On Metrics for Error Correction in Network Coding

On Metric s for Error Correction in Network Coding Danilo Silva and Frank R. Kschischang Abstract The problem of error cor rection in both co herent and non coheren t network cod ing is c onsidered under an adversarial mo del. For co herent network co ding, where knowledge of the n etwork top ology and netw ork code is assumed at the sou rce and destination nod es, the error cor rection cap ability of an (outer) cod e is succinctly d escribed by the rank metric; as a consequ ence, it is shown that un iv ersal network erro r correctin g codes achieving the Singleton bound can be easily co nstructed an d efficiently decoded . For nonco herent network co ding, wher e knowledge of the network topolog y and n etwork cod e is not assumed , the error cor rection capability of a (sub space) code is given exactly by a n ew metric, called the injec tion metric , which is closely related to, but different than , th e sub space metr ic of K ¨ otter and Kschischang. In particular, in th e case of a no n-constan t-dimension code, the decod er associated with the injection metric is sh own to correct more errors then a minimum-sub space-distance decod er . All of the se r esults are based on a general approach to adversarial er ror correction, wh ich could b e usefu l for other ad versarial chan nels b eyond network co ding. Index T erms Adversarial chann els, error co rrection, in jection distance, network cod ing, ran k distanc e, subsp ace codes. I . I N T RO D U C T I O N The problem of error correction for a network implementing linear network c oding has been a n activ e research area since 2002 [1]–[12 ]. The crucial motiv ation for the prob lem is the p henome non o f e rror This work was supported by C APES Foundation, Brazil, and by the Nat ural Sciences and Engineering Research Council of Canada. Portions of this paper were presented at t he I EEE Information T heory W orkshop, Bergen, Norway , July 2007, and at the 46th Annual Allerton Conference on Communications, Control, and Computing, Monticello, IL, September 2008. The authors are wi th The Edward S. R ogers S r . Department of Electrical and Computer Engineering, Univ ersit y of T oronto, T oron to, ON M5S 3G4, Canada (e-mail: danilo@comm.utoronto.ca, fr ank@comm.utoron to.ca). 2 propagation, which arises due to the recombina tion cha racteristic at the heart o f network co ding. A single corrupt packet oc curring in the ap plication layer (e.g ., introduced by a malicious user) may proceed undetected a nd contaminate othe r p ackets, caus ing poten tially drastic cons equen ces and essen tially ruling out classica l error correction approac hes. In the basic multicast model for linear n etwork coding, a source node transmits n pac kets, eac h consisting of m symbols from a finite fie ld F q . Eac h link in the network transports a packet free of errors, an d eac h node c reates ou tgoing pac kets as F q -linear c ombinations of incoming pac kets. T here are one o r more destination nod es that wish to obtain the original source pac kets. At a specific destination node, the received packets may be represented as the rows of an N × m matrix Y = AX , where X is the matrix whose rows are the s ource pa ckets and A is the transfer matrix of the network. Errors a re incorporated in the model by allowing u p to t error pa ckets to be adde d (in the vector s pace F m q ) to the packets sent over one or mo re links. The rece iv e d matrix Y at a spe cific destination no de may then be written as Y = AX + D Z (1) where Z is a t × m matrix whos e rows are the error pac kets, and D is the trans fer matrix from these packets to the des tination. Unde r this model, a coding-theoretic prob lem is how to des ign an ou ter c ode and the underlying network code su ch that reliable co mmunication (to all destinations) is pos sible. This coding p roblem ca n be posed in a n umber of ways d epending on the set o f assumptions made . For exa mple, we ma y assume that the ne twork topology a nd the network code are known at the s ource and at the destination nodes, in which case we call the system coherent network c oding . Alternati vely , we may a ssume tha t such information is unavail able, in which case we call the system noncoherent network cod ing . The error matrix Z may b e rando m or c hosen by an adversa ry , and there may be further assumptions on the knowledge or other cap abilities of the a dversary . Th e essential assumption, in order to pose a meaningful coding prob lem, is tha t the number of injected error pa ckets, t , is b ounded . Error correction for cohe rent network c oding was originally studied by Cai and Y eung [1]–[3]. Aiming to establish fundamental limits, they foc used on the fun damental case m = 1 . In [2], [3] (se e also [9], [10]), the a uthors deriv e a Sing leton bound in this context and cons truct codes that ac hiev e this bound. A drawback of their approa ch is tha t the field s ize required c an be very large (on the order of  |E | 2 t  , where |E | is the number of ed ges in the network), and no e f ficient dec oding method is giv en. Similar constructions, analyse s and bounds appea r also in [4], [5], [11], [12]. In S ection IV, we approac h this problem (for general m ) und er a d if feren t framework. W e assu me 3 the pess imistic situation in which the a dversary can not only inject up to t packets but can also freely choose the ma trix D . In this sce nario, it