Constant-Rank Codes and Their Connection to Constant-Dimension Codes

Constant-Rank Codes and Their Connection to Constant-Dimension Codes Maximilien Gadouleau, Member , IEEE, and Zhiyuan Y an, Senior Member , IEEE Abstract —Constant-dimension codes hav e recently r eceived attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achiev es the maximal cardinality both r emain open resear ch problems. In this paper , we introduce a new approach to solving these two problems. W e first establish a connection between constant-rank codes and constant-dimension codes. V ia this connection, we show that optimal constant- dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many ro ws. As such, the two afor e- mentioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectiv ely . T o this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant- rank code. Index T erms —Netw ork coding, random linear network coding, error control codes, subspace codes, constant-dimension codes, constant-weight codes, rank metric codes, subspace metric, in- jection metric. I . I N T RO D U C T I O N While random linear network coding [1]–[3] has proved to be a powerful tool for disseminating information in networks, it is highly susceptible to errors caused by various sources, such as noise, malicious or malfunctioning nodes, or insuf- ficient min-cut. If receiv ed packets are linearly combined at random to deduce the transmitted message, even a single error in one erroneous packet could render the entire transmission useless. Thus, error control for random linear network cod- ing is critical and has recei ved growing attention recently . Error control schemes proposed for random linear network coding assume two types of transmission models: some [4]– [8] depend on and take advantage of the underlying network This work was supported in part by Thales Communications Inc. and in part by a grant from the Commonwealth of Pennsylvania, Department of Com- munity and Economic De velopment, through the Pennsylvania Infrastructure T echnology Alliance (PIT A). The w ork of Zhiyuan Y an was supported in part by a summer extension grant from Air Force Research Laboratory , Rome, New Y ork, and the work of Maximilien Gadouleau was supported in part by the ANR project RISC. Part of the material in this paper was presented at 2008 IEEE International Symposium on Information Theory (ISIT 2008) in T oronto, Canada and 2008 IEEE International W orkshop on Wireless Network Coding (W iNC 2008) in San Francisco, California, USA. Maximilien Gadouleau was with the Department of Electrical and Com- puter Engineering, Lehigh University , Bethlehem, P A 18015 USA. Now he is with CReSTIC, Univ ersit ´ e de Reims Champagne-Ardenne, Reims 51100 France. Zhiyuan Y an is with the Department of Electrical and Com- puter Engineering, Lehigh Univ ersity , Bethlehem, P A, 18015 USA (e-mail: maximilien.gadouleau@univ-reims.fr; yan@lehigh.edu). topology or the particular linear network coding operations performed at v arious network nodes; others [9], [10] assume that the transmitter and receiver hav e no knowledge of such channel transfer characteristics. The contrast is similar to that between coherent and noncoherent communication systems. Error control for noncoherent random linear network coding was first considered in [9] 1 . Motiv ated by the property that random linear network coding is vector-space preserving, an operator channel that captures the essence of the noncoherent transmission model was defined in [9]. Similar to codes defined in complex Grassmannians for noncoherent multiple- antenna channels, codes defined in Grassmannians o ver a finite field [12], [13] play a significant role in error control for noncoherent random linear network coding. W e refer to these codes as constant-dimension codes (CDCs) henceforth. These codes can use either the subspace metric [9] or the injection metric [14]. The standard adv ocated approach to random linear network coding (see, e.g., [2]) in volves transmission of packet headers used to record the particular linear combination of the components of the message present in each received packet. From coding theoretic perspectiv e, the set of subspaces generated by the standard approach may be viewed as a suboptimal CDC with minimum injection distance 1 in a Grassmannian, because the whole Grassmannian forms a CDC with minimum injection distance 1 [9]. Hence, studying ran- dom linear network coding from coding theoretic perspective results in better error control schemes. General studies of subspace codes started only recently (see, for example, [15], [16]). On the other hand, there is a steady stream of works related to codes in Grassmanni- ans. For example, Delsarte [12] proved that a Grassmannian endowed with the injection distance forms an association scheme, and derived its parameters. The nonexistence of perfect codes in Grassmannians was prov ed in [13], [17]. In [18], it was shown that Steiner structures yield diameter- perfect codes in Grassmannians; properties and constructions of these structures were studied in [19]; in [20], it was shown that Steiner structures result in optimal CDCs. Related work on certain intersecting families and on byte-correcting codes can be found in [21] and [22], respectively . An application of codes in Grassmannians to linear authentication schemes was considered in [23]. In [9], a Singleton bound for CDCs and a family of codes that are nearly Singleton-bound-achieving are proposed, a recursiv e construction of CDCs which outperform the codes in [9] was giv en in [24], while a class of codes 1 A related work [11] considers security issues in noncoherent random linear network coding. 