Decoder Error Probability of MRD Codes
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1 Decoder Error Probability of MRD Codes Maximilien Gadolea Department of Electrical and Compter Engineering Lehigh University Bethlehem, PA USA Zhiyan Yan Department of Electrical and Compter Engineering Lehigh University Bethlehem, PA USA arxiv:cs/ v1 [cs.it] 26 Oct 2006 Abstract In this paper, we first introdce the concept of elementary linear sbspace, which has similar properties to those of a set of coordinates. Using this new concept, we derive properties of maximm rank distance (MRD) codes that parallel those of maximm distance separable (MDS) codes. Using these properties, we show that the decoder error probability of MRD codes with error correction capability t decreases exponentially with t 2 based on the assmption that all errors with the same rank are eqally likely. We arge that the channel based on this assmption is an approximation of a channel corrpted by crisscross errors. I. INTRODUCTION The Hamming metric has often been considered the most relevant metric for error-control codes so far. Recently, the rank metric [1] has attracted some attention de to its relevance to space-time coding [2] and storage applications [3]. In [4], space-time block codes with good rank properties have been proposed. Rank metric codes are sed to correct crisscross errors that can be fond in memory chip arrays and magnetic tapes [3]. Rank metric codes have been sed in pblic-key cryptosystems as well [5]. In [1], a Singleton bond on the minimm rank distance of rank metric codes was established, and codes that attain this bond were called maximm rank distance (MRD) codes. An explicit constrction of MRD codes (these codes are referred to as Gabidlin codes) was also given in [1], and this constrction was extended in [6]. Also, a decoding algorithm that parallels the extended Eclidean algorithm (EEA) was proposed for MRD codes. In this paper, we investigate the performance of MRD codes when sed to protect data from additive errors based on two assmptions. First, we assme all errors with the same rank are eqally likely. We arge that the channel based on this assmption is an approximation of a channel corrpted by crisscross errors (see Section IV for details). Second, we assme that a bonded rank distance decoder is sed, with error correction capability t. If the error has rank no more than t, the decoder gives the correct codeword. When the error has rank greater than t, the otpt of the decoder is either a decoding failre or a wrong codeword, which corresponds to a decoder error. Note that the decoder error probability of maximm distance separable (MDS) codes was investigated in [7], where all errors with the same Hamming weight were assmed to be eqiprobable. The main contribtions of this paper are: We introdce the concept of elementary linear sbspace (ELS). The properties of an ELS are similar to those of a set of coordinates. Using elementary linear sbspaces, we derive sefl properties of MRD codes. In particlar, we prove the combinatorial property of MRD codes, derive a bond on the rank distribtion 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. Using the properties of MRD codes, we derive a bond on the decoder error probability of MRD codes that decreases exponentially with t 2. Or simlation reslts are consistent with or bond. The rest of the paper is organized as follows. Section II gives a brief review of the rank metric, Singleton bond, and MRD codes. In Section III, we first introdce the concept of elementary linear sbspace and stdy its properties, and then obtain some important properties of MRD codes. Section IV derives the bond on the decoder error probability of MRD codes when all errors with the same rank are eqiprobable. Finally, in Section V, or bond on the decoder error probability is confirmed by simlation reslts. A. Rank metric II. PRELIMINARIES Consider an n-dimensional vector x = (x 0, x 1,...,x n 1 ) in GF(q m ) n. Assme 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 colmn 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 1,0 x 1,1... x 1,n 1 X =......, x m 1,0 x m 1,1... x m 1,n 1 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) [1]. The rank norm of x can also be viewed as