is essen tial to exploit the s tructure of the problem when m > 1 . The propos ed a pproach allows us to find a metric—the rank me tric—that suc cinctly describes the error correction capability of a code. W e quite eas ily obtain bo unds an d constructions analogou s to those of [2], [3], [9]–[11], and show that ma ny o f the results in [4], [12] can be reinterpreted an d simplified in this framew ork. Moreover , we find that ou r pes simistic assump tion ac tually inc urs no p enalty sinc e the codes we propose a chieve the Singleton bound of [2]. An advantage of this approach is that it is universal , in the sense that the outer code a nd the n etwork code may be d esigned independen tly o f ea ch o ther . More precisely , the ou ter code may be chose n as any rank-metric code with a good error- correction c apability , while the n etwork code ca n be de signed as if the network were error- free (and, in particular , the fie ld size ca n b e cho sen a s the minimum req uired for mu lticast). An a dditional a dvantage is that encod ing and decoding of properly chosen rank-metric codes can be p erformed very efficiently [8]. For nonco herent network coding , a combinatorial framework for error co ntrol was introdu ced by K ¨ otter and Kschisc hang in [7]. The re, the problem is formulated as the transmiss ion of subspa ces through an operator channe l, where the transmitted and received sub space s are the row spa ces of the matrices X and Y in (1), respe cti vely . They proposed a metric that is suitable for this chan nel, the so-called s ubspa ce distance [7]. They also presented a Singleton-like bo und for their metric and s ubspa ce c odes ac hieving this b ound. The main justifica tion for their metric is the fact that a minimum subs pace distanc e decod er seems to b e the neces sary and sufficient tool for optimally d ecoding the dis turbances impose d b y the operator cha nnel. Howe ver , wh en these disturba nces are translated to more concrete terms su ch as the number of error p ackets injec ted, only deco ding g uarantees can be ob tained for the minimum d istance decode r of [7], but no co n verse. More precisely , assu me that t error pac kets are injected an d a gene ral (not ne cess arily cons tant-dimension) subs pace code w ith minimum subspac e distance d is used. In this case, while it is pos sible to guara ntee suc cess ful dec oding if t < d/ 4 , and we k now of specific examples where decoding fails if this con dition is n ot me t, a general con verse is n ot k nown. In S ection V, we prove suc h a con verse for a new metric—called the injection distance —under a slightly different transmission model. W e assume that the adversary is allowed to arbitrarily s elect the matrices A and D , provided that a lower bound on the rank of A is resp ected. Under this pess imistic scena rio, we show that the injection d istance is the fundamen tal pa rameter be hind the error correc tion capability o f a c ode; that is, we c an g uarantee correction of t packet errors if and on ly if t is les s tha n h alf the minimum injection distance of the code. While this approach ma y se em too pess imistic, we provide a class of examples where a minimum-injection-distance decod er is able to correct mo re errors than a 4 minimum-subspac e-distance de coder . Moreover , the two approach es co incide when a constan t-dimension code is used. In order to gi ve a u nified treatment of both coherent a nd nonco herent network co ding, we first develop a general approac h to error correction over (certain) adversarial channels. Ou r treatment genera lizes the more ab stract po rtions of class ical coding the ory a nd has the main feature of ma thematical simplicity . The essen ce of our approach is to use a single func tion—called a discrepancy fun ction —to fully describe a n adversarial cha nnel. W e then propo se a distance-like func tion tha t is easy to handle analytically and (in many c ases , including a ll the channels considered in this pa per) precisely describes the error correction capability o f a code . The motiv a tion for this app roach is tha t, onc e such a d istance fun ction is found, one can virtually forget abou t the cha nnel mode l a nd fully concentrate on the co mbinatorial problem of finding the largest cod e with a spe cified minimum distance (just like in clas sical coding theory). Interestingly , our app roach is a lso useful to c haracterize the error detection capability o f a c ode. The remainde r of the pap er is or ga nized as follows. Se ction II e stablishes o ur notation and revie w some bas ic facts about matrices an d rank-metric code s. Section III-A pre sents our gen eral approach to adversarial error correction, which is subs equen tly specialized to coh erent an d nonco herent n etwork coding mod els. Section IV des cribes our main results for cohere nt network coding a nd discu sses their relationship with the work o f Y eung et al. [2]–[4]. Section V describes our ma in results for noncohe rent network coding and discusse s the ir relationship w ith the work of K ¨ otter an d Ksch ischang [7]. Section VI presents our conclusions . I I . P R E L I M I N A R I E S A. Basic Notation Define N = { 0 , 1 , 2 , . . . } a nd [ x ] + = max { x, 0 } . The following notation is used many times through out the paper . L et X be a s et, and let C ⊆ X . When ev er a function d : X × X → N is defined, denote d ( C ) , min x,x ′ ∈C : x 6 = x ′ d ( x, x ′ ) . If d ( x, x ′ ) is called a “distance” b etween x and x ′ , then d ( C ) is c alled the minimum “distance” of C . B. Matrices and Subspace s Let F q denote the fin ite field with q elements. W e use F n × m q to denote the se t of all n × m matrices over F q and use P q ( m ) to denote the s et of all subspac es of the vector sp ace F m q . 