Error Performance Analysis of Maximum Rank Distance Codes

Transcription

1 Error Performance Analysis of Maximum Rank Distance Codes arxiv:cs/ v1 [cs.it] 8 Dec 2006 Maximilien Gadouleau and Zhiyuan Yan Department of Electrical and Computer Engineering Lehigh University, PA 18015, USA s: {magc, yan}@lehigh.edu Abstract In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable (MDS) codes. Using these properties, we show that, for MRD codes with error correction capability t, the decoder error probability of bounded distance decoders decreases exponentially with t 2 based on the assumption that all errors with the same rank are equally likely. Finally, our simulation results show that our bounds seem applicable to other error models as well and that MRD codes are more resilient against crisscross errors than MDS codes. I. INTRODUCTION The idea of using the rank as a metric comes from Delsarte [1]. Error-control codes with the rank metric [2] [5] have been receiving some attention due to their applications in storage systems [3], public-key cryptosystems [4], and space-time coding [5]. In particular, as areal density of current storage equipments, which require that information be stored on two-dimensional surfaces, grows ever higher, it becomes more important to protect data against two-dimensional errors. Codes against several types of two-dimensional errors have been studied in the literature: Most two-dimensional codes are designed to correct a single cluster error of rectangular shape [6] [11] or other shapes [12], [13]; Other works consider the rank of the error array [2], [3], or crisscross errors [3], [14], [15]. Codes with the rank metric have been considered to protect data against crisscross errors (two-dimensional errors that are confined to a number of rows Part of the material in this paper was presented at IEEE ITW 2006 in Chengdu, China.

2 2 or columns or both), which occur in various storage applications [3], [16] [21] such as memory chip arrays, helical tapes, and linear magnetic tapes. The pioneering works in [2] and [3] have established many important properties of codes with the rank metric. In [2], a Singleton bound on the minimum rank distance of rank metric codes was established, and codes that attain the equality were called maximum rank distance (MRD) codes. An explicit construction for a subclass of MRD codes, which is referred to as Gabidulin codes henceforth, was also proposed in [2], and this construction was extended in [22]. In [3], an equivalent bound was established and an explicit construction was introduced, which essentially also leads to Gabidulin codes. In [3], it was shown that MRD codes are also optimal in the sense of Singleton bound in crisscross weight, a metric 1 considered in [3], [14] for crisscross errors. Decoding algorithms that parallel the extended Euclidean algorithm (EEA) and the Berlekamp-Massey algorithm were proposed for MRD codes in [2] and [3] respectively. In this paper, we investigate the error performance of MRD codes. The main results of this paper are new upper bounds on the error probability of bounded rank distance decoders for MRD codes. Our bounds indicate that the decoder error probability of MRD codes with error correction capability t decreases exponentially with t 2. To derive these bounds, we assume all errors with the same rank are equally likely. In Section VI-A, we argue that this error model approximates crisscross errors. Furthermore, although our bounds are derived based on this assumption on the error model, our simulation results in Section VI-B show that our bounds are applicable to other error models as well. Note that if the additive error has a rank no more than the error correction capability t, a bounded distance decoder produces the correct codeword. When the error has a rank greater than t, a bounded distance decoder produces either a decoder failure or a decoder error. Thus, our bounds on the decoder error probability when the error has rank greater than t characterizes the error performance of the code. We remark that our bounds are analogous to the upper bounds on the error probability of bounded Hamming distance decoders for maximum distance separable (MDS) codes in [23]. Assuming all errors with the same Hamming weight are equiprobable, the decoder error probability of MDS codes was bounded by 1 t! in [23]. We also note that the decoder error probability considered in this paper is different from the decoding error probability in [14] in several aspects: The latter refers to the probability that the decoder does not decode correctly, that is, it includes both decoder errors and decoder failures; The latter is based on a different channel (error) model. The derivation of our bounds is a nontrivial extension of the approach in [23]. Some by-products of the derivation are significant in themselves: 1 It is also referred to as the cover weight in [14].