2 the smallest nmber of rank 1 matrices B i sch that X = i B i [8]. All the ranks are over the base field GF(q) nless otherwise specified in this paper. To simplify notations, we denote the rank norm of x as rk(x) henceforth. Accordingly, x,y GF(q m ) n, d(x,y) def = rk(x y) is shown to be a metric over GF(q m ) n, referred to as the rank metric henceforth [1]. Hence, the minimm rank distance d of a code of length n is simply the minimm rank distance over all possible pairs of distinct codewords. A code with a minimm rank distance d can correct errors with rank p to t = (d 1)/2. B. The Singleton bond and MRD codes The minimm rank distance of a code can be specifically bonded. First, the minimm rank distance d of a code over GF(q m ) is obviosly bonded by m. Codes that satisfy d = m are stdied in [9]. Also, it can be shown that d d H [1], where d H is the minimm Hamming distance of the same code. De to the Singleton bond on the minimm Hamming distance of block codes, the minimm rank distance of an (n, M) block code over GF(q m ) ths satisfies d n log q m M + 1. (1) In this paper, we refer to the bond in (1) as the Singleton bond for rank metric codes and codes that attain the bond as MRD codes. Note that (1) implies that the cardinality of any MRD code is a power of q m. III. PROPERTIES OF MRD CODES In order to derive the bond on decoder error probability, we need some important properties of MRD codes, which will be established next sing the concept of elementary linear sbspace. A. Elementary linear sbspaces Many properties of MDS codes are established by stdying sets of coordinates. These sets of coordinates may be viewed as linear sbspaces which have a basis of vectors with Hamming weight 1. We first define elementary linear sbspaces, which are the conterparts of sets of coordinates. Definition 1 (Elementary linear sbspace (ELS)): A linear sbspace V of GF(q m ) n is said to be elementary if it has a basis B consisting of row vectors in GF(q) n. B is called an elementary basis of V. We remark that V is an ELS if and only if it has a basis consisting of vectors of rank 1. We also remark that not all linear sbspaces are elementary. For example, the span of a vector of rank > 1 has dimension 1, bt reqires vectors of rank 1 to span it. Next, we show that the properties of elementary linear sbspaces are similar to those of sets of coordinates. Definition 2 (Rank of a linear sbspace): The rank of a linear sbspace L of GF(q m ) n is defined to be the maximm rank among the vectors in L rk(l) def = max x L {rk(x)}. Proposition 1: If V is an ELS of GF(q m ) n with dim(v ) m, then rk(v ) = dim(v ). Proof: Let s first denote dim(v ) as v. Any vector x V can be expressed as the sm of at most v vectors of rank 1, hence its rank is pper bonded by v. Ths, rk(v ) v and it sffices to find a vector in V with rank eqal to v. Let B = {b j } v 1 j=0 be an elementary basis of V, and consider y = v 1 i=0 α ib i, where {α i } v 1 i=0 all belong to a basis set of GF(q m ) over GF(q). If we expand the coordinates of y with respect to the basis {α i } v 1 i=0, we get Y = ( b T 0,...,bT v 1,0T,...,0 T) T. Since the row vectors b 0,b 1,,b v 1 are linearly independent over GF(q), Y has rank v and rk(y) = v. Lemma 1: Any vector x GF(q m ) n with rank belongs to an ELS A of dimension. Also, x is not contained in any ELS of dimension less than. Proof: The vector x can be expressed as a linear combination x = 1 i=0 x ia i, where x i GF(q m ) and a i GF(q) n for 0 i 1. Let A be the ELS of GF(q m ) n spanned by a i s, then dim(a) = and x A. Also, sppose x B, where B is an ELS with dimension t <. Then by Proposition 1, rk(x) t <, which is a contradiction. Lemma 2: Let V be the set of all ELS s of GF(q m ) n with dimension v and V be the set of all linear sbspaces of GF(q) n with dimension v. Then there exists a bijection between V