5 Let dim V denote the d imension of a vector space V , let h X i de note the row space of a ma trix X , and let wt ( X ) de note the n umber of nonzero ro ws o f X . Recall that dim h X i = rank X ≤ wt ( X ) . Let U an d V be sub space s of some fixed vector spac e. Reca ll tha t the sum U + V = { u + v : u ∈ U , v ∈ V } is the sma llest vector spac e that con tains both U a nd V , wh ile the intersection U ∩ V is the lar gest vector space that is conta ined in both U and V . Rec all also that dim ( U + V ) = dim U + dim V − dim ( U ∩ V ) . (2) The ran k of a matrix X ∈ F n × m q is the smalles t r for which the re exist matrices P ∈ F n × r q and Q ∈ F r × m q such tha t X = P Q . Note that both matrices obtained in the de compos ition a re full-rank; accordingly , such a d ecompo sition is called a full-rank decomp osition [13]. In this c ase, note that, by partitioning P a nd Q , the matrix X ca n be further expan ded as X = P Q = h P ′ P ′′ i   Q ′ Q ′′   = P ′ Q ′ + P ′′ Q ′′ where rank ( P ′ Q ′ ) + rank ( P ′′ Q ′′ ) = r . Another useful property of the rank function is that, for X ∈ F n × m q and A ∈ F N × n q , we have [13] rank A + rank X − n ≤ rank AX ≤ m in { rank A, rank X } . (3) C. Rank-Metric Codes Let X, Y ∈ F n × m q be matrices. The ran k distanc e be tween X a nd Y is defined as d R ( X, Y ) , rank ( Y − X ) . It is well known that the rank d istance is indeed a metric; in p articular , it sa tisfies the triangle inequality [13], [14]. A rank -metric c ode is a matrix c ode C ⊆ F n × m q used in the context of the rank me tric. The Singleton bound for the ran k me tric [14 ] (see als o [8]) states that every rank-metric co de C ⊆ F n × m q with minimum rank distance d R ( C ) = d must satisfy |C | ≤ q max { n,m } (min { n,m }− d +1) . (4) Codes that achieve this b ound are called maximum-rank-distanc e (MRD) codes and they a re known to exist for a ll c hoices of parameters q , n , m a nd d ≤ min { n, m } [14]. 6 I I I . A G E N E R A L A P P R O AC H T O A D V E R S A R I A L E R R O R C O R R E C T I O N This se ction presents a ge neral approach to error co rrection over ad versarial c hannels. This approac h is specialized to coherent and noncoheren t n etwork coding in se ctions IV and V, res pectively . A. Adversarial C hannels An adv ersarial channel is specifie d by a finite inpu t alphab et X , a finite output alphab et Y and a collection of fan-out s ets Y x ⊆ Y for all x ∈ X . For each inpu t x , the ou tput y is constraine d to be in Y x but is othe rwise arbitrarily chos en by a n adversary . The constraint on the o utput is important: otherwise , the a dversary cou ld prevent c ommunication simply by mapping all inputs to the sa me output. No further restrictions a re imposed o n the adversary; in pa rticular , the ad versary is potentially omn iscient an d ha s unlimited computational power . A code for an adversarial ch annel is a sub set 1 C ⊆ X . W e say that a code is una mbiguous for a ch annel if the inpu t codeword can always be uniquely d etermined from the chan nel o utput. More precise ly , a code C is un ambiguous if the sets Y x , x ∈ C , a re pairwise d isjoint. The importance of this conce pt lies in the f act that, if the code is n ot una mbiguous, the n there exist codewords x, x ′ that a re indistinguishable at the d ecoder: if Y x ∩ Y x ′ 6 = ∅ , then the a dversary can (and will) exploit this ambiguity b y ma pping both x and x ′ to the same output. A decode r for a code C is any function ˆ x : Y → C ∪ { f } , wh ere f 6∈ C denotes a decod ing failure (detected error). When x ∈ C is transmitted and y ∈ Y x is received, a de coder is said to be succe ssful if ˆ x ( y ) = x . W e say that a d ecode r is infallible if it is suc cessful for all y ∈ Y x and all x ∈ C . Note that the existence of an infallible decod er for C implies that C is unambigu ous. Con versely , given any unambiguou s code C , one ca n a lways find (by definition) a decode r that is infalli ble. One example is the exhaustive deco der ˆ x ( y ) =      x if y ∈ Y x and y 6∈ Y x ′ for all x ′ ∈ C , x ′ = x f otherwise . In other words, an exhaustive deco der returns x if x is the un ique cod ew ord that could poss ibly have been transmitted when y is receiv ed, and returns a failure otherwise. Ideally , one would like to fin d a large (or largest) code that is u nambiguous for a given a dversarial channe l, together with a decod er that is infallible (and computationally-efficient to implement). 1 There is no l oss of generality in considering a single channel use, since the channel may be taken to correspond to multiple uses of a simpler channel. 