Refined Coding Bounds for Network Error Correction

Transcription

1 Refned Codng Bounds for Network Error Correcton Shenghao Yang Department of Informaton Engneerng The Chnese Unversty of Hong Kong Shatn, N.T., Hong Kong Raymond W. Yeung Department of Informaton Engneerng The Chnese Unversty of Hong Kong Shatn, N.T., Hong Kong Abstract Wth respect to a gven set of local encodng kernels defnng a lnear network code, refned versons of the Hammng bound, the Sngleton bound the Glbert-Varshamov bound for network error correcton are proved by the weght propertes of network codes. Ths refned Sngleton bound s also proved to be tght for lnear message sets. Index Terms Network error correcton codng, network Hammng weght, Sngleton bound, Glbert-Varshamov bound. I. INTRODUCTION The network error correcton problem studed n [1] [5] s to extend classcal error correcton codng theory to a general network settng. The concept of network error correcton codng was frst ntroduced by Ca Yeung [1] [3]. They generalzed the Hammng bound, the Sngleton bound the Glbert-Varshamov bound n classcal error correcton codng to network codng. Zhang [4] ntroduced the mnmum rank for lnear network codes, whch plays a role smlar to that of the mnmum dstance n decodng classcal error-correctng codes. Recently, network generalzatons of the Hammng weght, the Hammng dstance, the mnmum dstance of network codes have been obtaned by Yang Yeung [5]. In terms of the mnmum dstance, the capablty of a network code for error correcton, error detecton, erasure correcton can be fully characterzed. The relaton between network codng maxmum dstance separaton (MDS codes n classcal algebrac codng has been clarfed n [6]. In ths paper, we present stronger versons of the Hammng bound, the Sngleton bound the Glbert-Varshamov bound for network error correcton compared wth those bounds obtaned n [1] [3]. Our proofs of these bounds are based on the weght propertes of network codes [5]. An algorthm that constructs a lnear network codes achevng a refned Sngleton bound from a classcal lnear block code has recently been presented n [9]. In ths paper, we present a dfferent constructve proof to the tghtness of the Sngleton bound wth respect to a gven sets of global encodng kernels defnng a lnear network code. Ths paper s organzed as follows. Secton II formulates the network error correcton problem. Secton III revews the weght propertes of network codes. The codng bounds are proved n Secton IV. In the last secton we summarze our work. II. PROBLEM FORMULATION We study network transmsson n a drected acyclc communcaton network represented by G = (V, E, where V s the set of nodes E s the set of edges n the network. We assume an order on the edge set E whch s consstent wth the assocated partal order of the drected acyclc network G. An edge from node a to node b, say edge e, represents a communcaton channel from node a to node b. We call node a (node b the tal (head of edge e, denoted by tal(e (head(e. In(a = {e E : head(e = a} Out(a = {e E : tal(e = a} be the sets of nput edges output edges of node a, respectvely. There can be multple edges between a par of nodes, each edge can transmt one symbol n a fnte feld F. A multcast on G transmts nformaton from a source node s to a set of snk nodes T. n s = Out(s. The source node s modulates the nformaton to be multcast nto a row vector x F ns called the message vector. The set of all message vectors, a subset of F n s, s called the message set denoted by C. The source node s transmts the message vector by mappng ts n s components onto the edges n Out(s. Defne an n s E matrx A = [A,j ] as { 1 ej s the th edge n Out(s, A,j = (1 0 otherwse. By applyng the order on E to Out(s, the n s non-zero columns of A form an dentty matrx. An error vector z s an E -dmensonal row vector wth each component representng the error on the correspondng edge. An error pattern s a subset of E. An error vector s sad to match an error pattern f all the errors occur on the edges n the error pattern. The set of all error vectors that match error pattern ρ s denoted by ρ. ρ z be the error pattern correspondng to the non-zero components of an error vector z. For network G, a lnear network error-correctng code, or a lnear network code for brevty, s specfed by a set of local encodng kernels {k e,e : e, e E} the message set C. The local encodng kernel k e,e can be non-zero only f e In(tal(e. Defne the E E one-step transton matrx

