Characterizations of Network Error Correction/Detection and Erasure Correction

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1 Characerizaions of Nework Error Correcion/Deecion and Erasure Correcion Shenghao Yang Deparmen of Informaion Engineering The Chinese Universiy of Hong Kong Shain, N.T., Hong Kong Raymond W. Yeung Deparmen of Informaion Engineering The Chinese Universiy of Hong Kong Shain, N.T., Hong Kong Absrac In classical algebraic coding heory, he minimum disance of block code compleely deermines he abiliy of he code in erms of error correcion/deecion and erasure correcion. We have obained generalizaions of hese resuls for nework codes. I. INTRODUCTION In he previous sudies of nework coding, he ransmission over neworks is mosly assumed o be error-free [1]. However, in pracical communicaion neworks, ransmission suffers differen kinds of errors, such as random errors, link failures, raffic congesion and malicious modificaions. Some researchers have noiced ha nework coding can be used o deec and correc errors in neworks [2] [7]. The concep of nework error correcion coding, a generalizaion of classical error correcion coding, was firs inroduced by Cai and Yeung [4] [6]. They generalized he Hamming bound, he Singleon bound and he Gilber-Varshamov bound in classical error correcion coding o nework coding. Zhang [7] inroduced he minimum rank for linear nework codes, which plays a role similar o ha of he minimum disance in decoding classical error-correcing codes. The relaion beween nework coding and classical algebraic coding has been clarified in [1]. In his paper, he weigh properies of linear nework codes are invesigaed. We firs inroduce some new weigh definiions, called he nework Hamming weigh, for error vecors, receive vecors and message vecors. All hese nework Hamming weighs reduce o he usual Hamming weigh in he special case of classical error correcion. Wih hese nework Hamming weighs, he minimum disance of a nework code can be defined. The main conribuion of his paper is o characerize he abiliy of nework codes for error correcion, error deecion and erasure correcion in erms of he minimum disances of he codes. Le d be an ineger. Specifically, we show ha he following properies of a linear nework code are equivalen: 1) The mulicas minimum disance of he code is larger han or equal o d ) The code can correc all error vecors wih Hamming weigh less han (d + 1)/2. 3) The code can deec all non-zero error vecors wih Hamming weigh less han or equal o d. 4) The code can correc all erasures wih Hamming weigh less han or equal o d. This paper is organized as follows. Secion II formulaes he nework error correcion problem. Secion III defines he nework Hamming weighs and he minimum disances for nework codes. The error correcing and deecing abiliies of a nework code are characerized in Secion IV. Secion V discusses he relaion beween minimum rank decoding and minimum disance decoding for nework error correcion. The erasure correcion abiliy of a nework code is characerized in Secion VI. In he las secion we summarize our work and discuss opics for furher research. II. PROBLEM FORMULATION We sudy nework ransmission in a direced acyclic communicaion nework denoed by G = (V, E), where V is he se of nodes in he nework and E is he se of edges in he nework. We assume an order on he edge se E which is consisen wih he parial order induced by he acycliciy of G. An edge from node a o b, denoed by (a, b), represens a communicaion channel from node a o node b. We call node a (node b) he inpu node (oupu node) of edge (a, b), and edge (a, b) an inpu edge (oupu edge) of node b (node a). Le In(a) = {(b, a) : (b, a) E} and Ou(a) = {(a, b) : (a, b) E} be he ses of inpu edges and oupu edges of node a, respecively. There can be muliple edges beween a pair of nodes, and each edge can ransmi one symbol in a finie field F q. A mulicas on G ransmis informaion from a source node s o a se of sink nodes T. Le n s = Ou(s). The source node s modulaes he informaion o be mulicas ino a row vecor x F ns q called he message vecor. The vecor is sen in one use of he nework by mapping he n s componens of he vecor ono each edges in Ou(s). Define an n s E marix A = [A i,j ] as { 1 ej is he ih edge in Ou(s), A i,j = 0 oher wise. By applying he order on E o Ou(s), he n s nonzero columns of A form an ideniy marix. An error vecor z is an E -uple wih each componen represening he error on an edge.