2 with ev en greater cardinality was giv en in [25]. Despite the asymptotic optimality of the Singleton bound and the codes proposed in [9], neither is optimal in finite cases: upper bounds tighter than the Singleton bound exist and can be achiev ed in some special cases [20]. Thus, two research problems about CDCs remain open: the maximal cardinality of a CDC with finite dimension and minimum distance is yet to be determined, and it is not clear how to construct an optimal code that achiev es the maximal cardinality . In this paper , we introduce a no vel approach to solving the two aforementioned problems. Namely , we aim to solve these problems via constant-rank codes (CRCs), which are the counterparts in rank metric codes of constant Hamming weight codes. There are sev eral reasons for our approach. First, it is difficult to solve the two problems above directly based on CDCs since projectiv e spaces lack a natural group structure [10]. Also, the rank metric is very similar to the Hamming metric in many aspects, and hence familiar results from the Hamming space can be readily adapted. Furthermore, existing results for rank metric codes in the literature are more extensi ve than those for CDCs. Finally , the rank metric has been shown relev ant to error control for both noncoherent [10] and coherent [14] random linear network coding. Based on our approach, this paper makes two main con- tributions. Our first main contribution is that we establish a connection between CRCs and CDCs. V ia this connection, we sho w that optimal CDCs correspond to optimal CRCs ov er matrices with suf ficiently many ro ws. This connection con verts the aforementioned open research problems about CDCs into research problems about CRCs, thereby allowing us to take advantage of existing results on rank metric codes in general to tackle such problems. Despite previous works on rank metric codes, constant-rank codes per se unfortunately have receiv ed little attention in the literature. Our second main contribution is our in vestigation of CRCs. In particular, we deriv e upper and lower bounds on the maximum cardinality of a CRC, propose explicit constructions of optimal or asymptotically optimal CRCs, and establish asymptotic bounds on the maximum rate of CRCs. Our in vestigation of CRCs not only is important for our construction of CDCs, b ut also serves as a powerful tool to study CDCs and rank metric codes. The rest of the paper is organized as follows. Section II revie ws some necessary background. In Section III, we de- termine the connection between optimal CRCs and optimal CDCs. In Section IV, we study the maximum cardinality of CRCs, and present our results on the asymptotic beha vior of the maximum rate of a CRC. I I . P R E L I M I NA R I E S A. Rank metric codes Error correction codes with the rank metric [26]–[28] hav e been receiving steady attention in the literature due to their applications in storage systems [28], public-key cryptosystems [29], space-time coding [30], and network coding [9], [10]. Below we revie w some important properties of rank metric codes established in [26]–[28]. 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 , referred to as the rank metric henceforth. Please note that the rank metric for the vector representation of rank metric codes is defined dif ferently [27]. Since the connection between the matrix representation of rank metric codes and CDCs is more natural, we consider the matrix representation of rank metric codes henceforth. W e denote the number of matrices of rank r ( 0 ≤ r ≤ min { m, n } ) in GF( q ) m × n as N R ( q , m, n, r ) =  n r  α ( m, r ) [27], where α ( m, 0) def = 1 , α ( m, r ) def = Q r − 1 i =0 ( q m − q i ) , and  n r  def = α ( n, r ) /α ( r , r ) for r ≥ 1 . The term  n r  is often referred to as a Gaussian binomial [31], and satisfies q r ( n − r ) ≤  n r  < K − 1 q q r ( n − r ) (1) for all 0 ≤ r ≤ n , where K q = Q ∞ j =1 (1 − q − j ) [32]. K − 1 q decreases with q and satisfies 1 < K − 1 q ≤ K − 1 2 < 4 . W e denote the volume (i.e., the number of points) of the intersec- tion of two spheres in GF( q ) m × n of radii r and s and with rank distance d between their centers as J R ( q , m, n, r , s, d ) . A closed-form formula for J R ( q , m, n, r , s, d ) is determined in [33]. A rank metric code is a subset of GF( q ) m × n , and its minimum rank distance , denoted as d R , is simply the minimum rank distance over all possible pairs of distinct code words. It is shown in [26]–[28] that the minimum rank distance of a code of cardinality M in GF( q ) m × n satisfies d R ≤ n − log q m M +1 . In this paper, we refer to this bound as the Singleton bound for rank metric codes and codes that attain the equality as maximum rank distance (MRD) codes. W e refer to the subclass of MRD codes introduced in [34] as generalized Gabidulin codes. These codes are based on the vector view of rank metric codes, described as follows. The columns of a matrix X ∈ GF( q ) m × n can be mapped into elements of the field GF( q m ) according to a basis B m of GF( q m ) over GF( q ) . Hence X can be mapped into the v ector x ∈ GF( q m ) n , and the rank of X is equal to the maximum number of linearly independent coordinates of x . Generalized Gabidulin codes are linear MRD codes over GF( q m ) for m ≥ n . For all q , 1 ≤ d ≤ r ≤ n ≤ m , the number of codewords of rank r in an ( n, n − d + 1 , d ) linear MRD code over GF( q m ) is denoted by M ( q , m, n, d, r ) , and it is kno wn that [27] M ( q, m, n, d, r ) =  n r  r X j = d ( − 1) r − j  r j  q ( r − j )( r − j − 1) / 2  q m ( j − d +1) − 1  . (2) W e will omit the dependence of the quantities defined abov e on q , m , and n when there is no ambiguity in some proofs. B. Constant-dimension codes W e refer to the set of all subspaces of GF( q ) n with dimension r as the Grassmannian of dimension r and denote it as E r ( q , n ) , where | E r ( q , n ) | =  n r  ; we refer to E ( q , n ) = S n r =0 E r ( q , n ) as the projectiv