7 B. Discr e pancy It is useful to co nsider adversarial channe ls p arameterized by an ad versarial e f fort t ∈ N . Assume that the fan-out se ts a re of the form Y x = { y ∈ Y : ∆( x, y ) ≤ t } (5) for s ome ∆ : X × Y → N . The value ∆( x, y ) , wh ich we c all the discrepancy betwee n x a nd y , rep resents the minimum effort need ed for an adversary to transform an inp ut x into an ou tput y . The value of t represents the maximum adversarial effort (maximum disc repancy) allowed in the chann el. In principle, there is no los s of ge nerality in assu ming (5) s ince, by prop erly de fining ∆( x, y ) , one can always expres s any Y x in this form. For instan ce, on e cou ld s et ∆( x, y ) = 0 if y ∈ Y x , and ∆( x, y ) = ∞ otherwise. However , su ch a definition would be of no practical value sinc e ∆( x, y ) would b e merely an indicator fun ction. Thus , an eff ectiv e limitation o f our model is that it requires c hanne ls that are na turally characterized by some disc repancy function. In particular , one sho uld be able to interpret the maximum discrepancy t a s the level of “degraded ness” of the channel. On the other ha nd, the ass umption ∆( x, y ) ∈ N impos es eff ectiv e ly no c onstraint. Since |X × Y | is finite, given any “na turally define d” ∆ ′ : X × Y → R , one c an always shift, s cale and rou nd the image of ∆ ′ in order to produce some ∆ : X × Y → N that induces the same fan-out sets as ∆ ′ for all t . Example 1: Let us use the above no tation to d efine a t -error cha nnel, i.e ., a vector chan nel that introduces at mos t t symb ol errors (arbitrarily chosen b y an adversa ry). Assume that the channe l input and output alphabets are g i ven b y X = Y = F n q . It is easy to see that the chan nel can be cha racterized by a d iscrepancy fun ction that co unts the numbe r of compo nents in which a n input vec tor x and an output vec tor y dif fer . More p recisely , we have ∆( x, y ) = d H ( x, y ) , wh ere d H ( · , · ) d enotes the Hamming distance function. A main fea ture of ou r propos ed discrepan cy characterization is to a llow u s to s tudy a wh ole famil y of c hannels (with various lev els of d egradednes s) und er the same framework. For instance, we ca n use a single decode r for all channels in the same family . De fine the minimum-discr e pancy decode r given by ˆ x = argmin x ∈C ∆( x, y ) (6) where a ny ties in (6) are a ssumed to be broken arbitrarily . It is e asy to see that a minimum-discrepancy decode r is infallible provided that the co de is unambiguo us. Thus , we c an s afely restrict attention to a minimum-discrepancy dec oder , regardless of the maximum discrepa ncy t in the c hanne l. 8 C. Corr ection Cap ability Gi ven a fixed family o f channe ls—specifie d by X , Y and ∆( · , · ) , and pa rameterized by a maximum discrepancy t —we wish to identify the largest (worst) chann el parameter for which w e can guaran tee succe ssful decod ing. W e sa y that a code is t -disc r epancy -correcting if it is unambiguou s for a ch annel with maximum discre pancy t . The discrepancy-correction capability o f a cod e C is the lar gest t for which C is t -discrepancy-co rrecting. W e sta rt by gi ving a ge neral c haracterization o f the discrepancy-correc tion ca pability . Let the function τ : X × X → N be given by τ ( x, x ′ ) = m in y ∈Y max { ∆( x, y ) , ∆( x ′ , y ) } − 1 . (7) W e have the follo wing result. Pr opo sition 1: The discrep ancy-correction c apability of a code C is giv en exactly by τ ( C ) . In other words, C is t -discrepancy-correcting if and only if t ≤ τ ( C ) . Pr oof: Su ppose that the code is not t -discrepancy-correcting, i.e., that there exist some distinct x, x ′ ∈ C an d some y ∈ Y such that ∆( x, y ) ≤ t and ∆( x ′ , y ) ≤ t . Then τ ( C ) ≤ τ ( x, x ′ ) ≤ max { ∆( x, y ) , ∆( x ′ , y ) } − 1 ≤ t − 1 < t . In o ther words, τ ( C ) ≥ t implies tha t the code is t -discrepan cy- correcting. Con versely , su ppose that τ ( C ) < t , i.e., τ ( C ) ≤ t − 1 . The n there exist some dis tinct x, x ′ ∈ C suc h that τ ( x, x ′ ) ≤ t − 1 . This in turn implies that there exists some y ∈ Y suc h tha t max { ∆( x, y ) , ∆( x ′ , y ) } ≤ t . Since this implies that both ∆( x, y ) ≤ t and ∆ ( x ′ , y ) ≤ t , it follows tha t the code is not t -discrep ancy- correcting. At this p oint, it is temp ting to de fine a “distan ce-like” function g i ven by 2( τ ( x, x ′ ) + 1) , since this would enab le us to immediately obtain results analog ous to those of clas sical c oding theo ry (such as the error co rrection capab ility be ing half the minimum distance of the c ode). This ap proach has inde ed been taken in previous works, such as [12]. Note, however , that the terminology “dista nce” sugges ts a geometrical interpretation, which is not immediately c lear from (7). Moreover , the function (7) is n ot neces sarily mathematically tractable. It is the objective of this s ection to propos e a “dis tance” func tion δ : X × X → N that is motiv ated by geometrical cons iderations and is easier to h andle analytically , yet is useful to characte rize the correction ca pability of a code . In particular , we sha ll be able to ob tain the same results as [12] with much greater ma thematical simplicity—which will later turn out to be instrumental for code design. 