and V. Proof: Let V be an ELS of GF(q m ) n with B as an elementary basis. For any positive integer a, define { v 1 } def B q a = x i b i x i GF(q a ),b i B i=0 and a mapping f : V V given by V = B q m V = B q. This mapping takes V to a niqe V, which is the set of all linear combinations over GF(q) of the vectors in B. Note that dim(v ) = dim(v ). f is injective becase if V and W are distinct, then V and W are also distinct. f is also srjective since for all V V, there exists V V sch that f(v ) = V. For 0 v n, let [ n be the nmber of linear sbspaces of GF(q) n with dimension v [10]. For v = 0, [ n 0] = 1, and for 1 v n: [ ] n v 1 v = q n q i i=0 q v q. Lemma 2 implies that i Corollary 1: There are [ n ELS s in GF(q m ) n with dimension v. Proposition 2: Let V be an ELS of GF(q m ) n with dimension v, then there exists an ELS W complementary to V, i.e., V W = GF(q m ) n. Proof: Let V = f(v ), and let B a basis of V, then there exists W, with basis B, sch that V W = GF(q) n. Denote W = f 1 (W ). We now want to show that W V = GF(q m ) n. First, dim(w) is n v, hence we only need to show that V W = {0}. Let y 0 V W, then there is a non-trivial linear relationship over GF(q m ) among the
3 elements of B and those of B. This may be expressed as y i b i = 0. (2) b i B B Applying the trace fnction to each coordinate on both sides of (2), 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, W V = GF(q m ) n. Definition 3 (Restriction of a vector): Let L be a linear sbspace of GF(q m ) n and let L be complementary to L, i.e., L L = GF(q m ) n. Any vector x GF(q m ) n can then be represented as x = x L x L, where x L L and x L L. We will call x L and x L the restrictions of x on L and L, respectively. Note that for any given linear sbspace L, its complementary linear sbspace L is not niqe. Frthermore, the restriction of x on L, x L, depends on not only L bt also L. Ths, x L is well defined only when both L and L are given. All the restrictions in this paper are with respect to a fixed pair of linear sbspaces complementary to each other. Definition 4: Let x GF(q m ) n and V be an ELS. If there exists an ELS V complementary to V sch that x = x V 0, we say that x vanishes on V. Lemma 3: Given a vector x GF(q m ) n of rank, there exists an ELS with dimension n on which x vanishes. Also, x does not vanish on any ELS with dimension greater than n. Proof: By Lemma 1, x A, where A is an ELS with dimension. Let Ā be an ELS with dimension n that is complementary to A. Ths, x may be expressed as x = x A x Ā = x A 0. That is, x vanishes on Ā. Also, sppose x vanishes on an ELS B with dimension greater than n. Then there exists an ELS B with dimension < sch that x B, which contradicts Lemma 1. B. Properties of MRD codes We now derive some sefl properties for MRD codes, which will be sed in or derivation of the decoder error probability. In this sbsection, let C be an MRD code over GF(q m ) with length n (n m) and cardinality q mk. Note that C may be linear or nonlinear. First, we derive the basic combinatorial property of MRD codes. Lemma 4 (Basic combinatorial property): Let K be an ELS of GF(q m ) n with dimension k, and fix K, an ELS complementary to K. Then, for any vector k K, there exists a niqe codeword c C sch that its restriction on K satisfies c K = k. Proof: Sppose there exist c d C sch that c K = d K. Their difference c d is in K, and hence has rank at most n k by Proposition 1, which contradicts the fact that C is MRD. Then all the codewords lead to different restrictions on K. Also, C = K = q mk, ths for any k, there exists a niqe c sch that c K = k. This property allows s to obtain a bond on the rank distribtion of MRD codes. Lemma 5 (Bond on the rank distribtion): Let A be the nmber of codewords of C with rank. Then, for the redndancy r = n k and d, [ ] n A (q m 1) r. (3) Proof: From Lemma 3, any codeword c with rank d vanishes on an ELS with dimension v = n. Ths (3) can be established by first determining the nmber of codewords vanishing on a given ELS, and then mltiplying by the nmber of sch ELS s, [ n ]. Let V be an arbitrary ELS with dimension v. First, since v k 1, V is properly contained in an ELS K with dimension k. According to the combinatorial property, c is completely determined by c K. Hence, if we specify that a codeword vanishes on V, we may specify k v other nonzero components arbitrarily. There are at most (q m 1) k v = (q m 1) r ways to do so, implying that there are at most (q m 1) r vectors that vanish on V. Note that the exact formla for the rank distribtion of linear MRD codes was derived in [1]. However, the bond in (3) is more convenient for the present application. It is well known that a pnctred MDS code is an MDS code [11]. We will show that the restriction of an MRD code to an ELS is also MRD. Let V be an ELS with dimension v k, an elementary basis {b 0,b 1,,b v 1 }, and a complementary ELS V. For any codeword c, sppose c V where a i GF(q m ). Let s define a mapping r : GF(q m ) n GF(q m ) v given by c r(c) = (a 0,, a v 1 ). Then C V = {r(c) c C} is called the restriction of C on V. Lemma 6 (Restriction of an MRD code): For an ELS V with dimension v k, C V is an MRD code. Proof: Clearly, C V is a code over GF(q m ) with length v (v m) and cardinality q mk. Assme c d C, and consider x = c d. Then we have rk(r(c) r(d)) = rk(x V ) rk(x) rk(x V ) n k +1 (n v) = v k +1. The Singleton bond on C V completes the proof. = v 1 i=0 a ib i, IV. DECODER ERROR PROBABILITY OF MRD CODES IN CASE OF CRISSCROSS ERRORS Let C be a linear (n, k) MRD code over GF(q m ) (n m) with minimm rank distance d = n k + 1 and error correction capability t = (d 1)/2. We assme that C is sed to protect data from additive errors with rank. That is, the received word corresponding to a codeword c of C is c + e. We arge that the additive error with rank is an approximation of crisscross errors. Let s assme q = 2 and a codeword can be represented by an m n array of bits. Sppose some of the bits are recorded erroneosly, and the error patterns are sch that all corrpted bits are confined to a nmber of rows or colmns or both. Sch an error model, referred to as crisscross errors, occrs in memory chip arrays or magnetic tapes [3]. Sppose the errors are confined to a row (or colmn), then sch an error pattern can be viewed as the addition of an error array which has non-zero coordinates on only one row (or colmn) and hence has rank 1. We may reasonably assme each row is corrpted eqally likely and so is each colmn. Ths, all the errors that are restricted to
4 > 1 rows (or colmns) are eqally likely. Finally, if we assme the probability of a corrpted row is the same as that of a corrpted colmn, then all crisscross errors with weight [3] are eqally likely. The weight of the crisscross error is no less than the rank of the error [3]. However, in many cases the weight eqals the rank. Hence, assming all errors with the same rank are eqiprobable is an approximation of crisscross errors. A bonded distance decoder, which looks for a codeword within rank distance t of the received word, is sed to correct the error. Clearly, if e has rank no more than t, the decoder gives the correct codeword. When the error has rank greater than t, the otpt of the decoder is either a decoding failre or a decoder error. We denote the probabilities of error and failre for error correction capability t and a given error rank as (t; ) and P F (t; ) respectively. If t, then P F (t; ) = (t; ) = 0. When > t, (t; )+P F (t; ) = 1. In particlar, if t < < d t, then (t; ) = 0 and P F (t; ) = 1; Ths we only need to investigate the case where d t. Since C is linear and hence geometrically niform, we assme withot loss of generality that the all-zero codeword is sent. Ths, the received word can be any vector with rank with eqal probability. We call a vector decodable if it lies within rank distance t of some codeword. If D denotes the nmber of decodable vectors of rank, then for t+1 we have (t; ) = D N = D [ n ] A(m, ), (4) where N denotes the nmber of vectors of rank and A(m, ) def = 1 i=0 (qm q i ). Hence the main challenge is to derive pper bonds on D. We