3 3 We introduce the concept of elementary linear subspace (ELS) and derive its properties, which are similar to those of a set of coordinates. Using elementary linear subspaces, we derive useful properties of MRD codes. In particular, we prove a combinatorial property of MRD codes, derive a bound on the rank distribution of these codes, and show that the restriction of an MRD code on an ELS is also an MRD code. These properties parallel those of MDS codes. We also investigate the error performance of MRD codes by numerical simulations. Our simulation results show that our bounds seem applicable to the error probability of bounded rank distance decoders for MRD codes when other error models are used. Due to the widespread applications of MDS codes (Reed-Solomon codes in most cases) in storage systems, we also compare the error performance of MRD codes and MDS codes against crisscross errors by simulations. The rest of the paper is organized as follows. Section II gives a brief review of the rank metric, Singleton bound, and MRD codes. In Section III, we derive some combinatorial properties which are used in the derivation of our upper bounds. In Section IV, we first introduce the concept of elementary linear subspace and study its properties, and then obtain some important properties of MRD codes. In Section V, we derive our upper bounds on the decoder error probability of MRD codes. Section VI presents our simulation results for MRD and MDS codes, and the error performance of MRD codes is compared with that of MDS codes. A. Rank metric II. PRELIMINARIES In [2] and [3], the rank metric is defined in different approaches. In this paper we follow the approach in [2]. Consider an n-dimensional vector x = (x 0,x 1,...,x n 1 ) GF(q m ) n. Assume {α 0,α 1,...,α m 1 } is a basis set of GF(q m ) over GF(q), then for j = 0,1,...,n 1, x j can be written as x j = m 1 i=0 x i,jα i, where x i,j GF(q) for i = 0,1,...,m 1. Hence, x j can be expanded to an m-dimensional column vector (x 0,j,x 1,j,...,x m 1,j ) T with respect to the basis set {α 0,α 1,...,α m 1 }. Let X be the m n matrix obtained by expanding all the coordinates of x. That is, x 0,0 x 0,1... x 0,n 1 x X = 1,0 x 1,1... x 1,n ,. x m 1,0 x m 1,1... x m 1,n 1

4 4 where x j = m 1 i=0 x i,jα i. The rank norm of the vector x (over GF(q)), denoted as rk(x GF(q)), is defined to be the rank of the matrix X over GF(q), i.e., rk(x GF(q)) def = rank(x) [2]. All the ranks are over the base field GF(q) unless otherwise specified in this paper. To simplify notations, we denote the rank norm of x as rk(x) henceforth. The rank norm of x is also the number of coordinates in x that are linearly independent over GF(q). The field GF(q m ) may be viewed as a vector space over GF(q). The coordinates of x thus span a linear subspace of GF(q m ), denoted as S(x), and the rank of x is the dimension of S(x). For all x,y GF(q m ) n, it is easily verified that d(x,y) def = rk(x y) is a metric over GF(q m ) n, referred to as the rank metric henceforth [2]. Hence, the minimum rank distance d R of a code is simply the minimum rank distance over all possible pairs of distinct codewords. A code with a minimum rank distance d R can correct all errors with rank up to t = (d R 1)/2. The number of vectors of rank 0 u min{m,n} in GF(q m ) n is given by N u = [ n u] A(m,u), where A(m,u) is defined as follows: A(m,0) = 1 and A(m,u) = u 1 i=0 (qm q i ) for u 1. The [ ] n u term is the Gaussian binomial [24], defined as [ n [ u] = A(n,u)/A(u,u). Note that n v] is the number of v-dimensional linear subspaces of GF(q) n. B. The Singleton bound and MRD codes The minimum rank distance of a code can be specifically bounded. First, the minimum rank distance d R of a code of length n over GF(q m ) is obviously bounded above by min{m,n}. Codes that satisfy d R = m are studied in [25]. Also, it can be shown that d R d H [2], where d H is the minimum Hamming distance of the same code. Due to the Singleton bound on the minimum Hamming distance of block codes [26], the minimum rank distance of a block code of length n and cardinality M over GF(q m ) thus satisfies d R n log q m M +1. (1) In this paper, we refer to the bound in (1) as the Singleton bound for rank metric codes and codes that attain the equality as MRD codes. Note that although an MRD code is not necessarily linear, Equation (1) implies that its cardinality is a power of q m. III. COMBINATORIAL RESULTS In this section, we derive some combinatorial properties which will be instrumental in the derivation of our upper bounds on the decoder error probability of MRD codes in Section V. First, we derive bounds on A(m,u). Note that a trivial upper bound on A(m,u) is q mu. Let us now derive some lower bounds on A(m,u).