2 A nework code for nework G is specified by a se of local encoding funcions {k ei,e j : e i, e j E} and he message se C. Only linear local encoding funcions are considered in his paper. Define he E E one-sep ransiion marix K = [K i,j ] for nework G as { kei,e K i,j = j e i In(a), e j Ou(a) for some a V, 0 oher wise. For an acyclic nework, K N = 0 for some posiive ineger N. Define he ransfer marix of he nework by F = (I K) 1 [8], so ha he symbols ransmied on he edges are given by he componens of (xa + z)f. For a sink node T, wrie n = In(), and define an E n marix B = [B i,j ] for sink node as { 1 ei is he jh edge in In(), B i,j = 0 oher wise. The n nonzero rows of B form a permuaion marix. The received vecor for a sink node is y = (xa + z)f B, = xf s, + zf, (1) where F s, = AF B is he submarix of F given by he inersecion of he n s rows corresponding o he edges in Ou(s) and he n columns corresponding o he edges in In(), and F = F B is he submarix of F formed by he columns of F corresponding o he inpu edges of sink node. F s, and F are he ransfer marices of message ransmission and error ransmission, respecively, for he sink node. Equaion (1) is our formulaion of he mulicas nework error correcion problem. The classical error correcion problem is a special case in which boh of F s, and F reduce o ideniy marices. The message ransmission capaciy is measured by he rank of he ransfer marix F s,. Denoe he maximum flow beween source node s and sink node by maxflow(s,). Evidenly, for any linear nework code on G, he rank of F s, is upper bounded by maxflow(s,) [1]. Le C be he se of message vecors ha can be ransmied by he source and be decoded correcly. When he nework is error-free, he error correcion problem is reduced o he usual nework coding problem, for which he size of C is upper bounded by q min T maxflow(s,) [9]. In his paper, he ransmission problem formulaed by (1) is sudied in he scenario ha here can be errors in he edges (channels), which may be due o channel noise, link failures, raffic congesion, malicious modificaions, and so on. Classical coding heory refers o he message se C as he code. In nework error correcion coding, he code consiss of he message se C as well as he local encoding funcions of all he nodes in he nework. If C is a linear space, we say he nework code is linear, oher wise, non-linear. III. NETWORK HAMMING WEIGHTS The Hamming weigh and he Hamming disance are insrumenal for quanifying he abiliy of a classical block code for error correcion, error deecion and erasure correcion. We will inroduce similar definiions of weighs and disances for nework codes in his secion. The idea behind he definiion of he disance beween wo message vecors in he conex of neworks is ha he disance should be measured by he weigh of error vecors which can confuse he recepion of hese wo messages a he sinks. In classical error correcion, he weigh of an error vecor z is measured by he number of non-zero componens of he error vecor which is called he Hamming weigh of z and is denoed by w H (z). This disance measure, however, canno be applied o he nework case, because only he linear ransformaion of an error vecor affecs he recepion a he sinks. Thus, we measure he weigh of an error vecor by is affecion on he sink nodes. For any T, le Υ (y) = {z : zf = y} for a received vecor y Im(F ). Definiion 1: For any sink, he nework Hamming weigh of a received vecor y is defined by W rec (y) = min z Υ (y) w H(z). (2) Definiion 2: For any sink, he nework Hamming weigh of an error vecor z is defined by