have to distingish two cases: d and < d. The approach we se to bond D is similar to that in [7]. Proposition 3: For d, then [ ] n D (q m 1) r V t, (5) where V t = t i=0 N i is the volme of a ball of rank radis t. Proof: Each decodable vector can be written niqely as c + e, where c C and rk(e) t. For a fixed e, C + e is an MRD code, so it satisfies Eqation (3). Therefore, the nmber of decodable words of rank is at most [ n ] (q m 1) r mltiplied by the nmber of possible error vectors, V t. Lemma 7: Given y GF(q m ) v with rank w, there are at most [ ] s w A(m, s w)q w( s+w) vectors z GF(q m ) n v sch that x = (y,z) GF(q m ) n has rank s. Proof: The vector x has s linearly independent coordinates. Since w of them are in y, then s w of them are in z. Ths z has s w linearly independent coordinates which do not belong to S(y). Withot loss of generality, assme those coordinates are on the first s w positions of z, and denote these coordinates as z. For s w + 1 i n v, z i is a linear combination of the coordinates of y and the first s w coordinates of z. Hence, we have z i = a i + b i, where a i S(y) and b i S(z ). There are q w( s+w) choices for the vector a = (0,...,0, a s w+1,..., a ). The vector b has rank s w, so there are at most [ s w] A(m, s w) choices for b. Proposition 4: For d t < d, we have [ ] n [ ] v D (q m 1) w r w w=d [ ] A(m, s w)q w( s+w). (6) s w s=w Proof: Recall that a decodable vector of rank can be expressed as c + e, where c C and rk(e) t. This vector vanishes on an ELS V with dimension v = n by Lemma 3. Fix V, an ELS complementary to V. We have w def = rk(r V (c)) t. C V is an MRD code by Lemma 6, hence w d. By Lemma 5, and denoting r = r, the nmber of codewords of C V with rank w is at most [ v w] (q m 1) w r. For each codeword c sch that rk(r V (c)) = w, we mst cont the nmber of error vectors e sch that r V (c) + r V (e) = 0. Sppose that e has rank s w, and denote g = r V (e) and f = r V(e). Note that e is completely determined by f. The vector [ (g,f) has rank s, hence by Lemma 7, there are at most s w] A(m, s w) choices for the vector f. The total nmber D V of decodable vectors vanishing on V is then at most [ ] v D V (q m 1) w r w w=d [ ] A(m, s w)q w( s+w). (7) s w s=w The nmber of possible ELS s of dimension v is [ n. Mltiplying the bond on D V by [ n, the nmber of possible ELS s of dimension v, we get the reslt. We can obtain a bond similar to (5) which applies to the case d t < d. Corollary 2: For d t < d, then D < q2 n q 1[ ] (q m 2 1) r V t. Proof: We shall se Eqation (7). We have s [ ][ ] v D V (q m 1) r w s w s=w w=d q w( s+w) A(m, s w)(q m 1) w (8) < (q m 1) r q ms s w=d s=d [ ][ ] v q w( s+w). (9) w s w Using the following combinatorial relation [10]: s [ v [ w=d w][ s w] q w( s+w) = v+ ] s, we obtain D V < (q m 1) r t s=d qms[ n s]. It can be shown that q ms q2 q 2 1 A(m, s). Using this reslt, we find that D V <
5 q 2 q 2 1 (qm 1) r t s=d A(m, s)[ ] n s < q 2 q 2 1 (qm 1) r V t. We can eventally derive a bond on the decoder error probability. Proposition 5: For d t < d, the decoder error probability satisfies (t; ) < q2 q 2 1 (q m 1) r V t. (10) A(m, ) For d, the decoder error probability satisfies (t; ) < (qm 1) r V t. (11) A(m, ) Proof: Directly from Proposition 3 and Corollary 2. Before deriving an pper bond for (t; ), we need to establish two lemmas. Lemma 8: For 0 m, A(m, ) q m σ(q), where σ(q) = 1 1 ln(q) k=1 k(q k 1) is a decreasing fnction of q with σ(2) Proof: We have A(m, ) = q m M, where M = m j=m +1 log q(1 q j ). M is an increasing fnction of, with maximm eqal to M m = 1 ln(q) k=1 1 q mk k(q k 1) σ(q). Lemma 9: For m 1 and t m/2, V t q t(n+m t)+σ(q). Proof: First, we need to prove the following claim. [ Claim: For m 1 and i t m/2, we have m q i(t i) 1. An exhastive search proves the reslt for m < 4. We shall assme that m 4 herein. The case i = t being trivial, we hence assme that i < t. Using Lemma 8, we find that [ m q (t i)(m t+i) σ(q), hence [ m q i(t i) q (t i)(m t) σ(q) q m/2 σ(q) 1. The claim implies A(m, i) q mi [ m q i(m t+i). Since V t = t [ n ] i=0 i A(m, i), the bond on A(m, i) allows s to derive the bond on V t. The reslt in Proposition 5 may be weakened in order to find a bond on the decoder error probability which only depends on t. Proposition 6: For d t, the decoder error probability satisfies (t; ) < q t2 +2σ(q). (12) Proof: First sppose that d. From Proposition 5, we have for d t: (t; ) < (qm 1) r A(m,) V t. The bonds in Lemmas 8 and 9 lead to (t; ) < q mr+t(m+n t)+2σ(q). Since n m and 2t r, it follows that (t; ) < q t2 +2σ(q). For d t < d, we find that (t; ) < q 2 (q 1) q(q 2 1) q mr+t(m+n t)+2σ(q) < q mr+t(m+n t)+2σ(q). Using the same reasoning as above, we find the same conclsion. V. SIMULATION RESULTS In this section, we se nmerical simlations to verify or bond given in Proposition 6. In or simlations, we sed a special family of MRD codes called Gabidlin codes [1] with the following parameters: q = 2, m = n = 16, and d = 2t + 1 = n k + 1. The simlations were based on the following process: first a random message word in GF(q m ) k is encoded sing the generator matrix of the Gabidlin code, then a random error vector with rank > t is added to the codeword, and finally the EEA [1] is sed to decode the received word. Since > t, the decoding reslts in either a failre or an error. Similarly to the decoding of Reed- Solomon codes, decoder failre is declared based on the otpt of the EEA. Different vales for t and are sed in or simlations to verify or bond. Each vale of the decoder error probability is compted after at least 15 occrrences of decoder errors to ensre reliability of simlation reslts. Note that or bond given in Proposition 6 does not depend on, and decreases exponentially with t 2. In Figre 1, the decoder error probability is viewed as a fnction of t as t varies from 1 to 4 and is set to n = 16. Note that when t = 1, the bond in Proposition 6 is trivial. We observe that both the bond and the simlated decoder error probability decrease exponentially with t 2. In Figre 2, the decoder error probability is viewed as a fnction of as t is set to either 2 or 3 and varies from t + 1 to n = 16. Clearly, the decoder error probability varies with somewhat, bt the bond in Proposition 6 is applicable to all vales of. (t;16) (t;16) vs t, m=n=16 Bond Simlation reslts t Fig. 1. Decoder error probability of an MRD code as a fnction of t, with q = 2, m = n = 16, and = 16. REFERENCES [1] E. M. Gabidlin, Theory of codes with maximm rank distance, Problems on Information Transmission, vol. 21, no. 1, pp. 1 12, Jan [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless commnication: Performance criterion and code constrction, IEEE Trans. Info. Theory, vol. 44, pp , March [3] R. M. Roth, Maximm-rank array codes and their application to crisscross error correction, IEEE Trans. Info. Theory, vol. 37, no. 2, pp , March [4] P. Lsina, E. M. Gabidlin, and M. Bossert, Maximm Rank Distance codes as space-time codes, IEEE Trans. Info. Theory, vol. 49, pp , Oct
6 10 0 (t;) vs, m=n=16, t=2 or (t;) t=2, bond t=2, simlation t=3, bond t=3, simlation Fig. 2. Decoder error probability of an MRD code as a fnction of, with q = 2, m = n = 16, and t eqal to 2 or 3. [5] E. M. Gabidlin, A. V. Paramonov, and O. V. Tretjakov, Ideals over a non-commtative ring and their application in cryptology, LNCS, vol. 573, pp , [6] A. Kshevetskiy and E. M. Gabidlin, The new constrction of rank codes, Proc. IEEE Int. Symp. on Information Theory, pp , Sept [7] R. J. McEliece and L. Swanson, On the decoder error probability for Reed-Solomon codes, IEEE Trans. Info. Theory, vol. 32, no. 5, pp , Sept [8] E. M. Gabidlin and V. A. Obernikhin, Codes in the Vandermonde F-metric and their application, Problems of Information Transmission, vol. 39, no. 2, pp , [9] K. N. Manoj and B. Sndar Rajan, Fll Rank Distance codes, Technical Report, IISc Bangalore, Oct [10] G. E. Andrews, The Theory of Partitions, G.-C. Rota, Ed. Addison- Wesley, 1976, vol. 2. [11] R. Blaht, Theory and Practice of Error Control Codes. Addison- Wesley, 1983.
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