e space. For U, V ∈ E ( q , n ) , 3 their intersection U ∩ V is also a subspace in E ( q , n ) , and we denote the smallest subspace containing the union of U and V as U + V . Both the subspace metric [9, (3)] d S ( U, V ) def = dim( U + V ) − dim( U ∩ V ) = 2 dim( U + V ) − dim( U ) − dim( V ) and injection metric [14, Def. 1] d I ( U, V ) def = 1 2 d S ( U, V ) + 1 2 | dim( U ) − dim( V ) | = dim( U + V ) − min { dim( U ) , dim( V ) } are metrics over E ( q, n ) . The Grassmannian E r ( q , n ) endowed with either the sub- space metric or the injection metric forms an association scheme [9], [12]. Since d S ( U, V ) = 2 d I ( U, V ) for all U, V ∈ E r ( q , n ) and the injection distance provides a more nat- ural distance spectrum, i.e., 0 ≤ d I ( U, V ) ≤ r for all U, V ∈ E r ( q , n ) , we consider only the injection metric for Grassmannians and CDCs henceforth. W e denote the number of subspaces in E r ( q , n ) at distance d from a gi ven subspace as N C ( d ) = q d 2  r d  n − r d  [9]. A subset of E r ( q , n ) is called a constant-dimension code (CDC). W e denote the maximum cardinality of a CDC in E r ( q , n ) with minimum distance d as A C ( q , n, r , d ) . Construc- tions of CDCs and bounds on A C ( q , n, r , d ) hav e been gi ven in [9], [16], [20], [24], [25], [35]. In particular , A C ( q , n, r , 1) =  n r  and it is shown [9], [20] for r ≤  n 2  and 2 ≤ d ≤ r , q ( n − r )( r − d +1) ≤ A C ( q , n, r , d ) ≤  n r − d +1   r r − d +1  . (3) I I I . C O N N E C T I O N B E T W E E N C O N S TA N T - D I M E N S I O N C O D E S A N D C O N S TA N T - R A N K C O D E S In this section, we first establish some connections between the rank metric and the injection metric. W e then define constant-rank codes and we show how optimal constant-rank codes can be used to construct optimal CDCs. Let us denote the ro w space and the column space of X ∈ GF( q ) m × n ov er GF( q ) as R ( X ) and C ( X ) , respec- tiv ely . Follo wing the con vention of coding theory , a generator matrix of a subspace U is any full rank matrix whose ro w space is the subspace U . The notations introduced abov e are naturally extended to codes as follows: for C ⊆ GF( q ) m × n , C ( C ) def = { U ∈ E ( q , m ) : ∃ M ∈ C , C ( M ) = U } and R ( C ) def = { V ∈ E ( q , n ) : ∃ M ∈ C , R ( M ) = V } . Lemma 1: For U ∈ E r ( q , m ) , V ∈ E r ( q , n ) , and X ∈ GF( q ) m × n with rank r , C ( X ) = U and R ( X ) = V if and only if there e xist a generator matrix G ∈ GF( q ) r × m of U and a generator matrix H ∈ GF( q ) r × n of V such that X = G T H . The proof of Lemma 1 is straightforward and hence omitted. W e remark that X = G T H is referred to as a rank factoriza- tion [36]. W e now derive a relation between the rank distance between two matrices and the injection distances between their respectiv e ro w and column spaces. Theor em 1: For all X , Y ∈ GF( q ) m × n , d I ( R ( X ) , R ( Y )) + d I ( C ( X ) , C ( Y )) − | rk( X ) − rk( Y ) | ≤ d R ( X , Y ) ≤ min { d I ( R ( X ) , R ( Y )) , d I ( C ( X ) , C ( Y )) } + min { rk( X ) , rk( Y ) } . Pr oof: By Lemma 1, we hav e X = C T R and Y = D T S , where C ∈ GF( q ) rk( X ) × m , R ∈ GF( q ) rk( X ) × n , D ∈ GF( q ) rk( Y ) × m , S ∈ GF( q ) rk( Y ) × n are generator matrices of C ( X ) , R ( X ) , C ( Y ) , and R ( Y ) , respectiv ely . Hence X − Y = ( C T | − D T )( R T | S T ) T and rk( X − Y ) ≤ min { rk( C T | − D T ) , rk( R T | S T ) } . Sylvester’ s la w of nullity in [37, Corollary 6.1] or in [38, 0.4.5 (c)], states that rk( AB ) ≥ rk( A ) + rk( B ) − n for any matrices A with n columns and B with n rows. Therefore, rk( C T | − D T ) + rk( R T | S T ) − rk( X ) − rk( Y ) ≤ rk( X − Y ) ≤ min { rk( C T | − D T ) , rk( R T | S T ) } . Since rk( C T | − D T ) = d I ( C ( X ) , C ( Y )) + min { rk( X ) , rk( Y ) } and rk( R T | S T ) = d I ( R ( X ) , R ( Y )) + min { rk( X ) , rk( Y ) } , we obtain the claim. A constant-rank code (CRC) of constant rank r in GF( q ) m × n is a nonempty subset of GF( q ) m × n such that all elements hav e rank r . Proposition 1 below sho ws how a CRC leads to two CDCs with their minimum injection distance related to the minimum rank distance of the CRC. Pr oposition 1: Let C be a CRC of constant rank r and minimum distance d R in GF( q ) m × n . Then R ( C ) ⊆ E r ( q , n ) and C ( C ) ⊆ E r ( q , m ) have minimum distances at least d R − r . Proposition 1 follows directly from Theorem 1 and hence its proof is omitted. When the minimum rank distance of a CRC is greater than its constant rank, Proposition 2 belo w shows how the CRC leads to two CDCs with the same cardinality , and the relations between their distances can be further strengthened. Pr oposition 2: If C is a CRC of constant rank r and minimum rank distance d + r ( 1 ≤ d ≤ r ) in GF( q ) m × n , then R ( C ) ⊆ E r ( q , n ) and C ( C ) ⊆ E r ( q , m ) hav e car - dinality |C | and their minimum injection distances satisfy d I ( C ( C )) + d I ( R ( C )) ≤ d + r ≤ min { d I ( C ( C )) , d I ( R ( C )) } + r . Pr oof: Let X and Y be any two distinct codew ords in C . By Theorem 1, d I ( R ( X ) , R ( Y )) ≥ d R ( X , Y ) − r ≥ d > 0 , and hence d I ( R ( C )) ≥ d and | R ( C ) | = |C | . Similarly , d I ( C ( X ) , C ( Y )) ≥ d > 0 , and thus d I ( C ( C )) ≥ d and | C ( C ) | = |C | . Furthermore, if d R ( X , Y ) = d + r , then by Theorem 1, d + r ≥ d I ( C ( X ) , C ( Y )) + d I ( R ( X ) , R ( Y )) ≥ d I ( C ( C )) + d I ( R ( C )) . W e remark that the requirement of having a minimum distance greater than the constant rank is a strong condition on the CRC. Indeed, any codeword of a linear code has rank at least equal to the minimum distance of a code. Therefore, no set of codewords of a linear code (and, in particular , a linear MRD code) satisfies this condition. Therefore, while CRCs with minimum distance no