5 5 Lemma 1: For 0 u m, A(m,u) > q mu σ(q), where σ(q) = 1 ln(q) k=1 1 k(q k 1) function of q with σ(2) Also, for 0 u m 1, A(m,u) > q q 1 qmu σ(q). Proof: is a decreasing We have A(m,u) = q mu Mu, where M u = m j=m u+1 log q(1 q j ). Expanding the logarithms into Taylor series, we obtain M u = 1 q (m u)k q mk ln(q) k=1 k(q k 1). The first part of Lemma 1 follows from the fact that M u increases with u, with maximum M m = 1 1 q mk ln(q) k=1 k(q k 1) < σ(q). We have A(m,m 1) = q q 1 q m A(m,m), and hence for u m 1, we get M u M m 1 = ( ) ( ) log q q q 1 +M m < log q q q 1 +σ(q). The second part of Lemma 1 thus follows. Corollary 1: For n 0 and 0 t n, we have [ n t] < q t(n t)+σ(q). Proof: By definition, [ n t] = A(n,t)/A(t,t). Since A(n,t) q nt and by Lemma 1, A(t,t) > q t2 σ(q). Thus, [ n t] < q t(n t)+σ(q). Lemma 2: For m 2 and 0 u m/2, we have A(m,u) q2 1 q 2 q mu. Proof: Let us denote q mu A(m,u) function of u. Thus, it suffices to show that D(m, m/2 ) q2 q 2 1 odd, m = 2p+1, we have as D(m,u). It can be easily verified that D(m,u) is an increasing D(2p+1,p) = = < q (2p+1)p p 1 i=0 (q2p+1 q i ) q 2p2 p 1 i=0 (q2p q i 1 ) q 2p2 p 1 i=0 (q2p q i ) = D(2p,p). for m 2. First, notice that if m is Hence, we need to consider only the case where m = 2p, with p 1. Let us further show that D(2p,p) is a monotonically decreasing function of p since D(2p+2,p+1) = = q 2p+2 q 2p+2 1 The maximum of D(2p,p) is hence given by D(2,1) = q2 q 2 1. q 2p+2 q 2p+2 q q2p+2 q p+1 q 2p+2 D(2p,p) q 2p+1 (q 2p+2 q p+1 ) (q 2p+1 1)(q 2p+2 D(2p,p). (2) 1) IV. PROPERTIES OF MRD CODES Many properties of MDS codes are established by studying sets of coordinates. These sets of coordinates may be viewed as linear subspaces which have a basis of vectors with Hamming weight 1. Similarly, some properties of MRD codes may be established using elementary linear subspaces (ELS s), which can be considered as the counterparts of sets of coordinates.

6 6 A. Elementary linear subspaces Definition 1 (Elementary linear subspace): A linear subspace V of GF(q m ) n is said to be elementary if it has a basis set B consisting of row vectors in GF(q) n. B is called an elementary basis of V. For 0 v n, we define E v (q m,n) as the set of all ELS s with dimension v in GF(q m ) n. Lemma 3: A linear subspace V of GF(q m ) n is an ELS if and only if there exists a basis set consisting of vectors of rank 1 for V. Proof: The necessity is straightforward by definition. Let us show the sufficiency. Let A = {a i } v 1 i=0 be a basis set of V such that rk(a i ) = 1 for 0 i v 1. For 0 i v 1, all the coordinates of a i are equal to some nonzero coordinate a i,j scaled by some element of GF(q). Define b i = a 1 i,j a i, then b i GF(q) n. Applying this transformation to all the vectors in A, we obtain B = {b i }. Clearly, B is a basis set consisting of row vectors in GF(q) n for V. We also remark that not all linear subspaces are elementary. For example, the span of a vector of rank u > 1 has dimension 1, but requires u vectors of rank 1 to span it. Next, we show that the properties of elementary linear subspaces are similar to those of sets of coordinates. Lemma 4: There exists a bijection between E v (q m,n) and E v (q,n). Proof: Let V be an ELS of GF(q m ) n with an elementary basis B. For any positive integer a, denote the span of B over GF(q a ) as B q a def = } x i b i x i GF(q a ),b i B { v 1 i=0 and consider the mapping f : E v (q m,n) E v (q,n) given by V = B q m V = B q. This mapping takes V to a unique V, which is the span of B over GF(q). Note that dim(v ) = dim(v). Since for V W E v (q m,n) f(v) f(w), f is injective. Also, sincefor all V E v (q,n) there exists V E v (q m,n) such that f(v) = V, f is surjective. that Since the number of linear subspaces in GF(q) n with dimension v is given by [ n v], Lemma 4 implies Corollary 2: There are [ n v] ELS s in GF(q m ) n with dimension v. Proposition 1: For any V E v (q m,n) there exists V E n v (q m,n) such that V V = GF(q m ) n, where V V denotes the direct sum of V and V. Proof: We will provide a constructive proof. Let V = f(v) and let B be a basis of V, then there exists V with basis B such that V V = GF(q) n. Note that B B is a basis set of GF(q) n. Denote f 1 ( V ) as V. We now want to show that V V = GF(q m ) n. First, dim( V) = n v, hence we only