2 K = [K,j ] for network G as K,j = k e,e j. ( For an acyclc network, K N = 0 for some postve nteger N. Defne the transfer matrx of the network by F = (I K 1 [7], so that the symbols transmtted on the edges are gven by the components of (xa + zf. For a snk node t T, wrte n t = In(t, defne an E n t matrx B t = [B,j ] for snk node t as Defnton : For any snk node t, the network Hammng dstance between two receved vectors y 1 y s defned by Dt rec (y 1, y = Wt rec (y 1 y ; (8 the network Hammng dstance between two message vectors x 1 x s defned by D msg t (x 1, x = W msg t (x 1 x. (9 B,j = { 1 e s the jth edge n In(t, 0 otherwse. (3 Defnton 3: The uncast mnmum dstance of a network code wth message set C for snk node t s defned by Agan by applyng the order on E to In(t, the n t nonzero rows of B t form an dentty matrx. The recepton at a snk node t s gven by y t = (xa + zf B t, = xf s,t + zf t, (4 where F s,t = AF B t s the submatrx of F gven by the ntersecton of the n s rows correspondng to the edges n Out(s the n t columns correspondng to the edges n In(t, F t = F B t s the submatrx of F formed by the columns of F correspondng to the nput edges of snk node t. F s,t F t are the transfer matrces of message transmsson error transmsson, respectvely, for snk node t. Denote the rank of F s,t by m t. Euaton (4 s the formulaton of the multcast network error correcton problem gven n [4]. The classcal error correcton problem s a specal case n whch both of F s,t F t reduce to dentty matrces. The message transmsson capacty of the network code, n the absence of errors, s measured by the rank of the transfer matrx F s,t. Denote the maxmum flow between source node s snk node t by maxflow(s,t. Evdently, for any lnear network code on G, the rank of F s,t s upper bounded by maxflow(s,t [6]. When the network s error-free, the error correcton problem s reduced to the usual network codng problem, for whch the sze of the message set C s bounded by mn maxflow(s,t [8]. III. PREVIOUS RESULTS A. Weght Propertes of Network Codes For any t T, let Υ t (y = {z F E : zf t = y} for a receved vector y Im(F t = {z F t : z F E }. Defnton 1: For any snk node t, the network Hammng weghts of a receved vector y, of an error vector z, of a message vector x are defned by respectvely. W rec t (y = mn z Υ t (y w H(z, (5 Wt err (z = Wt rec (zf t, (6 W msg t (x = Wt rec (xf s,t, (7 d mn,t = mn{d msg t (x, x : x, x C, x x }. (10 Theorem 1 ( [5]: For a snk node t, the followng fve propertes of a network code are euvalent: 1 The code can correct any error vector z wth w H (z d/; The code can correct any error vector z wth Wt err (z d/; 3 The code can detect any error vector z wth 0 < w H (z d; 4 The code can detect any error vector z wth 0 < Wt err (z d; 5 The code has d mn,t d + 1; where d s a nonnegatve nteger. B. Codng Bounds d mn = mn d mn,t, (11 n = mn maxflow(s, t. (1 In terms of the noton of mnmum dstance, the Hammng bound the Sngleton bound for network codes obtaned n [] can be restated as C where r = d mn 1, =0 n ( n, (13 C n dmn+1. (14 The tghtness of (14 has been proved n [3]. IV. REFINED CODING BOUNDS In ths secton, we employ the tools of network Hammng weght to prove refned versons of the codng bounds n [1] [3]. The proofs presented here are consderably smpler more transparent.

3 A. Hammng Bound Sngleton Bound Theorem (Hammng bound Sngleton bound: A network code wth rank(f s,t = m t, message set C, uncast mnmum dstance d mn,t > 0 for any snk node t satsfes the 1 Hammng bound: C t =0 m t where r t = d mn,t 1, the Sngleton bound: C m t d mn,t +1, (15 (16 for all snk node t. Proof: Fx a snk node t. Fnd m t lnearly ndependent rows of F s,t let ρ t be the set of edges n Out(s that corresponds to these m t lnearly ndependent rows. Note that ρ t Out(s E, so that ρ t s an error pattern. Defne the set C t = {x F n s : x A ρ t, x F s,t = xf s,t for some x C}, (17 where the matrx A s defned as (1. Defne a mappng φ t : C C t (18 by φ t (x = x f x F s,t = xf s,t. Snce the rows of F s,t ndexed by ρ t form a bass for the row space of F s,t, φ t s