more than their constant- rank will be directly constructed from linear MRD codes in Section IV -B, designing CRCs with minimum distance greater than their constant-rank will require translates of codes instead, which are not as easy to manipulate. Propositions 1 and 2 show how to construct CDCs from a CRC. Alternatively , Proposition 3 below sho ws that we can construct a CRC from a pair of CDCs. Pr oposition 3: Let M be a CDC in E r ( q , m ) and N be a CDC in E r ( q , n ) such that |M| = |N | . Then there exists a CRC C ⊆ GF( q ) m × n with constant rank r and cardinality 4 |M| satisfying C ( C ) = M and R ( C ) = N . Furthermore, its minimum distance d R satisfies d I ( N ) + d I ( M ) ≤ d R ≤ min { d I ( N ) , d I ( M ) } + r . Pr oof: Denote the generator matrices of the component subspaces of M and N as G i and H i , respecti vely and define the code C formed by the codewords X i = G T i H i for 0 ≤ i ≤ |M|− 1 . Then C ( C ) = M and R ( C ) = N by Lemma 1 and the lower bound on d R follows from Theorem 1. Let X i and X j be distinct code words in C such that d I ( C ( X i ) , C ( X j )) = d I ( M ) . By Theorem 1, we obtain d R ≤ d R ( X i , X j ) ≤ d I ( M ) + r . Similarly , we also obtain d R ≤ d I ( N ) + r . The connections between general CRCs and CDCs deriv ed abov e naturally imply relations between optimal CRCs and op- timal CDCs. W e denote the maximum cardinality of a CRC in GF( q ) m × n with constant rank r and minimum rank distance d as A R ( q , m, n, d, r ) . If C is a CRC in GF( q ) m × n with constant rank r , then its transpose code C T forms a CRC in GF( q ) n × m with the same constant rank, minimum distance, and car - dinality . Therefore A R ( q , m, n, d, r ) = A R ( q , n, m, d, r ) , and henceforth in this paper we assume n ≤ m without loss of generality . W e further observe that A R ( q , m, n, d, r ) is a non- decreasing function of m and n , and a non-increasing function of d , and that A C ( q , n, r , d ) is a non-decreasing function of n and a non-increasing function of d . Pr oposition 4: For all q , 1 ≤ d ≤ r ≤ n ≤ m , and any 0 ≤ p ≤ r , min { A C ( q , n, r , d + p ) , A C ( q , m, r , r − p ) } ≤ A R ( q , m, n, d + r, r ) ≤ A C ( q , n, r , d ) . (4) Pr oof: Using the monotone properties of A R ( q , m, n, d R , r ) and A C ( q , n, r , d ) above, the upper bound follows from Proposition 2, while the lower bound follows from Proposition 3 for d I ( M ) = r − p and d I ( N ) = d + p . W e remark that the lower bound in (4) is trivial for d + p > min { r , n − r } or r − p > min { r , m − r } . Therefore, the lower bound in (4) is nontrivial when max { 0 , 2 r − m } ≤ p ≤ min { r − d, n − r − d } . Combining the bounds in (4), we obtain that the cardinal- ities of optimal CRCs over matrices with sufficiently many rows equal the cardinalities of CDCs with related distances. Furthermore, we show that optimal CDCs can be constructed from such optimal CRCs. Theor em 2: For all q , 2 r ≤ n ≤ m , and 1 ≤ d ≤ r , A R ( q , m, n, d + r, r ) = A C ( q , n, r , d ) if either d = r or m ≥ m 0 , where m 0 = ( n − r )( r − d + 1) + r + 1 . Furthermore, if C is an optimal CRC in GF( q ) m × n with constant rank r and minimum distance d + r for m ≥ m 0 or d = r , then R ( C ) is an optimal CDC in E r ( q , n ) with minimum distance d . Pr oof: First, the case where d = r directly follows from (4) for p = 0 . Second, if d < r and m ≥ m 0 , by (3) we obtain A C ( q , m, r , r ) ≥ q m − r ≥ q m 0 − r . Also, by [32, Lemma 1], we obtain q r ( r − d +1) − 1 < α ( r, r − d + 1) ≤ q r ( r − d +1) for all 2 ≤ d < r , and hence (3) yields A C ( q , n, r , d ) < q ( n − r )( r − d +1)+1 = q m 0 − r ≤ A C ( q , m, r , r ) . Thus, when p = 0 , the lower bound in (4) simplifies to A R ( q , m, n, d + r , r ) ≥ A C ( q , n, r , d ) . Combining with the upper bound in (4), we obtain A R ( q , m, n, d + r, r ) = A C ( q , n, r , d ) . The second claim immediately follows from Proposition 2. Theorem 2 implies that to determine A C ( q , n, r , d ) and to construct optimal CDCs, it is sufficient to determine A R ( q , m, n, d + r , r ) and to construct optimal CRCs ov er matrices with sufficiently many ro ws. W e observe that this implies that A R ( q , m, n, d + r , r ) remains constant for all m ≥ m 0 . When d = r , A R ( q , m, n, 2 r , r ) remains constant for m ≥ n . When d = 1 , m 0 = ( n − r + 1) r + 1 , but A R ( q , m, n, r + 1 , r ) remains constant for m ≥ n , and this is shown in Section IV -B. In comparison to existi ng constructions of CDCs [9], [10], [15], [20], [24], [35], our construction based on CRCs has two advantages. First and foremost, by Theorem 2, our construction leads to optimal CDCs for all parameter values. In contrast, none of previously proposed constructions lead to optimal CDCs for all parameter values. For example, the construction based on liftings of rank metric codes [9], [10] leads to suboptimal CDCs (though sometimes they may be nearly optimal). This is because CDCs of dimension r based on liftings of rank metric codes hav e the highest possible cov- ering radius r [39], which implies there exists a subspace that can be added to such CDCs without decreasing the minimum distance. The CDCs constructed in similar approaches [24] are not optimal for the same reason. The optimality for some constructions [15], [25] are not clear . The construction based on Steiner structures [20] and that based on computational techniques [35] lead to optimal CDCs, but are applicable to special cases only . The second advantage of our construction is an additional degree of freedom, which is the number m of rows of the matrices. By Theorem 2, optimal CRCs lead to optimal CDCs provided that m ≥ m 0 , and hence the parameter m may v ary an ywhere abo ve the lower bound m 0 . On the other hand, the constructions in the literature