W err (z) = W rec (zf ). (3) In oher words, W err (z) is he minimum Hamming weigh of any error vecor ha causes he same confusion a sink as he error vecor z. For any vecor z Υ (0), W err (z) = W rec (0) = min z Υ (0) w H (z) = w H (0) = 0. If error vecors z 1 and z 2 saisfy z 1 z 2 Υ (0), hen W err (z 1 ) = W rec (z 1 F ) = W rec (z 2 F ) = W err (z 2 ). Thus we have wo properies of he weigh of error vecors: 1) If zf = 0, hen he weigh of z is zero; 2) If he difference of wo error vecors is an error vecor wih weigh 0, hen hese wo error vecors have he same weigh. Definiion 3: For any sink, he nework Hamming weigh of a message vecor x is defined by (x) = W rec (xf s, ). (4) In oher words, W msg (x) is he minimum Hamming weigh of any error vecor ha has he same effec on sink (when he message vecor is 0) as he message vecor x (when he error vecor is 0). Definiion 4: For any T, he nework Hamming disance beween wo received vecors y 1 and y 2 is defined by W msg (y 1, y 2 ) = W rec (y 1 y 2 ). (5) Definiion 5: For any T, he nework Hamming disance beween wo message vecors x 1 and x 2 is defined by D msg (x 1, x 2 ) = W msg (x 1 x 2 ). (6) When F = F s, = I, hese definiions reduce o he usual Hamming weigh and Hamming disance. Theorem 1 (Basic properies): Le x, x 1, x 2 GF(q) ns be message vecors, y, y 1, y 2 Im(F ) be received vecors, and z F E q be an error vecor. Then 1) W err (z) w H (z), W msg (x) = W err ([x 0]), and W msg (x) w H (x), where [x 0] is he error vecor obained by concaenaing he message vecor x by he zero vecor.

3 2) D msg (x, x 1 ) = D msg (x 1, x) = (xf s,, x 1 F s, ) = (x 1 F s,, xf s, ). 3) (Triangle inequaliy) and D msg (y 1, y) + D rec (y, y 2 ) (y 1, y 2 ), (x 1, x) + D msg (x, x 2 ) D msg (x 1, x 2 ). The firs inequaliy in Propery 1) is a direc Proof: consequence of he definiions. The equaliy in Propery 1) can be obained from he fac ha [x 0]F = xf s,. The second inequaliy holds since W msg (x) = W err ([x 0]) w H ([x 0]) = w H (x). The symmery of he disances in Propery 2) is obvious. The oher par of his propery holds since D msg (x, x 1 ) = W msg (x x 1 ) = W rec ((x x 1 )F s, ) = (xf s,, x 1 F s, ). We now prove he hird propery. Consider z 1 Υ (y 1 y) and z 2 Υ (y y 2 ) such ha D rec (y 1, y) = w H (z 1 ) and D rec (y, y 2 ) = w H (z 2 ). Since z 1 + z 2 Υ (y 1 y 2 ), we have (y 1, y 2 ) = W rec (y 1 y 2 ) w H (z 1 + z 2 ) w H (z 1 ) + w H (z 2 ) = D rec (y 1, y) + (y, y 2 ). The riangle inequaliy for he disance of message vecors can be obained by considering D msg (x 1, x) + D msg (x, x 2 ) = D rec (x 1 F s,, xf s, ) + (xf s,, x 2 F s, ) (x 1 F s,, x 2 F s, ) = D msg (x 1, x 2 ). A message se C for a mulicas in nework G is a subse of he vecor space F n s q. Noe ha we do no require C o be a linear space. Definiion 6: The unicas minimum disance of a nework code wih message se C for sink node is defined by d min, = min{d msg (x, x ) : x, x C, x x }. Definiion 7: The mulicas minimum disance of a nework code wih message se C is defined by d min = min T d min,. IV. ERROR CORRECTION AND DETECTION CAPACITIES In his secion, we sudy he performance of a nework code for correcing and deecing errors. We assume ha a sink node knows he message se C as well as he ransfer marices F s, and F. A. Unicas Case Theorem 2: For a sink node, he following hree properies of a nework code are equivalen: 1) The code can correc any error vecor z wih w H (z) c. 2) The code can correc any error vecor z wih W err (z) c. 3) The code has d min, 2c + 1. Proof: To prove 3) 2), we assume d min, 