7 7 need to show that V V = {0}. Suppose there exists y 0 in V V, then there is a nontrivial linear relationship over GF(q m ) among the elements of B and those of B. This may be expressed as y i b i = 0. (3) b i B B Applying the trace function to each coordinate on both sides of (3), we obtain b i B B Tr(y i)b i = 0, which implies a linear dependence over GF(q) of the vectors in B and B. This contradicts the fact that B B is a basis. Therefore, V V = GF(q m ) n. We say that V is an elementary complement of V. Even though an elementary complement always exists, we remark that it may not be unique. The diameter of a code for the Hamming metric is defined in [26] as the maximum Hamming distance between two codewords. Similarly, we can define the rank diameter of a linear subspace. Definition 2: The rank diameter of a linear subspace L of GF(q m ) n is defined to be the maximum rank among the vectors in L, i.e., δ(l) def = max x L {rk(x)}. Proposition 2: For an ELS V GF(q m ) n, δ(v) dim(v). Furthermore, if dim(v) m, then δ(v) = dim(v). Proof: Let us denote dim(v) as v. Any vector x V can be expressed as the sum of at most v vectors of rank 1, hence its rank is upper bounded by v. Thus, δ(v) v by Definition 2. If v m, we show that there exists a vector in V with rank v. Let B = {b i } v 1 i=0 be an elementary basis of V, and consider y = v 1 i=0 α ib i, where {α i } m 1 i=0 is a basis set of GF(q m ) over GF(q). If we expand the coordinates of y with respect to the basis {α i } m 1 i=0, we get Y = ( b T 0,...,bT v 1,0T,...,0 T) T. Since the row vectorsb 0,b 1,,b v 1 are linearly independent overgf(q), Y has rankv andrk(y) = v. Lemma 5: Any vector x GF(q m ) n with rank u (u m) belongs to an ELS A of dimension u. Also, if rk(x) = u, then x is not contained in any ELS of dimension less than u. Proof: Suppose x = (x 0,x 1,...,x n 1 ) has rank u, then x has u linearly independent coordinates. Without loss of generality, assume the first u coordinates are linearly independent. Thus, for j = 0,,n 1, x j = u 1 i=0 a ijx i, where a ij GF(q). That is, x = (x 0,x 1,...,x u 1 )A, where A = {a ij } u 1,n 1 i=0,j=0 = (at 0,...,aT u 1 )T. Thus x = u 1 i=0 x ia i, with a i GF(q) n for 0 i u 1. Let A be the ELS of GF(q m ) n spanned by a i s, then dim(a) = u and x A. This proof can be easily adapted to the case where rk(x) < u. Suppose x B with rk(x) = u, where B is an ELS with dim(b) = t < u. Then, δ(b) rk(x) = u > t = dim(b), which contradicts Proposition 2.