use fix ed dimensions and do not introduce any new parameter . For instance, in order to obtain a CDC in E r ( q , n ) by lifting a rank metric code, the original code must be in GF( q ) r × ( n − r ) . This additional de gree of freedom is significant for code design, as it may be easier to construct optimal CRCs with larger m . Thus our construction is a very promising approach to solving the two open research problems mentioned in Section I. I V . C O N S TA N T - R A N K C O D E S Having pro ved that optimal CRCs over matrices with suf- ficiently many rows lead to optimal CDCs, in this section we in vestigate the properties of CRCs. A. Bounds W e now deriv e bounds on the maximum cardinality of CRCs. W e first remark that the bounds on A R ( q , m, n, d, r ) deriv ed in Section III can be used in this section. Also, since A R ( q , m, n, 1 , r ) = N R ( q , m, n, r ) and A R ( q , m, n, d, r ) = 1 for d > 2 r , we shall assume 2 ≤ d ≤ 2 r henceforth 2 . 2 Since the minimum distance of a code is defined using pairs of distinct codew ords, the minimum distance for a code of cardinality one is defined to be zero sometimes. 5 W e first deri ve the counterparts of the Gilbert and the Hamming bounds for CRCs in terms of intersections of spheres with rank radii. Pr oposition 5: For all q , 1 ≤ r, d ≤ n ≤ m , and t = b d − 1 2 c , N R ( q , m, n, r ) P d − 1 i =0 J R ( q , m, n, i, r , r ) ≤ A R ( q , m, n, d, r ) ≤ min 1 ≤ s ≤ n ( N R ( q , m, n, s ) P t i =0 J R ( q , m, n, i, s, r ) ) . Pr oof: The proof of the lower bound is straightforward and hence omitted. Let C = { c k } K − 1 k =0 be a CRC with constant rank r and minimum distance d in GF( q ) m × n . F or all 0 ≤ k ≤ K − 1 and 0 ≤ s ≤ n − 1 , if we denote the set of matrices in GF( q ) m × n with rank s and distance ≤ t from c k as R k,s , then | R k,s | = P t i =0 J R ( i, s, r ) for all k . Clearly R k,s ∩ R l,s = ∅ for all k 6 = l , and hence N R ( s ) ≥ | S K − 1 k =0 R k,s | = K | R k,s | , which yields the upper bound. W e no w derive upper bounds on A R ( q , m, n, d, r ) . W e begin by proving the counterpart in rank metric codes of a well- known bound on constant-weight codes proved by Johnson in [40]. Pr oposition 6 (J ohnson bound for rank metric codes): For all q , 1 ≤ r , d < n ≤ m , A R ( q , m, n, d, r ) ≤ q n − 1 q n − r − 1 A R ( q , m, n − 1 , d, r ) . Pr oof: Let C be an optimal CRC in GF( q ) m × n with constant rank r and minimum distance d . For all C ∈ C and all V ∈ E n − 1 ( q , n ) , we define f ( V , C ) = 1 if R ( C ) ⊆ V and f ( V , C ) = 0 otherwise. For any C , the row space of C is contained in  n − r 1  subspaces in E n − 1 ( q , n ) and hence P V ∈ E n − 1 ( q ,n ) f ( V , C ) =  n − r 1  ; for all V , P C ∈C f ( V , C ) = |{ C ∈ C : R ( C ) ⊆ V }| . Summing over all possible pairs, we obtain X V ∈ E n − 1 ( q ,n ) X C ∈C f ( V , C ) = X V ∈ E n − 1 ( q ,n ) |{ C ∈ C : R ( C ) ⊆ V }| , X C ∈C X V ∈ E n − 1 ( q ,n ) f ( V , C ) =  n − r 1  A R ( q , m, n, d, r ) . Hence there exists U ∈ E n − 1 ( q , n ) such that |{ C ∈ C : R ( C ) ⊆ U }| = P C ∈C f ( U, C ) ≥ [ n − r 1 ] [ n 1 ] A R ( q , m, n, d, r ) . By Lemma 1, all the code words C i with R ( C i ) ⊆ U can be expressed as C i = G T i H i U , where H i ∈ GF( q ) r × ( n − 1) and U ∈ GF( q ) ( n − 1) × n is a generator matrix of U . Therefore, the code { G T i H i } forms a CRC in GF( q ) m × ( n − 1) with constant rank r , minimum distance d , and cardinality |{ C ∈ C : R ( C ) ⊆ U }| , and hence q n − r − 1 q n − 1 A R ( q , m, n, d, r ) ≤ |{ C ∈ C : R ( C ) ⊆ U }| ≤ A R ( q , m, n − 1 , d, r ) . The Singleton bound for rank metric codes yields upper bounds on A R ( q , m, n, d, r ) . For any I ⊆ { 0 , 1 , . . . , n } , let A R ( q , m, n, d, I ) denote the maximum cardinality of a code in GF( q ) m × n with minimum rank distance d such that all code- words have ranks belonging to I . Then A R ( q , m, n, d, r ) ≤ q m ( n − d +1) − A R ( q , m, n, d, P r ) , where P r def = { i : 0 ≤ i ≤ n, | i − r | ≥ d } . W e now determine the counterpart of the Singleton bound for CRCs. Pr oposition 7 (Singleton bound for CRCs): For all 0 ≤ i ≤ min { d − 1 , r } , A R ( q , m, n, d, r ) ≤ A R ( q , m, n − i, d − i, J i ) , where J i = { r − i, r − i + 1 , . . . , min { n − i, r }} . Pr oof: Let C be an optimal CRC in GF( q ) m × n with constant rank r and minimum distance d , and consider the code C i obtained by puncturing i coordinates of the codewords in C . Since i ≤ r , the codewords of C i all hav e ranks between r − i and min { n − i, r } . Also, since i < d , any two codew ords have distinct puncturings, and we obtain |C i | = |C | and d R ( C i ) ≥ d − i . Hence A R ( q , m, n, d, r ) = |C | = |C i | ≤ A R ( q , m, n − i, d − i, J i ) . W e no w combine the counterpart of the Johnson bound in Proposition 6 and that of the Singleton bound in Proposition 7 in order to obtain an upper bound on A R ( q , m, n, d, r ) for d ≤ r . Pr oposition 8: For all q , 1 ≤ d ≤ r ≤ n ≤ m , A R ( q , m, n, d, r ) ≤  n r  α ( m, r − d + 1) . Pr oof: Applying Proposition 6 n − r times successiv ely , we obtain A R ( q , m, n, d, r ) ≤  n r  A R ( q , m, r , d, r ) . For n = r and i = d − 1 , J i = { r − d + 1 } and hence Proposition 7 yields A R ( q , m, r , d, r ) ≤ A R ( q , m, r − d + 1 , 1 , r − d + 1) = N R ( q , m, r − d + 1 , r − d + 1) = α ( m, r − d + 1) . Thus A R ( q , m, n, d, r ) ≤  n r  α ( m, r − d + 1) . W e now deriv e the counterpart in rank metric codes of the Bassalygo-Elias bound [41] and we also tighten the bound when d > r + 1 . For a code C ⊆ GF( q ) l × k ( k ≤ l ), A i def = |{ C ∈ C : rk( C ) = i }| for 0 ≤ i ≤ l ; we refer to A i ’ s as the rank distribution of C . Pr oposition 9 (Bassalygo-Elias bound for rank metric codes): For max { r, d } ≤ k ≤ n , 0 ≤ s ≤ k , k ≤ l ≤ m , and any code C ⊆ GF( q ) l × k with