2c + 1. When he message vecor is x and he error vecor is z, he received vecor a sink is y = xf s, + zf. We hen declare he ransmied message vecor o be ˆx = arg min x C Drec (xf s,, y ). (7) We will show ha his decoding algorihm can always decode correcly for any message vecor x and any error vecor z wih W err (z) c. To his end, by means of he riangle inequaliy, we obain (xf s,, y ) + D rec (x F s,, y ) (xf s,, x F s, ), where x is any oher message vecor no equal o x. Since D rec (xf s,, x F s, ) = D msg (x, x ) d min, 2c + 1 and D rec (xf s,, y ) = W rec (xf s, y ) = W rec (zf ) = (z) c, we have W err (x F s,, y ) (xf s,, x F s, ) (xf s,, y ) c + 1 > (xf s,, y ). So he decoding algorihm gives ˆx = x. Thus 2) is rue. For any error vecor z, W err (z) w H (z). Thus, 2) 1). Now we prove 1) 3). Assume 3) does no hold, i.e., d min, 2c. We will show ha he nework code canno correc all error vecors z wih w H (z) c. Firs, we find wo messages x 1, x 2 such ha D msg (x 1, x 2 ) 2c. Since W rec ((x 1 x 2 )F s, ) = D rec (x 1 F s,, x 2 F s, ) = D msg (x 1, x 2 ) 2c, here exiss an error vecor z such ha (x 1 x 2 )F s, = zf and w H (z) 2c. Thus, we can consruc wo new error vecors z 1 and z 2 which saisfy w H (z 1 ) c, w H (z 2 ) c, and z 2 z 1 = z. I follows ha x 1 F s, + z 1 F = x 2 F s, + z 2 F. Then under he condiion ha he message vecor is x 1 and he error vecor is z 1, or he condiion ha he message vecor is x 2 and he error vecor is z 2, y = x 1 F s, + z 1 F = x 2 F s, + z 2 F is received a sink node. Therefore, no maer wha decoding algorihm is used, he algorihm canno decode correcly in boh cases, i.e., 1) does no hold. Hence, 1) 3). This complees he proof. Equaion (7) gives a decoding algorihm for nework codes. The proof of Theorem 2 verifies ha his algorihm can correc any vecor z wih W err (z) c a sink node if he code has d min, 2c + 1. This decoding algorihm can be regarded as he minimum disance decoding in he nework case. Theorem 3: For a sink node, he following hree properies of a nework code are equivalen:

4 1) The code can deec any error vecor z wih 0 < w H (z) d. 2) The code can deec any error vecor z wih 0 < W err (z) d. 3) The code has d min, d + 1. Remark: In classical coding, an error vecor wih Hamming weigh zero means no error has occurred. However, in he nework case, an error vecor z wih W err (z) = 0 does no imply ha z = 0. Raher, i means zf = 0, i.e., he error vecor z has no effec on sink node. Such an invisible error vecor can by no means (perhaps does no need o) be deeced a sink node. Proof: To prove 3) 2), we assume d min, d + 1. Le x C be a message vecor, and z be an error vecor wih (z) d. A sink node, if he received vecor y x F s, for any x C, we declare ha a leas one error has occurred during he ransmission. If he error vecor z canno been deeced, hen here exiss x C such ha y = xf s, + W err zf (x, x ) = (xf s,, x F s, ) = W rec (z) d < d min,. This is a conradicion o he definiion of d min,. So we conclude ha all he error vecors z wih W err (z) d can be deeced. The proof of 2) 1) is immediae because W err (z) w H (z). To prove 1) 3), we assume 3) does no hold, i.e., d min, d. Similar o he proof of Theorem 2, we can find wo message = x F s,. Thus D msg (zf ) = W err vecors x, x and an error vecor z wih w H (z) d and (x x )F s, = zf. This means when he message vecor x is ransmied, he error vecor z canno be deeced. Thus 1) does no hold. Hence, 1) 3). The proof is complee. B. Mulicas Case In he mulicas case, for a paricular nework code, an error vecor may have differen