9 For x, x ′ ∈ X , define the ∆ -distance between x and x ′ as δ ( x, x ′ ) , min y ∈Y  ∆( x, y ) + ∆( x ′ , y )  . (8) The following interpretation h olds. Conside r the co mplete bipartite graph with verte x s ets X and Y , and assume tha t e ach edg e ( x, y ) ∈ X × Y is labeled by a “length” ∆( x, y ) . T hen δ ( x, x ′ ) is the length o f the shortest path between vertices x, x ′ ∈ X . Rou ghly speaking, δ ( x, x ′ ) giv es the minimum total effort that an adversary would have to spend (in ind epende nt channel realizations) in order to make x a nd x ′ both plausible explanations for some received output. Example 2: Let us co mpute the ∆ -distance for the ch annel of Exa mple 1. W e h av e δ ( x, x ′ ) = min y { d H ( x, y ) + d H ( x ′ , y ) } ≥ d H ( x, x ′ ) , since the Hamming distan ce satisfies the triangle inequa lity . This bo und is achiev a ble by taking, for instance, y = x ′ . T hus, δ ( x, x ′ ) = d H ( x, x ′ ) , i.e. , the ∆ -distance for this channel is g i ven precise ly b y the Hamming distance. The follo wing res ult justifies our definition of the ∆ -distance. Pr opo sition 2: For any code C , τ ( C ) ≥ ⌊ ( δ ( C ) − 1) / 2 ⌋ . Pr oof: This follo ws from the fact that ⌊ ( a + b + 1) / 2 ⌋ ≤ max { a, b } for all a, b ∈ Z . Proposition 2 shows that δ ( C ) gives a lower bound on the correction capability of a c ode—therefore providing a c orrection guarantee. The con verse result, howev er , is not ne cessa rily true in general. Thus, up to this point, the propo sed function is on ly partially useful: it is co nceiv a ble tha t the ∆ -distance might be too conservati ve an d giv e a guarantee d co rrection capab ility that is lower than the actual one. Nev ertheless, it is easier to deal with addition, as in (8), rathe r than maximization, as in (7). A special case where the c on verse is true is for a family of chan nels whos e disc repancy function satisfies the follo wing c ondition: Definition 1: A discrepancy function ∆ : X × X → N is sa id to be nor mal if, for all x, x ′ ∈ X and all 0 ≤ i ≤ δ ( x, x ′ ) , there exists s ome y ∈ Y such that ∆( x, y ) = i and ∆( x ′ , y ) = δ ( x, x ′ ) − i . Theorem 3: Suppose that ∆( · , · ) is no rmal. For every code C ⊆ X , we hav e τ ( C ) = ⌊ ( δ ( C ) − 1) / 2 ⌋ . Pr oof: W e just n eed to show that ⌊ ( δ ( C ) − 1) / 2 ⌋ ≥ τ ( C ) . T ake any x, x ′ ∈ X . Since ∆( · , · ) is normal, there exists s ome y ∈ Y s uch tha t ∆ ( x, y ) + ∆( x ′ , y ) = δ ( x, x ′ ) and either ∆( x, y ) = ∆( x ′ , y ) or ∆( x, y ) = ∆( x ′ , y ) − 1 . Th us, δ ( x, x ′ ) ≥ 2 max { ∆( x, y ) , ∆( x ′ , y ) } − 1 a nd therefore ⌊ ( δ ( x, x ′ ) − 1) / 2 ⌋ ≥ τ ( x, x ′ ) . Theorem 3 s hows that, for certain families of ch annels, our propose d ∆ -distan ce ac hiev es the go al of this section: it is a (seemingly) tractable function tha t precisely d escribes the c orrection capability of a c ode. In pa rticular , the basic result of classica l coding theo ry—that the Ha mming distan ce precis ely 10 describes the error co rrection c apability of a cod e—follo ws from the fact tha t the Hamming distan ce (as a discrepa ncy function) is normal. As we s hall see, much of our e f fort in the next sections reduce s to showing that a spec ified discrepa ncy function is normal. Note that, for normal discrepa ncy fun ctions, we actually have τ ( x, x ′ ) = ⌊ ( δ ( x, x ′ ) − 1) / 2 ⌋ , so Theorem 3 may also be regarded a s providing an alternati ve (and more tractable) expression for τ ( x, x ′ ) . Example 3: T o giv e a non tri vial example, let us con sider a binary vector cha nnel that introduces a t most ρ erasures (arbitrarily chosen by an a dversary). The input alphabet is gi ven by X = { 0 , 1 } n , while the output alphabet is gi ven by X = { 0 , 1 , ǫ } n , wh ere ǫ deno tes an erasure. W e may define ∆( x, y ) = P n i =1 ∆( x i , y i ) , where ∆( x i , y i ) =              0 if y i = x i 1 if y i = ǫ ∞ otherwise . The fan-out sets are the n given by Y x = { y ∈ Y : ∆( x, y ) ≤ ρ } . In order to c ompute δ ( x, x ′ ) , ob serve the minimization in (8). It is eas y to see that we should cho ose y i = x i when x i = x ′ i , a nd y i = ǫ when x i 6 = x ′ i . It follows that δ ( x, x ′ ) = 2 d H ( x, x ′ ) . No te tha t ∆( x, y ) is normal. It follows from Theo rem 3 that a code C can correct all the ρ erasure s introduce d by the channel if and only if 2 d H ( C ) > 2 ρ . This result precisely matches the we ll-known result of classical coding theory . It is worth clarifying that, while we ca ll δ ( · , · ) a “d istance, ” this fun ction may no t n ecess arily be a metric. While symmetry and non-negati v ity follow from the d efinition, a ∆ -distance may not always satisfy “ δ ( x, y ) = 0 ⇐ ⇒ x = y ” or the triangle inequ ality . Ne vertheless, we keep the terminology for con venien ce. Although this is not ou r main interest in this paper , it is worth pointing out that the framework of this s ection is also useful for obtaining results on error de tection . Namely , the ∆ -distance gi ves, in general, a lower bou nd o n the discrepa ncy detection capab ility of a code unde r a bounded discrepancy- correcting decoder; whe n the disc repancy func tion is normal, then the ∆ -distance p recisely ch aracterizes this detection capability (similarl y as in classica l coding theory). For more de tails on this top ic, see Appendix A. 11 I V . C O H E R E N T N E T W O R K C O D I N G A. A W orst-Case Mod el a nd the Rank Metric The basic channe l mode l for co herent ne twork co ding with ad versarial errors is a matrix chan nel with input X ∈ F n × m q , output Y ∈ F N × m q , and chann el law giv en by (1), wh ere A ∈ F N × n q is fixed an d known to the receiv er , and Z ∈ F t × m q is arbitrarily chose n b y a n adversary . Here, we make the follo wing additional assumptions: • T he adversary has unlimited c omputational power and is omniscient; in pa rticular , the a dversary knows both A and X ; • T he matrix D ∈ F N × t q is arbitrarily chosen b y the adversary . W e a lso a ssume tha t t < n (more prec isely , we sh ould a ssume t < rank A ); otherwise, the adversa ry may alw ays choose D Z = − AX , lea ding to a tri vial commun