well-defned. The mappng φ t s onto by the defnton of C t. The mappng φ t s also one-to-one because otherwse there exsts x C t such that x F s,t = x 1 F s,t = x F s,t for dstnct x 1, x C, a contradcton to the assumpton that d mn,t > 0. Thus the mappng φ t s a one-to-one onto mappng, whch mples that C t = C. Z t = {z ρ t : w H (z r t }. (19 By Theorem 1, the network code wth C beng the message set can correct all the errors n Z t at snk node t. Snce snk node t has the same recepton for the transmsson of ether x C or φ t (x C t for the same error vector, the network code wth C t beng the message set can also correct all the errors n Z t at snk node t. Consder the problem of fndng a subset of ρ t as an errorcorrectng code that can correct all the errors n Z t. Ths problem s euvalent to the problem n classcal algebrac codng of fndng a block code wth codeword length m t that can correct r t errors. The vectors n the set C t = {xa : x C t} must form such a code, otherwse the network code wth C t beng the message set cannot possbly correct all the error vectors n Z t at snk node t. Applyng the Hammng bound the Sngleton bound for classcal error-correctng codes to C t, we have C t t =0 mt, (0 C t mt dmn,t+1. (1 The proof s completed by notng that C = C t = C t. The Hammng bound the Sngleton bound n Theorem are more refned than those n [1], [] because as we wll show, they mply (13 (14 but not vce versa. The Hammng bound n Theorem mples C t =0 =0 mt mt maxflow(s,t =0 ( maxflow(s,t ( (3 (4 for all snk nodes t, where (3 follows from r r t (4 follows from m t maxflow(s, t the neualty proved n the Appendx. By the same neualty, upon mnmzng over all snk nodes t T, we obtan (13. To verfy that the condton for the neualty n the Appendx apples n the above, by consderng the Sngleton bound n (16, we obtan or for all t T. Then 1 C (5 mt dmn,t+1 (6 d mn,t 1 m t (7 r = d mn 1 (8 d mn,t 1 (9 d mn,t 1 (30 m t (31 for all t T. For the Sngleton bound n Theorem, we frst note that t s maxmzed when m t = maxflow(s, t for all t T. Ths can be acheved by a lnear broadcast whose exstence was proved n [10], [6]. To show that the Sngleton bound n Theorem mples (14, consder C m t d mn,t +1 m t d mn +1 maxflow(s,t d mn+1 (3 (33 (34 for all snk nodes t. Then (14 s obtaned upon mnmzng over all t T. B. Glbert Bound Varshamov Bound t (x, d = {x F ns For x = 0, we can wrte t (0, d = {x F ns : D msg t (x, x d}. (35 : W msg t (x d}. (36

4 Then t s readly seen that t (0, d s closed under scalar multplcaton,.e., α t (0, d = {αx : x t (0, d} = t (0, d, (37 where α F α 0. For two subsets V 1, V F ns, ther sum s the set defned by V 1 + V = {v 1 + v : v 1 V 1, v V }. (38 For v F ns V F ns, we also wrte {v} + V as v + V. Theorem 3 (Glbert bound: Gven a set of local encodng kernels, let C max be the maxmum possble sze of the message set such that the network code has uncast mnmum dstance greater than or eual to d t > 0 for each snk node t. Then, where C max ns (0, (39 (0 = t (0, d t 1. (40 Proof: C be the message set wth the maxmum possble sze. Then for any x F n s, there exsts a codeword c C a snk node t such that D msg t (x, c d t 1, (41 snce otherwse we could add x to the message set whle keepng the mnmum dstance larger than or eual to d t for each snk node t, whch s a contradcton on the maxmalty of C. (c = t (c, d t 1. (4 Hence, the whole space F n s s contaned n the unon of (c over all messages c C,.e., F n s c C (c. (43 Snce (c = c + (0, we have (c = (0. So we deduce that ns C (0, that s C ns (0. (44 Theorem 4 (Varshamov bound: Gven a set of local encodng kernels, let ω max be the maxmum possble dmenson of the lnear message set such that the network code has uncast mnmum dstance larger than or eual to d t > 0 for each snk node t. Then, ω max n s log (0, (45 where (0 s defned n (40. Proof: C be the lnear message set wth the maxmum possble dmenson. We clam that F ns (0 + C. (46 If the clam s true, then ns = (0 + C (47 (0 C (48 = (0 ω max, (49 provng (45. The clam s proved by contradcton. g F ns \ ( (0 + C, (50 C = C + g. Then C s a subspace wth dmenson ω max +1. If C (0 {0}, then there exsts a non-zero vector c + αg (0, (51 where c C α F. Here, α 0, otherwse we have c = 0 because C (0 = {0}. Snce t (0, d t 1 s closed under scalar