weighs for differen sink nodes, so ha he code may have differen unicas minimum disance for differen sink nodes. Applying Theorem 2 o all he sink nodes, we obain ha a nework code can correc all he error vecors in he se Θ = {z : W err (z) < d min, /2 for all u T }. (8) In pracice, we are very ofen concerned abou correcing all error vecors whose Hamming weighs do no exceed a cerain hreshold. However, i is no clear how he condiion specifying he se Θ in (8) is relaed o he Hamming weighs of he error vecors in ha se. Therefore, we also obain he following heorem which is he mulicas version of Theorem 2. The mulicas Hamming weigh of an error vecor z is defined by W err (z) = max T W err (z). Theorem 4: The following hree properies of a nework code are equivalen: 1) The code can correc any error vecor z wih w H (z) c a all he sink nodes. 2) The code can correc any error vecor z wih W err (z) c a all he sink nodes. 3) The code has d min 2c + 1. Proof: Assume 3) holds. Then d min, 2c + 1 a all he sink nodes. If z is any error vecor wih W err (z) c, i.e., W err (z) c a all he sink nodes, hen by Theorem 2, he nework code can correc error vecor z a all he sink nodes. Hence, 3) 2). Since W err (z) = max u T W err (z) w H (z), 2) 1) is immediae. To prove 1) 3), assume 1) holds. By Theorem 2, we have d min, 2c + 1 a all he sink nodes. Thus, d min 2c + 1, i.e., 3) holds. This complees he proof. Remark: From Theorem 4, we see ha a nework code can correc all he error vecors in he se Θ = {z : W err < d min /2}. We now show ha Θ Θ. Consider any error vecor z Θ, i.e., z saisfies W err (z) < d min /2, or max u T W err (z) < (1/2) min u T d min,, which implies W err (z) < d min, /2 for all he sink nodes. Therefore z Θ, and hence Θ Θ. However, Θ Θ does no hold in general. Using a similar argumen, we can prove he following error deecion heorem for mulicas. The deails are omied here. Theorem 5: The following hree properies of a nework code are equivalen: 1) The code can deec any error vecor z wih 0 < w H (z) d a all he sink nodes. 2) The code can deec any error vecor z wih 0 < W err (z) d a all he sink nodes. 3) The code has d min d + 1. V. RELATION BETWEEN THE MINIMUM RANK AND THE MINIMUM DISTANCE Zhang [7] has defined he minimum rank for linear nework codes and presened a minimum rank decoding algorihm based on his noion. In his secion, we will generalize he noion of minimum rank o non-linear message ses and prove ha he minimum rank is equal o he minimum disance. We will also prove ha under cerain condiions, minimum rank decoding is equivalen o minimum disance decoding. An error paern is a subse of E, denoed by ρ. An error vecor is said o mach an error paern if all he errors occur on he edges in he error paern. Noe ha if an error vecor z maches an error paern ρ, i also maches any error paern ρ if ρ ρ. The se of all error vecors ha mach ρ is denoed by ρ. Le ρ z be he error paern corresponding o he nonzero componens of an error vecor z. Define and (ρ) = {zf : z ρ } Φ = {(x x )F s, : x, x C, x x }. The rank of an error paern ρ for he sink node, denoed by rank (ρ), is defined as he dimension of he subspace (ρ). For ρ ρ, since (ρ ) (ρ), we have rank (ρ ) rank (ρ). The unicas minimum rank of a nework code for sink node is defined by r min, = min{rank (ρ) : (ρ) Φ }. (9)