ications s cenario. The first as sumption above allows us to use the a pproach of Se ction III. The second a ssumption may seem so mewhat “pe ssimistic, ” but it has the analytical advantage of eliminating from the problem any further depe ndence on the network cod e. (Recall that, in principle, D would be de termined by the network code and the choice o f links in error .) The p ower of the approach of Se ction III lies in the fact tha t the c hanne l mode l defined above c an be completely described by the following discrepancy function ∆ A ( X, Y ) , min r ∈ N , D ∈ F N × r q , Z ∈ F r × m q : Y = AX + D Z r . (9) The discrepan cy ∆ A ( X, Y ) re presents the minimum numb er of error packets that the adversary nee ds to inject in order to transform an input X into an output Y , given that the transfer ma trix is A . The subscript in ∆ A ( X, Y ) is to emph asize the dependen ce on A . For this discrepancy f unction, the minimum- discrepancy de coder becomes ˆ X = argmin X ∈C ∆ A ( X, Y ) . (10) Similarly , the ∆ -distance induced by ∆ A ( X, Y ) is gi ven b y δ A ( X, X ′ ) , min Y ∈ F N × m q  ∆ A ( X, Y ) + ∆ A ( X ′ , Y )  (11) for X, X ′ ∈ F n × m q . W e now wish to find a s impler expression for ∆ A ( X, Y ) a nd δ A ( X, X ′ ) , a nd show that ∆ A ( X, Y ) is normal. 12 Lemma 4: ∆ A ( X, Y ) = rank ( Y − AX ) . (12) Pr oof: Conside r ∆ A ( X, Y ) as giv en by (9). For any fea sible triple ( r , D , Z ) , we have r ≥ rank Z ≥ rank D Z = rank ( Y − AX ) . This bound is a chiev able by setting r = rank ( Y − AX ) and letting D Z be a full-rank decompos ition of Y − AX . Lemma 5: δ A ( X, X ′ ) = d R ( AX, AX ′ ) = rank A ( X ′ − X ) . Pr oof: From (11) an d Lemma 4, we have δ A ( X, X ′ ) = min Y { d R ( Y , AX ) + d R ( Y , AX ′ ) } . Sinc e the rank metric satisfie s the triangle inequ ality , we have d R ( AX, Y ) + d R ( AX ′ , Y ) ≥ d R ( AX, AX ′ ) . This lo we r bound can be a chieved by cho osing, e.g., Y = AX . Note that δ A ( · , · ) is a metric if and only if A has full column rank—in which c ase it is precisely the rank metric. (If rank A < n , then there exist X 6 = X ′ such that δ A ( X, X ′ ) = 0 .) Theorem 6: The d iscrepancy func tion ∆ A ( · , · ) is normal. Pr oof: Let X , X ′ ∈ F n × m q and let 0 ≤ i ≤ d = δ A ( X, X ′ ) . Then ran k A ( X ′ − X ) = d . By performing a full-rank decompos ition o f A ( X ′ − X ) , we ca n always fin d two ma trices W and W ′ such that W + W ′ = A ( X ′ − X ) , rank W = i an d ra nk W ′ = d − i . T aking Y = AX + W = AX ′ − W ′ , we have tha t ∆ A ( X, Y ) = i and ∆ A ( X ′ , Y ) = d − i . Note tha t, und er the discrep ancy ∆ A ( X, Y ) , a t -discrepa ncy-correcting cod e is a c ode that can c orrect any t pac ket errors injected by the a dversary . Using Th eorem 6 and Theo rem 3, we have the follo wing result. Theorem 7: A c ode C is guarante ed to correct any t pa cket errors if and only if δ A ( C ) > 2 t . Theorem 7 shows that δ A ( C ) is indee d a fundame ntal pa rameter ch aracterizing the error correction capability of a c ode in o ur mod el. Note that, if the condition of Theorem 7 is violate d, then the re exists at least one codewor d for which the adversary can ce rtainly induce a decoding fail ure. Note that the error c orrection c apability of a code C is d epend ent on the n etwork code through the matrix A . Let ρ = n − rank A b e the column-rank d eficiency of A . Since δ A ( X, X ′ ) = rank A ( X ′ − X ) , it follo w s from (3) tha t d R ( X, X ′ ) − ρ ≤ δ A ( X, X ′ ) ≤ d R ( X, X ′ ) and d R ( C ) − ρ ≤ δ A ( C ) ≤ d R ( C ) . (13) 13 Thus, the e rror c orrection cap ability of a code is strongly tied to its minimum rank distance; in particular , δ A ( C ) = d R ( C ) if ρ = 0 . While the lo wer bound δ A ( C ) ≥ d R ( C ) − ρ may not be tight in gen eral, we sho uld expect it to be tight when C is sufficiently large. This is indeed the case for MRD c odes, as dis cusse d in Section IV -C. Thu s, a rank deficiency of A will typically reduce the error c orrection ca pability o f a code. T a king into ac count the worst case, we can use Theorem 7 to giv e a correction guarantee in terms of the minimum rank distance o f the code. Pr opo sition 8: A c ode C is guaranteed to co rrect t packet errors, under rank deficien cy ρ , if d R ( C ) > 2 t + ρ . Note that the g uarantee o f Proposition 8 d epend s only o n ρ an d t ; in pa rticular , it is independe nt of the network c ode or the specific transfer matrix A . B. Reinterpreting the Mod el of Y eung et al. In this subse ction, we in vestigate the mod el for coh erent network coding stud ied by Y eun g et al. in [1]–[4], which is similar to the one con sidered in the pre v ious subs ection. Th e mode l is that of a matrix channe l with input X ∈ F n × m q , output Y ∈ F N × m q , and c hanne l law gi ven by Y = AX + F E (14) where A ∈ F N × n q and F ∈ F N ×|E | q are fixed and kn own to the receiver , and E ∈ F |E |× m q is arbitrarily chosen by a n adversa ry provided wt ( E ) ≤ t . (Recall that |E | is the n umber of edges in the network.) In addition, the adversary ha s un limited co mputational power and is o mniscient, knowing, in particular , A , F and X . W e now s how that some of the con cepts defined in [4], su ch as “network Hamming distance, ” can be reinterpreted in the frame work of Se ction III. As a conseq uence , we can easily recover the results of [4] on error correction and detection gu arantees. First, note that the current model c an be