minimum rank distance d and rank distribution A i ’ s, A R ( q , m, n, d, r ) ≥ max s, { A i } ,k,l P n i =0 A i J R ( q , l, k , s, r , i ) N R ( q , l, k , s ) . (5) Furthermore, if r + 1 < d ≤ 2 r , then A R ( q , m, n, d, r ) ≥ max s, { A i } ,k,l P n i =0 A i J R ( q , l, k , s, r , i ) N R ( q , l, k , s ) − P n i =0 A i P d − r − 1 t =0 J R ( q , l, k , s, t, i ) . (6) The proof of Proposition 9 is gi ven in Appendix A. Although the RHS of (5) and (6) can be maximized ov er { A i } , it is dif ficult to do so since { A i } is not a vailable for most rank metric codes with the exception of linear MRD codes. Thus, we deriv e a bound using the rank weight distribution of linear MRD codes. Cor ollary 1: For all q , 1 ≤ r , d ≤ n ≤ m , A R ( q , m, n, d, r ) ≥ N R ( q , m, n, r ) q m ( − d +1) . Pr oof: Applying (5) to an ( n, n − d +1 , d ) MRD code over GF( q m ) , we obtain N R ( s ) A R ( d, r ) ≥ P n i =0 M ( d, i ) J R ( s, r , i ) . Summing for all 0 ≤ s ≤ n , we obtain A R ( d, r ) ≥ N R ( r ) q m ( − d +1) since P n s =0 J R ( s, r , i ) = N R ( r ) . 6 The RHS of (5) and (6) decrease rapidly with increasing d , rendering the bounds in (5) and (6) tri vial for d approaching 2 r . Proposition 10 below sho ws that the bound in Corol- lary 1 is tight up to a scalar for d ≤ r . T o mea- sure the tightness, we introduce a ratio C ( q, m, n, d, r ) def = A R ( q , m, n, d, r ) / [ N R ( q , m, n, r ) q m ( − d +1) ] for 2 ≤ d ≤ r ≤ n ≤ m . Pr oposition 10: For all q , 2 ≤ d ≤ r ≤ n ≤ m , C ( q , m, n, d, r ) ≤ q 2 q 2 − 1 for r + d − 1 ≤ m and C ( q , m, n, d, r ) < q − 1 q K − 1 q otherwise. Pr oof: By Proposition 8, C ( d, r ) ≤ q m ( d − 1) α ( m, r − d + 1) /α ( m, r ) = q ( m − r + d − 1)( d − 1) /α ( m − r + d − 1 , d − 1) . Since α ( n, l ) > q q − 1 K q q nl for all 1 ≤ l ≤ n − 1 [32, Lemma 1], we obtain C ( d, r ) < q − 1 q K − 1 q . Finally , α ( n, l ) ≥ q 2 − 1 q 2 q nl for l ≤ n − l [32, Lemma 1] yields C ( d, r ) ≤ q 2 q 2 − 1 for r + d − 1 ≤ m . The proof of Proposition 10 indicates that the upper bound in Proposition 8 is also tight up to a scalar for d ≤ r . Howe ver , these bounds are not constructiv e. Below we derive constructiv e bounds on A R ( q , m, n, d, r ) . B. Constructions of CRCs W e first gi ve a construction of asymptotically optimal CRCs when d ≤ r . W e assume the matrices in GF( q ) m × n are mapped into vectors in GF( q m ) n according to a fixed basis B m of GF( q m ) over GF( q ) . Pr oposition 11: For all q , 2 ≤ d ≤ r ≤ n ≤ m , A R ( q , m, n, d, r ) ≥ M ( q , m, n, d, r ) >  n r  q m ( r − d ) . Pr oof: The codew ords of rank r in an ( n, n − d + 1 , d ) lin- ear MRD code over GF( q m ) form a CRC in GF( q ) m × n with constant rank r and minimum distance d . Thus, A R ( d, r ) ≥ M ( d, r ) . W e now prov e the lower bound on M ( d, r ) . First, for d = r , M ( r, r ) =  n r  ( q m − 1) >  n r  . Second, suppose d < r . By (2), M ( d, r ) can be expressed as M ( d, r ) =  n r  P r j = d ( − 1) r − j µ j , where µ j def = q ( r − j )( r − j − 1) / 2  r j  ( q m ( j − d +1) − 1) . It can be easily sho wn that µ j > µ j − 1 for d + 1 ≤ j ≤ r , and hence M ( d, r ) ≥  n r  ( µ r − µ r − 1 ) . Therefore, M ( d, r ) ≥  n r  [( q m ( r − d +1) − 1) −  r 1  ( q m ( r − d ) − 1)] >  n r  q m ( r − d ) . Cor ollary 2: For all q , 1 ≤ r ≤ n ≤ m , A R ( q , m, n, r , r ) =  n r  ( q m − 1) . Pr oof: By Proposition 8, A R ( r , r ) ≤  n r  ( q m − 1) , and by Proposition 11, A R ( r , r ) ≥ M ( r, r ) =  n r  ( q m − 1) . By Corollary 2, the codewords of rank r in an ( n, n − r + 1 , r ) linear MRD code are optimal CRCs with minimum distance r . Proposition 12 shows that for all but one case, the codew ords of rank r in an ( n, n − d + 1 , d ) MRD code form a code whose cardinality is close to that of an optimal CRC up to a scalar which tends to 1 for lar ge q . T o measure the optimality , we introduce a ratio B ( q, m, n, d, r ) def = A R ( q , m, n, d, r ) / M ( q , m, n, d, r ) for 1 ≤ d < r ≤ n ≤ m . Pr oposition 12: For all q , 1 ≤ d < r ≤ n ≤ m and m ≥ 3 , B (2 , m, m, m − 1 , m ) ≤ 2 m − 1 − 1 (7) B ( q , m, m, m − 1 , m ) < q − 1 q − 2 for q > 2 (8) B ( q , m, m, m − 2 , m ) < ( q 2 − 1)( q − 1) ( q 2 − 1)( q − 2) + 1 (9) B ( q , m, m, d, m ) < ( q 3 − 1)( q 2 − 1)( q − 1) ( q 3 − 1)( q 2 − 1)( q − 2) + q 3 − 2 for d < m − 2 (10) B ( q , m, n, d, r ) < q q − 1 for r < m. (11) The proof of Proposition 12 is gi ven in Appendix B. W e now construct CRCs for d > r using generalized Gabidulin codes [34]. Let g ∈ GF( q m ) n hav e rank n , and for 0 ≤ i ≤ m − 1 , denote the vector in GF( q m ) n obtained by raising each coordinate of g to the q ai -th power , g [ i ] , where a and m are coprime. Let C be the ( n, n − d + 1 , d ) generalized Gabidulin code over GF( q m ) generated by  g [0] T , g [1] T , . . . , g [ n − d ] T  T , and C 0 be the ( n, d − r , n − d + r + 1) generalized Gabidulin code generated by  g [ n − d +1] T , g [ n − d +2] T , . . . , g [ n − r ] T  T . W e consider the coset C + c 0 , where c 0 ∈ C 0 , and we denote the number of codew ords of rank r in C + c 0 as σ r ( c 0 ) . Lemma 2: For all d > r , there exists c 0 ∈ C 0 such that σ r ( c 0 ) ≥  n r  q m ( r − d +1) . Pr oof: Any code word c 0 ∈ C 0 can be expressed as c 0 = c n − d +1 g [ n − d +1] + c n − d +2 g [ n − d +2] + . . . + c n − r g [ n − r ] , where c i ∈ GF( q m ) for n − d + 1 ≤ i ≤ n − r . If c n − r = 0 , then ( C + c 0 ) ⊂ D , where D is the ( n, n − r, r + 1) generalized Gabidulin code generated by  g [0] T , g [1] T , . . . , g [ n − r − 1] T  T . Therefore σ r ( c 0 ) = 0 if c n − r = 0 . Denote the number of codewords of rank r in C ⊕ C 0 as τ r . Since S c 0 ∈C 0 ( C + c 0 ) = C ⊕ C 0 , we hav e τ r = P c 0 ∈C 0 σ r ( c 0 ) . Also, C ⊕ C 0 forms an ( n, n − r + 1 , r ) MRD code, and hence τ r = M ( q , m, n, r , r ) =  n