8 8 Let L be a linear subspace of GF(q m ) n and let L be complementary to L, i.e., L L = GF(q m ) n. We denote the projection of x on L along L as x L [27]. Remark that x = x L +x L. Note that for any given linear subspace L, its complementary linear subspace L is not unique. Thus, x L depends on both L and L, and is well-defined only when both L and L are given. All the projections in this paper are with respect to a pair of fixed linear subspaces complementary to each other. Definition 3: Let x GF(q m ) n and L be a linear subspace. The vector x vanishes on L if there exists a linear subspace L complementary to L such that x = x L. Lemma 6: Given a vector x GF(q m ) n of rank u, there exists an ELS with dimension n u on which x vanishes. Also, x does not vanish on any ELS with dimension greater than n u. Proof: By Lemma 5, there exists A E u (q m,n) such that x A. Let Ā be an elementary complement of A. Thus, x vanishes on Ā by definition. Also, suppose x vanishes on an ELS B with dimension greater than n u. Then there exists an ELS B with dimension < u such that x B, which contradicts Lemma 5. The decomposition over two complementary ELS s induces a mapping from GF(q m ) n to GF(q m ) v. Definition 4: Let V E v (q m,n) an elementary basis B = {b 0,...,b v 1 } and an elementary complement V. For anyx GF(q m ) n, we definer V (x) = (r 0,...,r v 1 ) GF(q m ) v such thatx V = v 1 i=0 r ib i. Lemma 7: Ther V function is linear: x,y GF(q m ) n and a GF(q m ),r V (x+y) = r V (x)+r V (y) and r V (ax) = ar V (x). Also, for any x GF(q m ) n, rk(r V (x)) = rk(x V ). Proof: The linearity is straightforward. We can express x V = r V (x)b where B = (b T 0,...,bT v 1 )T is a v n matrix over GF(q) with full rank. Therefore rk(x V ) = rk(r V (x)). Definition 5: ForV E v (q m,n) with an elementary basisb = {b 0,...,b v 1 }, let V be an elementary complement of V with an elementary basis B = {b v,...,b n 1 }. For any x GF(q m ) n, we define s V, V(x) = (r V (x),r V (x)) GF(qm ) n. Lemma 8: For any x GF(q m ) n, rk(s V, V(x)) = rk(x). Proof: Note that x = x V +x V = r V (x)(b T 0,bT 1,...,bT v 1 )T +r V(x)(b T v,bt v+1,...,bt n 1 )T = s V, V(x)B, where B = (b T 0,...,bT n 1 )T is an n n matrix over GF(q) with full rank. Therefore rk(s V, V(x)) = rk(x). Corollary 3: For any x GF(q m ) n and two complementary ELS s V and V, 0 rk(x V ) rk(x) and rk(x) rk(x V )+rk(x V ).

9 9 Corollary 3 illustrates the difference between ELS s and sets of coordinates. When V and V are two complementary sets of coordinates, the Hamming weight of a vector x GF(q m ) n is the sum of the Hamming weights of the projections of x on V and V. B. Properties of MRD codes We now derive some useful properties for MRD codes, which will be instrumental in Section V. These properties are similar to those of MDS codes. Let C be an MRD code over GF(q m ) with length n (n m), cardinality q mk, redundancy r = n k, and minimum rank distance d R = n k +1. Note that C may be linear or nonlinear. First, we derive the basic combinatorial property of MRD codes. Lemma 9 (Basic combinatorial property): For K E k (q m,n) and its elementary complement K and any vector k K, there exists a unique codeword c C such that c K = k. Proof: Suppose there exist c,d C,c d such that c K = d K. Then c d K, and 0 < rk(c d) n k by Proposition 2, which contradicts the fact that C is MRD. Then all the codewords lead to different projections on K. Since C = K = q mk, for any k K there exists a unique c such that c K = k. Lemma 9 allows us to bound the rank distribution of MRD codes. Lemma 10 (Bound on the rank distribution): Let A u be the number of codewords in C with rank u. Then, for u d R, [ ] n A u (q m 1) u r. (4) u Proof: By Lemma 6, any codeword c with rank u d R vanishes on an ELS with dimension v = n u. Thus (4) can be established by first determining the number of codewords vanishing on a given ELS of dimension v, and then multiplying by the number of such ELS s, [ n v] = [ n u]. For V Ev (q m,n), V is properly contained in an ELS K with dimension k since v k 1. By Lemma 9, c is completely determined by c K. Given an elementary basis set of K, it suffices to determine r K (c). However, c K vanishes on V, hencev of the coordinates of r K (c) must be zero. By Lemma 6, the other k v coordinates must be nonzero. Hence, a codeword that vanishes on V is completely determined by k v arbitrary nonzero coordinates. There are at most (q m 1) k v = (q m 1) u r choices for these coordinates, and hence at most (q m 1) u r codewords that vanish on V. Note that the exact formula for the rank distribution of linear MRD codes was derived in [2]. However, the bound in (4) is more instrumental for the present application. Definition 6 (Restriction of a code): For V E v (q m,n) (n v k) and its elementary complement V, C V = {r V (c) c C} is called the restriction of C to V.