5 Lemma 1 ( [7]): For any sink node and any error paern ρ, here exiss a subse ρ ρ such ha ρ = rank (ρ ) and (ρ ) = (ρ). Proof: The subspace (ρ) is spanned by he row vecors in F which correspond o he edges in ρ. Therefore, here exiss a subse ρ ρ such ha hese vecors corresponding o he edges in ρ form a basis of (ρ). This implies ha (ρ ) = (ρ) and rank (ρ ) = ρ. Theorem 6: r min, = d min,. Proof: Fix a sink node. Le Ω = {ρ : (ρ) Φ } and Γ = {z : zf Φ }. I is obvious ha r min, = min{rank (ρ) : ρ Ω} and d min, = min{w H (z) : z Γ}. Consider any z Γ. Since zf (ρ z ), we have ρ z Ω and rank (ρ z ) ρ z = w H (z). Now we will show ha w H (z) r min, by conradicion. By assuming ha w H (z) < r min,, we have rank (ρ z ) w H (z) < r min,, which is a conradicion o he definiion of he minimum rank. Thus we mus have w H (z) r min,. Hence, d min, = min z Γ w H (z) r min,. On he oher hand, for any ρ Ω, here exiss an error vecor z ρ such ha zf Φ. Thus z Γ, which means ha he se ρ Γ is no empy. By Lemma 1, here exiss an error paern ρ ρ such ha ρ = rank (ρ) and (ρ) = (ρ ). Furhermore, ρ Ω. Now we will show ha rank (ρ) w H (z) for some z ρ Γ by conradicion. Assume rank (ρ) < w H (z) for all z ρ Γ. Then we have ρ < w H (z) for all z ρ Γ. If here exiss z ρ Γ ρ Γ, hen w H (z ) ρ which conradics o wha we have obained. Thus, we have ρ Γ =, which is a conradicion o ρ Ω. So, rank (ρ) w H (z) d min, for some z ρ Γ. Hence, r min, = min ρ Ω rank (ρ) d min,. The proof is complee. Zhang [7] presened a minimum rank decoding algorihm and proved ha for any error paern ρ wih rank (ρ) (r min, 1)/2, he algorihm always decodes correcly. We now show ha for an error paern ρ wih rank (ρ) (r min, 1)/2, an error vecor z ρ has W err (z) (d min, 1)/2, so ha by Theorem 2 he minimum disance decoding can also decode correcly. By Lemma 1, here exiss ρ ρ such ha ρ = rank (ρ) and (ρ ) = (ρ). Thus for any z ρ, here exiss z ρ such ha zf = z F. Hence, W err (z) = W err (z ) w H (z ) ρ = rank (ρ) (r min, 1)/2 = (d min, 1)/2. For any error vecor z wih W err (z) (d min, 1)/2, we canno deermine wheher rank (ρ z ) (r min, 1)/2 holds or no. Neverheless, we know ha here exiss z Υ (zf ) wih w H (z ) (d min, 1)/2. Thus, rank (ρ z ) ρ z = w H (z ) (d min, 1)/2. When he error vecor is z, minimum rank decoding always decodes correcly. Now if he error vecor is z insead, since z F = zf, he impac o he sink node is exacly he same as if he error vecor is z. Thus minimum rank decoding also decodes correcly. Therefore, minimum rank decoding always decodes correcly no only for z wih rank (ρ z ) (d min, 1)/2 (as shown in [7]), bu more generally for z wih W err (z) (d min, 1)/2. By Theorem 2, minimum disance decoding always decodes correcly for error vecor z wih W err (z) (d min, 1)/2. Hence, under his condiion, minimum rank decoding and minimum disance decoding are equivalen. VI. NETWORK ERASURE In classical algebraic coding, erasure correcion is equivalen o error correcion wih he poenial posiions of he errors in he codewords known by he decoder. In his secion, we exend his heme o nework coding by assuming ha he se of channels in each of which an error may have occurred during he ransmission is known by he sink nodes, and we refer o his se of channels as he erasure paern. As before, we assume ha each sink node knows he message se C as well as he ransfer marices F s, and F. Two quaniies will be employed o characerize he abiliy of a nework code for erasure correcion. The firs one is he Hamming weigh of an erasure paern ρ, denoed by ρ. The second one, called he nework Hamming weigh of an error paern ρ, is defined as W esr (ρ) = max z ρ W err (z) w H (z) ρ for any z ρ, we have W err (z). Since W esr (ρ) ρ. (10) Theorem 7: A a sink node, he following hree properies of a nework code are