completely described by the following dis crepancy function ∆ A,F ( X, Y ) , min E ∈ F |E |× m q : Y = AX + F E wt ( E ) . (15) 14 The ∆ -distance induced b y this discrepan cy function is given by δ A,F ( X 1 , X 2 ) , min Y ∆ A,F ( X 1 , Y ) + ∆ A,F ( X 2 , Y ) = min Y , E 1 ,E 2 : Y = AX 1 + F E 1 Y = AX 2 + F E 2 { wt ( E 1 ) + wt ( E 2 ) } = min E 1 ,E 2 : A ( X 2 − X 1 )= F ( E 1 − E 2 ) { wt ( E 1 ) + wt ( E 2 ) } = min E : A ( X 2 − X 1 )= F E wt ( E ) where the last equality follows from the fact that wt ( E 1 − E 2 ) ≤ wt ( E 1 ) + wt ( E 2 ) , a chiev able if E 1 = 0 . Let us now examine so me of the con cepts define d in [4]. For a spec ific sink no de, the dec oder proposed in [4, Eq. (2)] has the form ˆ X = argmin X ∈C Ψ A,F ( X, Y ) . The d efinition o f the objectiv e func tion Ψ A,F ( X, Y ) requires several other de finitions p resented in [4]. Specifically , Ψ A,F ( X, Y ) , D r ec ( AX, Y ) , where D r ec ( Y 1 , Y 2 ) , W r ec ( Y 2 − Y 1 ) , W r ec ( Y ) , min E ∈ Υ( Y ) wt ( E ) , an d Υ( Y ) , { E : Y = F E } . Su bstituting all these values into Ψ A,F ( X, Y ) , we obtain Ψ A,F ( X, Y ) = D r ec ( AX, Y ) = W r ec ( Y − AX ) = min E ∈ Υ( Y − AX ) wt ( E ) = min E : Y − AX = F E wt ( E ) = ∆ A,F ( X, Y ) . Thus, the decoder in [4] is prec isely a minimum-discrep ancy de coder . In [ 4], the “network Ha mming distance” between tw o messag es X 1 and X 2 is defined as D msg ( X 1 , X 2 ) , W msg ( X 2 − X 1 ) , where W msg ( X ) , W r ec ( AX ) . Again, simply substituting the correspon ding defini- tions yields D msg ( X 1 , X 2 ) = W msg ( X 2 − X 1 ) = W r ec ( A ( X 2 − X 1 )) = min E ∈ Υ( A ( X 2 − X 1 )) wt ( E ) 15 = min E : A ( X 2 − X 1 )= F E wt ( E ) = δ A,F ( X 1 , X 2 ) . Thus, the “network Hamming distance” is precisely the ∆ -distance ind uced by the discre pancy function ∆ A,F ( X, Y ) . Finally , the “u nicast minimum distance ” of a network code with message set C [4] is precisely δ A,F ( C ) . Let us return to the problem o f characterizing the correction c apability of a code. Pr opo sition 9: The discrepa ncy function ∆ A,F ( · , · ) is normal. Pr oof: Let X 1 , X 2 ∈ F n × m q and let 0 ≤ i ≤ d = δ A,F ( X 1 , X 2 ) . L et E ∈ F |E |× m q be a solution to the minimization in (15). Then A ( X 2 − X 1 ) = F E an d wt ( E ) = d . By pa rtitioning E , we can always find two ma trices E 1 and E ′ 2 such that E 1 + E ′ 2 = E , ra nk E 1 = i and rank E ′ 2 = d − i . T aking Y = AX 1 + F E 1 = AX 2 − F E 2 , we have that ∆ A,F ( X 1 , Y ) ≤ i an d ∆ A,F ( X 2 , Y ) ≤ d − i . Sinc e d ≤ ∆ A,F ( X 1 , Y ) + ∆ A,F ( X 2 , Y ) , it follows that ∆ A,F ( X 1 , Y ) = i and ∆ A,F ( X 2 , Y ) = d − i . It follo w s that a c ode C is gua ranteed to co rrect any t p acket e rrors if and only if δ A,F ( C ) > 2 t . T hus, we recover theorems 2 and 3 in [4] (for error detection, see App endix A). The analog ous results for the multicast case can be obtained in a s traightforward man ner . W e now wish to c ompare the parameters devised in this su bsection with those of Section IV -A. From the des criptions of (1) a nd (14), it is intuitiv e that the model of this sub section should be equiv alent to that of the previous s ubsec tion if the matrix F , rather than fixed and known to the receiver , is a rbitrarily and secretly chosen b y the adversary . A formal proof of this fact is given in the following proposition. Pr opo sition 10: ∆ A ( X, Y ) = min F ∈ F N ×|E | q ∆ A,F ( X, Y ) δ A ( X, X ′ ) = min F ∈ F N ×|E | q δ A,F ( X, X ′ ) δ A ( C ) = min F ∈ F N ×|E | q δ A,F ( C ) . Pr oof: Consider the minimization min F ∈ F N ×|E | q ∆ A,F ( X, Y ) = min F ∈ F N ×|E | q , E ∈ F |E |× m q : Y = AX + F E wt ( E ) . For any feasible ( F , E ) , we have wt ( E ) ≥ ra nk E ≥ rank F E = rank ( Y − AX ) . T his lower bound can be achieved by taking F = h F ′ 0 i and E =   E ′ 0   16 where F ′ E ′ is a full-rank de compos ition of Y − AX . This proves t he first statemen t. The seco nd statement follo ws from the first b y noticing that δ A ( X, X ′ ) = ∆ A ( X, AX ′ ) and δ A,F ( X, X ′ ) = ∆ A,F ( X, AX ′ ) . The third statement is immed iate. Proposition 10 shows that the model of S ection IV -A is indee d more pessimistic, as the a dversary has additional power to ch oose the worst pos sible F . It follows that a ny c ode that is t -error-correcting for that model must also be t -e rror -correc ting for the model of Y eung et al. C. Optimality of MRD Codes Let us no w ev aluate t he performance of a n MRD co de under the models of the two previous subse ctions. The Singleton bound of [2] (see a lso [9]) states that |C | ≤ Q n − ρ − δ A,F ( C )+1 (16) where Q is the size of the alphabet 2 from which packets are drawn. Note that Q = q m in our se tting, since each packet con sists of m s ymbols from F q . Using Proposition 10, we ca n also o btain |C | ≤ Q n − ρ − δ A ( C )+1 = q m ( n − ρ − δ A ( C )+1) . (17) On the other hand, the size of a n MRD code, for m ≥ n , is giv en by |C | = q m ( n − d R ( C )+1) (18) ≥ q m ( n − ρ − δ A ( C )+1) (19) ≥ q m ( n − ρ − δ A,F ( C )+1) where (19 ) follows from (13). S ince Q = q m , both (16 ) a nd (17) are achieved in this cas e. Thu s, we have the follo wing res ult. Theorem 11: When m ≥ n , an MRD c ode C ⊆ F n × m q achieves maximum cardinality with respect to both δ A and δ A,F . Theorem 11 shows that, if an alphabet of s ize Q = q m ≥ q n is a llo wed (i.e., a p acket size of a t least n log 2 q bits), then MRD codes turn out to be op timal under both mod els of sections IV -A an d IV -B. Remark: It is straightforward to extend the resu lts of Section IV -A for the case of multiple he teroge- neous receiv ers, where each receiver u expe riences a rank deficien cy ρ ( u ) . In this case, it ca n be shown that an MRD code with m ≥ n achieves the refined Sing leton bound of [9]. 