10 10 It is well known that a punctured MDS code is an MDS code [28]. We now show that the restriction of an MRD code to an ELS is also MRD. Lemma 11 (Restriction of an MRD code): For an ELS V with dimension n v k, C V is an MRD code with length v, cardinality q mk, and minimum rank distance d R = v k +1 over GF(q m ). Proof: Clearly, C V is a code over GF(q m ) with length v and cardinality q mk. For c d C, consider x = c d. By Lemma 7, we have rk(r V (c) r V (d)) = rk(r V (c d)) = rk(x V ) rk(x) rk(x V) n k+1 (n v) = v k+1. The Singleton bound on C V completes the proof. V. PERFORMANCE OF MRD CODES We evaluate the error performance of MRD codes using a bounded rank distance decoder. We assume that the errors are additive and that all errors with the same rank are equiprobable. A bounded rank distance decoder produces a codeword within rank distance t of the received word if it can find one, and declares a decoder failure if it cannot. In the following, we first derive bounds on the decoder error probability assuming the error has rank u. In the end, we derive a bound on the decoder error probability that does not depend on u at all. Thus, it is independent of the weight of the error. Clearly, if u t, the decoder produces the correct codeword. When the error has rank greater than t, the decoder produces either a decoder failure or a decoder error. We denote the probabilities of decoder error and failure for the bounded distance decoder for error correction capability t and an error of rank u as P E (t;u) and P F (t;u) respectively. If u t, then P F (t;u) = P E (t;u) = 0. When u > t, P E (t;u)+p F (t;u) = 1. In particular, if t < u < d R t, then P E (t;u) = 0 and P F (t;u) = 1. Thus we investigate the case where u d R t and P E (t;u) characterizes the performance of the code. We assume that the MRD code is linear, and without loss of generality that the all-zero codeword is sent. Thus, the received word can be any vector with rank u with equal probability. We call a vector decodable if it lies within rank distance t of some codeword. If D u denotes the number of decodable vectors of rank u, then for u d R t we have P E (t;u) = D u N u = D u [ n u] A(m,u), (5) Hence the main challenge is to derive upper bounds on D u. We have to consider two cases, u d R and d R t u < d R, separately. Proposition 3: For u d R, D u where V t = t i=0 N i is the volume of a ball of rank radius t. [ ] n (q m 1) u r V t, (6) u

11 11 Proof: Any decodable vector can be uniquely written as c+e, where c C and rk(e) t. For a fixed e, C+e is an MRD code, which satisfies Equation (4). Therefore, the number of decodable words of rank u is at most [ n u] (q m 1) u r, by Lemma 10, multiplied by the number of possible error vectors, V t. Lemma 12: Suppose y = (y 0,...,y v 1 ) GF(q m ) v has rank w, and let u = n v. Then there exist at most [ u s w] A(m,s w)q w(u s+w) vectors z = (z 0,...,z u 1 ) GF(q m ) u such that x = (y,z) GF(q m ) n has rank s. Proof: Without loss of generality, suppose the w linearly independent coordinates of y are y = (y 0,...,y w 1 ). Since x and y have s and w linearly independent coordinates respectively, z has s w coordinates (without loss of generality, assume these are the first s w coordinates of z) denoted as z = (z 0,...,z s w 1 ) such that {y 0,...,y w 1,z 0,...,z s w 1 } are linearly independent. For 0 i u 1, z i = a i + b i, where a i S(y) and b i S(z ) and z = a + b = (a 0,a 1,...,a u 1 ) + (b 0,b 1,...,b u 1 ). There are q w(u s+w) choices for the vector a = (0,...,0,a s w,...,a u 1 ). The vector b = (z 0,...,z s w 1,b s w,...,b u 1 ) has rank s w, so there are at most [ u s w] A(m,s w) choices for b. Proposition 4: For d R t u < d R, we have [ ] n t [ ] v t [ ] u D u (q m 1) w r A(m,s w)q w(u s+w). (7) u w s w w=d u s=w Proof: Recall that a decodable vector of rank u can be expressed as c + e, where c C and rk(e) t. This decodable vector vanishes on an ELS V with dimension v = n u by Lemma 6. We have w def = rk(r V (c)) t by Corollary 3. C V is an MRD code by Lemma 11, hence w d R u. A codeword c C is completely determined by c V by Lemma 11. Denoting r = r u, the number of codewords in C V with rank w is at most [ v w] (q m 1) w r by Lemma 10. For each codeword c such that rk(r V (c)) = w, we count the number of error vectors e such that r V (c) + r V (e) = 0. Suppose that e has rank s (w s t), then s (e) = ( r V, V V(c),r V (e)) has rank s by Lemma 8. By Lemma 12, there are at most [ u s w] A(m,s w) choices for r V (e), and hence as many choices for e. The total number D V of decodable vectors vanishing on V is then at most t [ ] v t [ ] u D V (q m 1) w r A(m,s w)q w(u s+w). (8) w s w w=d R u s=w The number of possible ELS s of dimension v is [ n v]. The bound in (7) follows by multiplying the bound on D V by [ n v], the number of possible ELS s of dimension v. We also obtain a bound similar to (6) for d R t u < d R.

