r ) ≤  n r  q m ( r − d R +1) < q r ( n − r )+ σ ( q )+ m ( r − d R +1) . Asymptotically , this b ecomes a R ( ν, δ R , ρ ) ≤ ρ ( ν − ρ ) − δ R + ρ fo r ρ ≥ δ R . Secon d, suppose r < d R . B y the same token, we obtain A R ( q m , n, d R , r ) ≤ [ n r ] [ d R r ] A R ( q m , d R , d R , r ) ≤ q r ( n − d R )+ σ ( q )+ m , and h ence a R ( ν, δ R , ρ ) ≤ ρ ( ν − δ R ) for ρ ≤ δ R . W e observe that the asymptotic b ehavior of the maximal cardinality of constant-d imension codes dep ends on whether ρ = r m ≤ 1 2 , while the asymp totic behavior of the maximal cardinality of constant-ran k cod es depend s on whether ν = n m ≤ 1 . This is d ue to the different behaviors o f ran k metric cod es o f length n over GF ( q m ) fo r m ≥ n and m < n respectively . Th e construction of an asymp totically optimal co nstant-dime nsion code in E r ( q , m ) given in [4] and revie wed in Section III-B is based on a rank metric code of length m − r over GF( q r ) . Hence r ≥ m − r fo r the ra nk metric code is equiv alent to r ≥ m/ 2 (or ρ ≥ 1 / 2 ) for th e constant-d imension cod e. By the Single ton bo und on rank metric codes, the asymp- totic b ehavior of the c ardinality of a n ( n, n − d R + 1 , d R ) linear MRD code over GF( q m ) with ν ≤ 1 is given by ν − d R . 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