r  ( q m − 1) . Suppose that for all c 0 ∈ C 0 , σ r ( c 0 ) <  n r  q m ( r − d +1) . Then τ r = P c 0 : c n − r 6 =0 σ r ( c 0 ) <  n r  ( q m − 1) , which contradicts τ r =  n r  ( q m − 1) . Although Lemma 2 prov es the existence of a v ector c 0 for which the translate C + c 0 has high cardinality , it does not indicate how to choose c 0 . For d = r + 1 , it can be shown that all c 0 ∈ C 0 satisfy the bound, and that they all lead to optimal codes. Cor ollary 3: If d = r + 1 , then σ r ( c 0 ) =  n r  for all c 0 ∈ C 0 . Pr oof: First, by Proposition 4, σ r ( c 0 ) ≤ A R ( q , m, n, r + 1 , r ) ≤ A C ( q , n, r , 1) =  n r  for all c 0 ∈ C 0 . Suppose there exists c 0 such that σ r ( c 0 ) <  n r  . Then τ r <  n r  ( q m − 1) , which contradicts τ r =  n r  ( q m − 1) . Pr oposition 13: For all q , 1 ≤ r < d ≤ n ≤ m , A R ( q , m, n, d, r ) ≥  n r  q n ( r − d +1) , and a class of codes that satisfy this bound can be constructed from Lemma 2. Pr oof: The code words of rank r in a code considered in Lemma 2 form a CRC in GF( q ) m × n with constant rank r , minimum distance d , and cardinality ≥  n r  q m ( r − d +1) . There- 7 fore, A R ( q , m, n, d, r ) ≥  n r  q m ( r − d +1) . The proof is con- cluded by noting that A R ( q , m, n, d, r ) ≥ A R ( q , n, n, d, r ) ≥  n r  q n ( r − d +1) . Cor ollary 4: For all q , 1 ≤ r < n ≤ m , A R ( q , m, n, r + 1 , r ) =  n r  = A C ( q , n, r , 1) . This can be shown by combining Propositions 4 and 13. W e note that  n r  is independent of m . W e also remark that the lower bound in Proposition 13 is also tri vial for d approaching 2 r . Since the proof is only partly constructiv e, computer search can be used to help find better results for small parameter values. By Proposition 4, the lo wer bounds on A R ( q , m, n, d, r ) deriv ed in this section for d > r can be viewed as lower bounds on the maximum cardinality of a corresponding CDC. Although in Corollary 4, we obtain a tight bound for d = r + 1 , we remark that the bound in Proposition 13 does not improv e on the lo wer bounds on A C ( q , n, r , d − r ) previously deri ved in the literature when d > r + 1 . Howe ver , the construction of good CDCs from CRCs is an interesting topic for future work. C. Asymptotic r esults W e study the asymptotic behavior of CRCs using the following set of normalized parameters: ν = n m , ρ = r m , and δ = d m . By definition, 0 ≤ ρ, δ ≤ ν , and since we assume n ≤ m , ν ≤ 1 . W e consider the asymptotic rate de- fined as a R ( ν, δ, ρ ) def = lim m →∞ sup h log q m 2 A R ( q , m, n, d, r ) i . W e now inv estigate ho w A R ( q , m, n, d, r ) behaves as the parameters tend to infinity . W ithout loss of generality , we only consider the case where 0 ≤ δ ≤ min { ν , 2 ρ } , since a R ( ν, δ, ρ ) = 0 for δ > 2 ρ . Pr oposition 14: For 0 ≤ δ ≤ ρ , a R ( ν, δ, ρ ) = ρ (1 + ν − ρ ) − δ . For ρ ≤ δ , we have to distinguish three cases. First, for 2 ρ ≤ ν , max  (1 − ρ )( ν − ρ ) 1 + ν − 2 ρ (2 ρ − δ ) , ρ (2 ν − ρ ) − ν δ  ≤ a R ( ν, δ, ρ ) ≤ ( ν − ρ )(2 ρ − δ ) . (12) Second, for ν ≤ 2 ρ ≤ 1 , max { ρ (1 − ρ )( ν − δ ) , ρ (2 ν − ρ ) − ν δ } ≤ a R ( ν, δ, ρ ) ≤ ρ ( ν − δ ) . (13) Third, for 2 ρ ≥ 1 , max n ρ 2 (1 + ν − 2 ρ − δ ) , ρ (2 ν − ρ ) − ν δ, 0 o ≤ a R ( ν, δ, ρ ) ≤ ρ ( ν − δ ) . (14) The proof of Proposition 14 is gi ven in Appendix C. Proposition 12 indicates that the codewords of a gi ven rank in a linear MRD code form asymptotically optimal CRCs. In particular , Proposition 14 shows that the set of code words with rank n in an ( n, n − d + 1 , d ) linear MRD code constitutes a CRC of rank n and asymptotic rate of ν − δ , which is equal to the asymptotic rate of an optimal rank metric code [42]. W e can split the range of δ into tw o re gions: when δ ≤ ρ , the asymptotic rate of CRCs is determined due to the construction of good CRCs when d ≤ r ; when δ ≥ ρ , we only hav e bounds 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 δ a R ( ν , δ , ρ ) Fig. 1. Asymptotic bounds on the maximal rate of a CRC as a function of δ , with ν = 3 / 4 and ρ = 1 / 5 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 δ a R ( ν , δ , ρ ) Fig. 2. Asymptotic bounds on the maximal rate of a CRC as a function of δ , with ν = 3 / 4 and ρ = 2 / 5 . on the asymptotic rate of CRCs. Also, the lower bounds based on the connection between CDCs and CRCs (the first lower bound in the LHS of (12), (13), and (14) are tighter for 2 ρ ≤ ν and on the other hand become trivial for ρ approaching 1 . The bounds on a R ( ν, δ, ρ ) in the three cases in (12), (13), and (14) are illustrated in Figures 1, 2, and 3 for ν = 3 / 4 and ρ = 1 / 5 , 2 / 5 , 3 / 5 , respecti vely . V . C O N C L U S I O N Rank metric codes and CDCs have been considered for error control in noncoherent random linear network coding. It has been shown that these two classes of codes are related by the lifting operation, which turns an optimal rank metric code into a nearly optimal constant-dimension code. Ho wev er , liftings of rank metric codes are not optimal constant-dimension codes. In this paper , we first established a nov el connection between CRCs and CDCs, by showing that optimal CRCs over matrices with sufficiently many rows lead to optimal CDCs with a related minimum injection distance. In comparison to previously proposed constructions of CDCs, our construction based on CRCs guarantees the optimality of CDCs, and hence is a promising approach. Despite previous works on rank metric codes in general, CRCs hav e received little attention in 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 δ a R ( ν , δ , ρ ) Fig. 3. Asymptotic bounds on the maximal rate of a CRC as a function of δ , with ν = 3 / 4 and ρ = 3 / 5 . the literature. W e hence in vestigated the properties of CRCs, deriv ed bounds on their cardinalities, and proposed explicit constructions of CRCs