Decoder Error Probability of MRD Codes

Transcription

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 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 [3], two-dimensional codes with the rank metric were stdied: a Singleton bond in a different form was established and a decoding algorithm that is analogos to the Berlekamp-Massey algorithm was proposed 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 distance decoder is sed That is, the decoder looks for a codeword within rank distance t of the received word, where t denotes the error correction capability of MRD codes If sch a codeword exists, the decoder finds it; otherwise, the decoder reports a decoding failre Clearly, 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 i0 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

2 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 i0 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), ie, rk(x GF(q)) def rank(x) [1] The rank norm of x can also be viewed as 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 referred to as fll rank distance codes and 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 linear block codes, the minimm rank distance of an (n, k) linear block code over GF(q m ) ths satisfies d n k + 1 (1) In general, for a block code over GF(q m ) with length n and cardinality M, a more general Singleton bond is given by d n log q m M + 1 (2) In this paper, we refer to the bond in (2) as the Singleton bond for rank metric codes and codes that attain the bond as MRD codes Note that (2) implies that the cardinality of any nonlinear MRD codes 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 {rk(x)} x L 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 j0 be an elementary basis of V, and consider y v 1 i0 α ib i, where {α i } v 1 i0 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 i0, we get b 0 Y b v 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 i0 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 } B q a def { v 1 x i b i x i GF(q a ), b i B i0

3 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 i0 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, ie, 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 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 fnction 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, 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, ie, 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: Let 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 that C is MRD Then all q mk codewords lead to different restrictions on K The niqeness is de to C K q mk 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, A (q m 1) r (4) Proof: From Lemma 3, any codeword c with rank d vanishes on an ELS with dimension v n Ths (4) 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 (4) 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 an complementary ELS V For any codeword c, sppose c V v 1 i0 a ib i, 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 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

4 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 > 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, ), (5) where N denotes the nmber of vectors of rank and A(m, ) def 1 i0 (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 D (q m 1) r V t, (6) where V t t i0 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 (4) 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 Proposition 4: For d t < d, we have [ ] v [ ] D (q m 1) w r A(m, s w) w s w wd sw Proof: The approach sed here is similar to that in the proof of Lemma 5 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(c V ) t C V is an MRD code by Lemma 6, hence w d By (4), the nmber of codewords with rank w on V is at most [ v w] (q m 1) w r, where r r For each codeword c with rank w on V, we mst cont the nmber of error vectors e sch that c + e vanishes on V Sppose that e has rank s w On V, e V c V, bt e V is an arbitrary vector of rank s w Ths the total nmber of error vectors, for a given codeword of rank w on V, is t [ sw s w] A(m, s w) The total nmber D V of decodable vectors vanishing on V is at most [ ] v [ ] D V (q m 1) w r A(m, s w) w s w wd sw s [ ][ ] v (q m 1) r w s w sd wd A(m, s w)(q m 1) w (7) Mltiplying the bond on D V by [ n, the nmber of possible ELS s of dimension v, we get the reslt By relaxing the reslt in Proposition 4, the bond for D in (6) also applies to the case d t < d Corollary 2: For d t, then D [ n (q m 1) r V t Proof: The reslt has already been proved for d To prove it for < d, we shall se Eqation (7) First, we remark that A(m, s w)(q m 1) w A(m, s)q w( s+w) Replacing this bond on A(m, s w) in (7), and sing the following combinatorial relation [10]: s [ v ] w0 w][ s w q w( s+w) [ ] v+ s, we obtain D V (q m 1) r V t Mltiplying this bond by the nmber of choices for V, we get the desired reslt Before deriving an pper bond for (t; ), we need to establish two lemmas, whose proofs are omitted de to limited space Lemma 7: For 0 m, A(m, ) q m σ(q), where σ(q) 1 ln(q) k1 σ(2) k(q k 1) is a decreasing fnction of q with

5 Lemma 8: For m 1 and t m/2, V t q t(n+m t)+σ(q) Proposition 5: For d t, the decoder error probability satisfies (t; ) q t2 +2σ(q) (8) Proof: From Corollary 2, we have for d t: (t;16) vs t, mn16 Bond Simlation reslts (t; ) (qm 1) r V t A(m, ) The bonds in Lemmas 7 and 8 lead to (t; ) q mr+t(m+n t)+2σ(q) Since n m and 2t r, it follows that (t; ) q m(2t r) t2 +2σ(q) q t2 +2σ(q) (t;16) V SIMULATION RESULTS In this section, we se nmerical simlations to verify or bond given in Proposition 5 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 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 5 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 5 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 5 is applicable to all vales of REFERENCES [1] E M Gabidlin, Theory of codes with maximm rank distance, Problems on Information Transmission, vol 21, no 1, pp 1 12, Jan 1985 [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 1998 [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 1991 [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 2003 [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 , 1991 [6] A Kshevetskiy and E M Gabidlin, The new constrction of rank codes, Proc IEEE Int Symp on Information Theory, pp , Sept t Fig 1 Decoder error probability of an MRD code as a fnction of t, with q 2, m n 16, and 16 (t;) (t;) vs, mn16, t2 or 3 t2, bond t2, simlation t3, bond t3, 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 [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 1986 [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 , 2003 [9] K N Manoj and B Sndar Rajan, Fll Rank Distance codes, Technical Report, IISc Bangalore, Oct 2002 [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