12 12 Proposition 5: For d R t u < d R, then [ ] D u < q2 n q 2 (q m 1) u r V t. (9) 1 u Proof: Equation (8) implies t s [ ][ ] v u D V (q m 1) u r q w(u s+w) A(m,s w)(q m 1) w w s w < (q m 1) u r t s=ww=d R u s=d R u q ms s w=d R u [ ][ ] v u q w(u s+w). (10) w s w Using the following combinatorial relation [24]: s [ v u ] w=d R u w][ s w q w(u s+w) = [ v+u] s, we obtain DV < (q m 1) u r t s=d R u qms[ n] s. By Lemma 2, we find that DV < (q m 1) u r t q 2 s=d R u q 2 1 A(m,s)[ n s] < q 2 q 2 1 (qm 1) u r V t. Finally, we can derive our bounds on the decoder error probability. Proposition 6: For d R t u < d R, the decoder error probability satisfies P E (t;u) < q2 q 2 1 For u d R, the decoder error probability satisfies Proof: (q m 1) u r V t. (11) A(m, u) P E (t;u) < (qm 1) u r V t. (12) A(m, u) The bound in (11) follows directly from Equation (5) and Proposition 5, while the bound in (12) follows directly from Equation (5) and Proposition 3. The result may be weakened in order to find a bound on the decoder error probability which depends on t only. But, we need a bound on V t first. [ n t def Lemma 13: Let V t = t i=0 N i be the volume of a ball in GF(q m ) n with rank radius t. Then V t ] q mt. Proof: Without loss of generality, assume the ball is centered at zero. From Lemma 5, every vector x in the ball belongs to some ELS V with dimension t. Since V = q mt and E t (q m,n) = [ n t], it follows that V t [ n t] q mt. Proposition 7: For u d R t, the decoder error probability satisfies Proof: P E (t;u) < q t2 +2σ(q). (13) First suppose that u d R. Applying A(m,u) > q mu σ(q) in Lemma 1, Lemma 13, and Corollary 1 to (12), we obtain P E (t;u) < q mr+t(m+n t)+2σ(q). Since n m and 2t r, it follows that P E (t;u) < q t2 +2σ(q). For d R t u < d R, applying A(m,u) > q q 1 q mr+t(m+n t)+2σ(q) in

13 13 Lemma 1, Lemma 13, and Corollary 1 to (11), we obtain P E (t;u) < q2 (q 1) q(q 2 1) q mr+t(m+n t)+2σ(q) < q mr+t(m+n t)+2σ(q) q t2 +2σ(q). Note that the bound in Proposition 7 does not depend on the rank of the error at all. This implies that the bound applies to any error vector provided the errors with the same rank are equiprobable. Based on conditional probability, we can easily establish Corollary 4: For an MRD code with d R = 2t+1 and any additive error, the decoder error probability of bounded rank distance decoders satisfies P E (t) < q t2 +2σ(q). VI. SIMULATION RESULTS A. Error Models In Section V, we assumed that all errors with same rank were equiprobable. We argue that this error model is an approximation of crisscross errors. Let us assume q = 2 for simplicity. A codeword of MRD codes of length n over GF(2 m ) can be represented by an m n array of bits. Suppose some of the bits are corrupted by crisscross errors. If the errors are confined to a row (or column), then such an error pattern can be viewed as the addition of an error array which has non-zero bits on only one row (or column) and hence has rank 1. It is reasonable to assume each row is corrupted equally likely and so is each column. Thus, all the errors that are restricted to u > 1 rows (or columns) are equally likely. Finally, if we assume the probability of a corrupted row is the same as that of a corrupted column, then all crisscross errors with crisscross weight u [3] are equally likely. The weight u of the crisscross error is no less than the rank of the error [3]. However, in many cases the weight u equals the rank. Hence, assuming all errors with the same rank are equiprobable is an approximation of crisscross errors. In this section, we want to evaluate our bounds with respect to the decoder error probability for three different error models. We consider errors with fixed rank error model on which our bounds are based as well as crisscross errors and errors with fixed Hamming weight. The latter two error models may be more relevant to some applications. Note that MRD codes are viewed as two-dimensional codes over GF(q) for crisscross errors, and they are considered as block codes over GF(q m ) when errors with either fixed rank or fixed weight are considered. For crisscross errors, we assume any row (column respectively) has a probability p r (p c respectively) of being in error. If a row (or column) error occurs, a random nonzero row (or column) vector is added to the row (or column) considered. Note that this crisscross error model is similar to but different from that in [14]. In this paper we restrict ourselves to