in some cases. Although we hav e not been able to propose constructions of optimal CRCs in all cases, we hope our no vel connection between CRCs and CDCs and in vestigation of CRCs can lead to constructions of optimal CDCs, which is the topic of our future work. V I . A C K N O W L E D G M E N T The authors are grateful to the anonymous revie wers and the Associate Editor Dr . Ludo T olhuizen for their constructive comments, which have resulted in improvements in both the results and the presentation of the paper . In particular, comments by one of the revie wers hav e slightly improved the results in Theorem 1 and Propositions 2, 3, 4, and 14. A P P E N D I X A. Pr oof of Pr oposition 9 (Bassalygo-Elias bound for rank metric codes) Pr oof: For all X ∈ GF( q ) l × k with rank s and C ∈ C , we define f r ( X , C ) = 1 if d R ( X , c ) = r and f r ( X , C ) = 0 otherwise. Note that P X :rk( X )= s f r ( X , C ) = J R ( q , l, k , s, r , rk( C )) for all C ∈ C and P c ∈ C f r ( X , C ) = |{ Y ∈ C − X : rk( Y ) = r }| ≤ A R ( q , l, k , d, r ) for all X ∈ GF( q ) l × k . W e obtain X C ∈C X X :rk( X )= s f r ( X , C ) = n X i =0 A i J R ( q , l, k , s, r , i ) , (15) X X :rk( X )= s X C ∈C f r ( X , C ) ≤ N R ( q , l, k , s ) A R ( q , l, k , d, r ) . (16) Combining (15) and (16), we obtain A R ( q , l, k , d, r ) ≥ P n i =0 A i J R ( q , l, k , s, r , i ) N R ( q , l, k , s ) . (17) Suppose d > r + 1 . F or all C ∈ C , let us denote the set of matrices with rank s at distance at most d − r − 1 from C as S C , and S def = S C ∈C S C . For X ∈ S C , we have d R ( X , C ) ≤ d − r − 1 < r . W e have for C 0 ∈ C and C 0 6 = C , d R ( X , C 0 ) ≥ d R ( C , C 0 ) − d R ( X , C ) ≥ r + 1 ; and hence f r ( X , C 0 ) = 0 for all C 0 ∈ C . Therefore, P C ∈C f r ( X , C ) = 0 for all X ∈ S and X X :rk( X )= s X C ∈C f r ( X , C ) = X X ∈ S X C ∈ C f r ( X , C ) + X X / ∈ S rk( X )= s X C ∈C f r ( X , C ) ≤ [ N R ( q , l, k , s ) − | S | ] A R ( q , l, k , d, r ) . (18) Since d − r − 1 < d 2 , the balls with radius d − r − 1 around the codew ords are disjoint and hence | S | = P n i =0 A i P d − r − 1 t =0 J R ( q , l, k , s, t, i ) . Combining (15) and (18), we obtain A R ( q , l, k , d, r ) ≥ P n i =0 A i J R ( q , l, k , s, r , i ) N R ( q , l, k , s ) − P n i =0 A i P d − r − 1 t =0 J R ( q , l, k , s, t, i ) . (19) Note that (17) and (19) both hold for any s and rank spectrum { A i } . Furthermore, since A R ( q , l, k , d, r ) is a non-decreasing function of l and k , A R ( q , m, n, d, r ) ≥ A R ( q , l, k , d, r ) for all max { r , d } ≤ k ≤ n and k ≤ l ≤ m . Thus, we hav e (5) and (6). B. Pr oof of Pr oposition 12 Pr oof: By Proposition 8, we obtain A R ( q , m, m, d, m ) ≤ α ( m, m − d + 1) for r = n = m and A R ( q , m, n, d, r ) ≤  n r  α ( m, r − d + 1) <  n r  q m ( r − d +1) otherwise. W e no w deri ve lower bounds on M ( q , m, n, d, r ) . Again, M ( q , m, n, d, r ) =  n r  P r j = d ( − 1) j µ j where µ j > µ j − 1 for d + 1 ≤ j ≤ r . Therefore, when needed, we shall only consider the last terms in the summation. First, M ( q , m, m, m − 1 , m ) = ( q 2 m − 1) − q m − 1 q − 1 ( q m − 1) > q − 2 q − 1 ( q 2 m − 1) > q − 2 q − 1 α ( m, 2) , which leads to (8). For q = 2 , M (2 , m, m, m − 1 , m ) = 2(2 m − 1) = (2 m − 1 − 1) − 1 α ( m, 2) , which results in (7). Second, when r = n = m and d = m − 2 , M ( q , m, m, m − 2 , m ) = ( q 3 m − 1) − α ( m, 1) q − 1 ( q 2 m − 1) + α ( m, 2) ( q 2 − 1)( q − 1) ( q m − 1) > q − 2 q − 1 α ( m, 1)( q 2 m − 1) + 1 ( q 2 − 1)( q − 1) α ( m, 2)( q m − 1) > ( q 2 − 1)( q − 2) + 1 ( q 2 − 1)( q − 1) α ( m, 1) , which leads to (9). Third, when r = n = m and d < m − 2 , 9 by considering the last four terms in the summation, we obtain M ( q , m, m, d, m ) > ( q m ( m − d +1) − 1) − α ( m, 1) q − 1 ( q m ( m − d ) − 1) + α ( m, 2) ( q 2 − 1)( q − 1) ( q m ( m − d − 1) − 1) − α ( m, 3) ( q 3 − 1)( q 2 − 1)( q − 1) ( q m ( m − d − 2) − 1) >  q − 2 q − 1 + q 3 − 2 ( q 3 − 1)( q 2 − 1)( q − 1)  α ( m, m − d + 1) , which results in (10). Fourth, when d < r < m , by considering the last two terms in the summation, we obtain M ( q , m, n, d, r ) ≥  n r   ( q m ( r − d +1) − 1) −  r 1  ( q m ( r − d ) − 1)  ≥  n r   q m ( r − d +1) − 1 − q m ( r − d )+ r + q r  ≥  n r  q m ( r − d +1) (1 − q r − m ) . Therefore, since r < m , B ( q , m , n, d, r ) < (1 − q r − m ) − 1 ≤ q q − 1 , which leads to (11). C. Pr oof of Pr oposition 14 Pr oof: W e first deri ve a lo wer bound on a R ( ν, δ, ρ ) . For d ≤ r , Proposition 11 yields A R ( d, r ) ≥ q r ( n − r )+ m ( r − d ) , which asymptotically becomes a R ( ν, δ, ρ ) ≥ ρ (1 + ν − ρ ) − δ for δ ≤ ρ . Similarly , for d > r , Proposition 13 yields A R ( q , m, n, d, r ) ≥ q r ( n − r )+ n ( r − d +1) , which asymptotically becomes a R ( ν, δ, ρ ) ≥ ρ (2 ν − ρ ) − ν δ for δ ≥ ρ . Proposition 4 and (3) yield log q A R ( d, r ) ≥ min { ( n − r )(2 r − d − p + 1) , ( m − r )( p + 1) } for d > r and 2 r ≤ n . T reating the two terms as functions and assuming that p is real, the lo wer bound is maximized when p = ( n − r )(2 r − d +1) − m + r m + n − 2 r . Using p = j ( n − r )(2 r − d +1) − m + r m + n − 2 r k , asymptotically we obtain a R ( ν, δ, ρ ) ≥ (1 − ρ )( ν − ρ ) 1+ ν − 2 ρ (2 ρ − δ ) for 2 ρ ≤ ν . For d > r and n ≤ 2 r ≤ m , Proposition 4 and (3) lead to log q A R ( d, r ) ≥ min { r ( n − d − p + 1) , ( m − r )( p + 1) } . After maximizing this expression over p , we asymptotically obtain a R ( ν, δ, ρ ) ≥ ρ (1 − ρ )( ν − δ ) for ν ≤ 2 ρ ≤ 1 . For d > r and 2 r ≥ m , Proposition 4 and (3) lead to log q A R ( d, r ) ≥ min { r ( n − d − p + 1) , r ( m − 2 r + p + 1) } . After maximizing this expression over p , we asymptotically obtain a R ( ν, δ, ρ ) ≥ ρ 2 (1 + ν − 2 ρ − δ ) for 2 ρ ≥ 1 . W e no w derive an upper bound on a R ( ν, δ, ρ ) . 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