equivalen: 1) The code can correc any erasure paern ρ wih ρ d. 2) The code can correc any erasure paern ρ wih W esr (ρ) d. 3) The code has d min, d + 1. Proof: To prove 3) 2), assume d min, d + 1. Le ρ be an erasure paern wih W esr (ρ) d. We ry o find a message vecor z C and an error vecor z ρ ha saisfy he equaion y = xf s, + zf ρ,. We call such a (x, z) pair a soluion. If here exiss only one x C such ha his equaion is solvable, we claim ha x is he decoded message vecor. If his equaion has wo soluions (x 1, z 1 ) and (x 2, z 2 ), where x 1 x 2, we can check ha D msg (x 1, x 2 ) = W rec ((x 1 x 2 )F s, ) = W rec ((z 2 z 1 )F ) = W err (z 2 z 1 ) d since z 2 z 1 ρ. This a conradicion o d min, d + 1. Hence, he code can correc any erasure paern ρ wih W esr (ρ) d, i.e., 2) holds. Since W esr (ρ) ρ for (10), 2) 1) is immediae. Finally we prove 1) 3) by conradicion. Assume a code has d min, d. We will find an erasure paern ρ wih ρ r ha canno be correced. Since d min, d, here exiss an error vecor z wih w H (z) d such ha D msg (x 1, x 2 ) = w H (z), where x 1, x 2 C, x 1 x 2. Thus we can consruc z 1, z 2 ρ z such ha z 2 z 1 = z. If y = x 1 F s, + z 1 F = x 2 F s, +z 2 F is received, by means of an argumen similar o ha in Theorem 2, we see ha sink node canno correc he erasure paern ρ z wih ρ z d. This complees he proof. For an erasure paern ρ, define he mulicas weigh as W esr (ρ) = max W esr u T (ρ). Using Theorem 7, we can easily obain he mulicas heorem for erasure correcion.

6 Theorem 8: The following hree properies of a nework code are equivalen: 1) The code can correc any erasure paern ρ wih ρ d a all he sink nodes. 2) The code can correc any erasure paern ρ wih W esr (ρ) d a all he sink nodes. 3) The code has d min d + 1. VII. CONCLUDING REMARKS In his work, we have defined he minimum disance of a nework code. Based on his minimum disance, he abiliy of a nework code in erms of error correcion/deecion and erasure correcion can be fully characerized. These resuls are nework generalizaions of he corresponding resuls in classical algebraic coding heory. Wih he inroducion of nework error correcion in [5], [6] and our resuls, any quesion ha may be raised in classical algebraic coding heory can be raised in he more general seing of nework coding. Thus here is a hos of problems o be invesigaed in he direcion of our work. REFERENCES [1] R. W. Yeung, S.-Y. R. Li, N. Cai, and Z. Zhang, Nework coding heory, Foundaion and Trends in Communicaions and Informaion Theory, vol. 2, no. 4 and 5, pp , [2] T. Ho, B. Leong, R. Koeer, M. Medard, M. Effros, and D. R. Karger, Byzanine modificaion deecion in mulicas neworks using randomized nework coding, in Proc. IEEE ISIT 04, June [3] S. Jaggi, M. Langberg, T. Ho, and M. Effros, Correcion of adversarial errors in neworks, in Proc. IEEE ISIT 05, July [4] N. Cai and R. W. Yeung, Nework coding and error correcion, in Proc. IEEE ITW 02, [5] R. W. Yeung and N. Cai, Nework error correcion, par I: basic conceps and upper bounds, Communicaions in Informaion and Sysems, vol. 6, no. 1, pp , [6] N. Cai and R. W. Yeung, Nework error correcion, par II: lower bounds, Communicaions in Informaion and Sysems, vol. 6, no. 1, pp , [7] Z. Zhang, Nework error correcion coding in packeized neworks, in Proc. IEEE ITW 06, Oc [8] R. Koeer and M. Medard, An algebraic approach o nework coding, IEEE/ACM Trans. Neworking, vol. 11, no. 5, pp , Oc [9] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, Nework informaion flow, IEEE Trans. Inform. Theory, vol. 46, no. 4, pp , July 2000.