14 14 the case where p r = p c for simplicity. For errors with fixed Hamming weight, we also consider assume that all errors with same Hamming weight are equiprobable as in [23]. B. Simulation Results Let us denote our bounds given in Proposition 6 and Corollary 4 as B 1 (t;u) and B 2 (t) respectively. In our simulations, we used Gabidulin codes [2] with the following parameters: q = 2, m = n = 16, and d R = 2t+1 = n k+1. Our simulations are based on the following process: first a random message word in GF(q m ) k is encoded using the generator matrix of the Gabidulin code, then an error generated based on the error model is added to the codeword, and finally a bounded rank distance decoder based on the EEA in [2] is used to decode the received word. Similar to the decoding of Reed-Solomon codes, decoder failures are declared based on the output of the EEA. Each decoder error probability is computed after at least 100 occurrences of decoder errors to ensure reliability of simulation results. In Figure 1, our bounds on the decoder error probability and simulation results with different t values are shown. For the fixed rank error model, the rank of the error is set to 6, and u = 6 is also used to calculate B 1 (t;u). For the fixed Hamming weight error model, the Hamming weight of the error is set to 6. For the crisscross error model, p r and p c are both set to 6/32. In Figure 2, t is set to 3 and our bounds and the simulated decoder error probabilities are shown as functions of u. For the three error models, the u axis in Figure 2 has different meanings. For the fixed rank error model, u denotes the rank of the error, which varies between t+1 = 4 and 16. For the fixed Hamming weight error model, u denotes the Hamming weight of the error, which is also between t+1 = 4 and 16. For the crisscross error model, u represents p r and p c since p r and p c are both set to 1/2 in Figure 2. u m+n, which varies between 4/32 and We observe that B 2 (t) is tighter than B 1 (t;u) in all cases. Also, B 1 (t;u) is quite tight in some cases (see Figure 1). Under the fixed rank error model, our bounds and the simulated decoder error probability decrease exponentially with t 2. It is interesting to note that the decoder error probabilities under the other two error models also decrease exponentially with t 2. Also, B 2 (t) bounds from above the decoder error probabilities for the other two error models, and even the bound B 1 (t;u) is applicable in all except one case in Figure 2. Thus, although B 2 (t) is derived based on the fixed rank error model, it seems applicable to the decoder error probability of MRD codes under other error models.

15 Decoder error probability Fixed rank Fixed Hamming weight Crisscross t 10 4 B 2 (t) B 1 (t;6) Fig. 1. Decoder error probability of an MRD code as a function of t, with q = 2, m = n = 16. For fixed rank error, u = 6; for fixed Hamming weight error, w H = 6; for crisscross error, p r = p c = 6/32. Decoder error probability B 2 (t) B 1 (t;u) Fixed rank Fixed Hamming weight Crisscross u Fig. 2. Decoder error probability of an MRD code as a function of u, with q = 2, m = n = 16, and t = 3. u = rk(e) for fixed rank errors, u = w H(e) for fixed Hamming weight errors, and u = (m+n)p r for crisscross errors.

16 16 C. Comparison with MDS codes In this section we compare the error performance of MRD codes with that of MDS codes against crisscross errors. Recall that the rank of a crisscross error is not more than its Hamming weight. Thus, if the rank of the crisscross error is no more than t, then an (n,k,2t+1) MRD code will always decode successfully, whereas the Hamming weight of the error may be greater than t, thus overwhelming an (n,k,2t+1) MDS code with a bounded Hamming distance decoder. In this simulation, we use a (16,8) Gabidulin code with a bounded rank distance decoder and a (16,8) shortened RS code with a bounded Hamming distance decoder. As illustrated in Figure 3, the decoder success probability of the MRD code is approximately one order of magnitude higher than that of the MDS code. The difference may be more meaningful when the expected value of the error rank is at most (m+n)p r since for p r t m+n we expect a high decoder success probability for MRD codes. For example, when p r = p c are 0.1 and respectively, the decoding success probability for the MRD code is approximately 0.8 and 0.6 respectively, whereas the decoding success probability for the MDS code is less than 0.2 for the same two cases. Our simulation results suggest that the MRD code are more resilient against crisscross errors than MDS codes MRD MDS 10 1 Decoder success probability p r Fig. 3. t = 4. Decoder success probability of an MRD code and of an MDS code as a function of p r, with q = 2, m = n = 16, and

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