multplcaton for all t T, see from (40 that the same holds for (0. Thus from (51, g (0 α 1 c (5 (0 + C, (53 whch s a contradcton to (50. Therefore, C (0 = {0},.e., C s a message set such that the network code has uncast mnmum dstance larger than or eual to d t, whch s a contradcton on the maxmalty of C. The proof s completed. C. Tghtness of the Sngleton Bound Theorem 5: Gven a set of local encodng kernels over a fnte feld wth sze where s suffcently large, for every 0 ω mn m t, (54 there exsts a message set C wth C = ω such that d mn,t = + 1 (55 for all snk nodes t. Proof: we start wth any gven set of local encodng kernels whch defnes a lnear network code. Ths determnes m t for all snk nodes t. Fx an ω whch satsfes (54. We wll then construct an ω-dmensonal lnear message set whch together wth the gven lnear network code consttute a lnear network error-correctng code that satsfy (55 for all t. Note that (54 (55 mply d mn,t 1. (56 We now construct the message set C. g 1,, g ω F n s be a seuence of vectors obtaned as follows. For each, 1 ω, choose g such that g / t (0, + g 1,, g 1, (57 t (0, g 1,, g = {0}, (58

5 for each snk node t. If such g 1,, g ω exst, then C = g 1,, g ω s the desred message set snce (58 holdng for = ω means d mn,t + 1 for any snk node t. We frst prove that g satsfyng (57 exsts f the feld sze s suffcently large. Observe that t (0, + g 1,, g 1 t (0, 1 (59 ( E mt ω n s m t 1 (60 ( E = ns ω+ 1. (61 Thus, when consderng all the snk nodes, we have at most ( E n s ω+ 1 (6 vectors that cannot be chosen as g. If > ( E, (63 then there exsts a vector that can be chosen as g for = 1,, ω. Fx g 1,, g ω that satsfy (57. We proof by nducton that (58 holds for these g any snk node t. If (58 does not hold for = 1, then there exsts a non-zero vector αg 1 t (0,, where α F. Snce t (0, s closed under scalar multplcaton α 0, we have g 1 t (0,, a contradcton to (57 holdng for g 1. Assume (58 holds for k 1. If (58 does not hold for = k, then there exsts a non-zero vector k α g t (0,, (64 =1 where α F. If α k = 0, k 1 α g t (0,, (65 =1 a contradcton to the assumpton that (58 holds for = k 1. Thus α k 0. Agan, by t (0, beng closed under scalar multplcaton, we have k 1 g k t (0, α 1 k α g (66 =1 t (0, + g 1,, g k 1, (67 a contradcton to g k satsfyng (57. The proof s completed. The neualty m ( m =0 APPENDIX PROOF OF AN INEQUALITY < m+1 r =0 for r m/ can be establshed by consderng m ( m =0 = < =0 m+1 (m +1 m+1 m+1 =0 (68 (69, (70 where (70 holds because (m +1 m+1 > 1 gven that r m/. REFERENCES [1] N. Ca R. W. Yeung, Network codng error correcton, n Proc. IEEE ITW 0, 00. [] R. W. Yeung N. Ca, Network error correcton, part I: basc concepts upper bounds, Communcatons n Informaton Systems, vol. 6, no. 1, pp , 006. [3] N. Ca R. W. Yeung, Network error correcton, part II: lower bounds, Communcatons n Informaton Systems, vol. 6, no. 1, pp , 006. [4] Z. Zhang, Network error correcton codng n packetzed networks, n Proc. IEEE ITW 06, Oct [5] S. Yang R. W. Yeung, Characterzatons of network error correcton/detecton erasure correcton, n Proc. NetCod 07, Jan [6] R. W. Yeung, S.-Y. R. L, N. Ca, Z. Zhang, Network codng theory, Foundaton Trends n Communcatons Informaton Theory, vol., no. 4 5, pp , 005. [7] R. Koetter M. Medard, An algebrac approach to network codng, IEEE/ACM Trans. Networkng, vol. 11, no. 5, pp , Oct [8] R. Ahlswede, N. Ca, S.-Y. R. L, R. W. Yeung, Network nformaton flow, IEEE Trans. Inform. Theory, vol. 46, no. 4, pp , July 000. [9] S. Yang, C. K. Nga, R. W. Yeung, Constructon of lnear network codes that acheve a refned Sngleton bound, submtted to ISIT 07. [10] S.-Y. R. L, R. W. Yeung, N. Ca, Lnear network codng, IEEE Trans. Inform. Theory, vol. 49, no., pp , Feb [11] S. Jagg, P. Sers, P. A. Chou, M. Effros, S. Egner, K. Jan, L. Tolhuzen, Polynomal tme algorthms for multcast network code constructon, IEEE Trans. Inform. Theory, vol. 51, no. 6, pp , June 005. V. CONCLUDING REMARKS Refned versons of the Hammng bound, the Sngleton bound the Glbert-Varshamov bound for network error correcton n [1] [3] are obtaned. Ths refned Sngleton bound s also shown to be tght for lnear message sets. By employng the tools of network Hammng weght, the proofs presented here are consderably smpler more transparent.