2 This alphabet is usually assumed a finite field, but, for the Singleton bound of [2], it is sufficient t o assume an abelian group, e.g., a vector space over F q . 17 Note tha t, due to (17), (18) and (19 ), it follows that δ A ( C ) = d R ( C ) − ρ for a n MRD code with m ≥ n . Thus, in this c ase, we can restate Theorem 7 in terms of the minimum rank distance of the code. Theorem 12: An MRD code C ⊆ F n × m q with m ≥ n is guaran teed to c orrect t pac ket errors, u nder rank deficiency ρ , if and only if d R ( C ) > 2 t + ρ . Observe that Theo rem 12 holds regardless of the s pecific trans fer ma trix A , de pending o nly on its column-rank deficiency ρ . The results of this s ection imply that, wh en des igning a linear ne twork code, we may focus s olely on the objec ti ve o f ma king the network c ode feas ible, i.e., maximizing ra nk A . If a n error correction guarantee is d esired, then an outer code can b e applied end-to-end without req uiring any modifica tions o n (or e ven knowledge of) the underlying n etwork cod e. Th e de sign of the outer code is essentially trivial, as any MRD code can b e used, with the on ly requirement tha t the numb er of F q -symbols per pa cket, m , is at least n . Remark: Consider the d ecoding rule (10). The fact tha t (10) together with (12) is equ i valent to [8, Eq. (20)] implies that the dec oding problem can be solved by exactly the sa me rank-metric technique s proposed in [8]. In particular , for certain MRD code s with m ≥ n a nd minimum rank dista nce d , there exist e f ficient enc oding and decoding algorithms both requ iring O ( dn 2 m ) ope rations in F q per codeword. For more details, see [15]. V . N O N C O H E R E N T N E T W O R K C O D I N G A. A W orst-Case Mod el a nd the Injection Metric Our mod el for noncoh erent network c oding with ad versarial errors differs from its c oherent cou nterpart of Sec tion IV -A only with resp ect to the transfer matrix A . Namely , the matrix A is unknown to the receiv er and is free ly chosen b y the adversary wh ile respecting the constraint rank A ≥ n − ρ . The parameter ρ , the maximum column rank defic iency of A , is a p arameter of the system that is known to all. Note that, as d iscuss ed a bove for the matrix D , the a ssumption that A is chose n by the adversa ry is what provides the cons ervati ve (worst-case) n ature of the model. The constraint on the ran k of A is required for a mea ningful coding problem; o therwise, the adversa ry cou ld prevent co mmunication by simply choos ing A = 0 . As before, we assume a minimum-discrepancy deco der ˆ X = argmin X ∈C ∆ ρ ( X, Y ) (20) 18 with discrepancy function giv en by ∆ ρ ( X, Y ) , min A ∈ F N × n q ,r ∈ N ,D ∈ F N × r q ,Z ∈ F r × m q : Y = AX + D Z rank A ≥ n − ρ r (21) = min A ∈ F N × n q : rank A ≥ n − ρ ∆ A ( X, Y ) . Again, ∆ ρ ( X, Y ) repres ents the minimum numbe r o f error packets neede d to produce an output Y given an inpu t X under the current a dversarial model. The subscript is to e mphasize that ∆ ρ ( X, Y ) is still a function of ρ . The ∆ -distance induced by ∆ ρ ( X, Y ) is defined below . For X , X ′ ∈ F n × m q , let δ ρ ( X, X ′ ) , min Y ∈ F N × m q  ∆ ρ ( X, Y ) + ∆ ρ ( X ′ , Y )  . (22) W e now prove that ∆ ρ ( X, Y ) is normal and the refore δ ρ ( C ) c haracterizes the correction capability of a code. First, obse rve tha t, using Lemma 4, we may rewrite ∆ ρ ( X, Y ) as ∆ ρ ( X, Y ) = min A ∈ F N × n q : rank A ≥ n − ρ rank ( Y − AX ) . (23) Also, note that δ ρ ( X, X ′ ) = m in Y  ∆ ρ ( X, Y ) + ∆ ρ ( X ′ , Y )  = min A,A ′ ∈ F N × n q : rank A ≥ n − ρ rank A ′ ≥ n − ρ { min Y rank ( Y − AX ) + rank ( Y − A ′ X ′ ) } = min A,A ′ ∈ F N × n q : rank A ≥ n − ρ rank A ′ ≥ n − ρ rank ( A ′ X ′ − AX ) (24) where the last equality follo ws from the fact that d R ( AX, Y ) + d R ( A ′ X ′ , Y ) ≥ d R ( AX, A ′ X ′ ) , achiev able by choosing, e.g., Y = AX . Theorem 13: The discrepancy function ∆ ρ ( · , · ) is normal. Pr oof: Let X , X ′ ∈ F n × m q and let 0 ≤ i ≤ d = δ ρ ( X, X ′ ) . Let A, A ′ ∈ F N × n q be a solution to the minimization in (24). Then rank ( A ′ X ′ − AX ) = d . B y performing a full-rank decomp osition of A ′ X ′ − AX , we can al ways find two matrices W and W ′ such that W + W ′ = A ′ X ′ − AX , rank W = i and rank W ′ = d − i . T aking Y = AX + W = AX ′ − W ′ , we have that ∆ ρ ( X, Y ) ≤ i an d ∆ ρ ( X ′ , Y ) ≤ d − i . Since d ≤ ∆ ρ ( X, Y ) + ∆ ρ ( X ′ , Y ) , it follo ws that ∆ ρ ( X, Y ) = i and ∆ ρ ( X ′ , Y ) = d − i . 19 As a consequ ence of Theorem 13, we hav e the following result. Theorem 14: A code C is guaranteed to correct any t pac ket errors if and only if δ ρ ( C ) > 2 t . Similarly as in Sec tion IV -A, The orem 14 shows that δ ρ ( C ) is a fundamental parameter cha racterizing the error correction capability o f a code in the current mod el. In contrast to Section IV -A, howev er , the expression for ∆ ρ ( X, Y ) (and, co nseque ntly , δ ρ ( X, X ′ ) ) doe s not seem mathematica lly appealing since it in volves a minimization. W e now proc eed to finding simpler expres sions for ∆ ρ ( X, Y ) and δ ρ ( X, X ′ ) . The minimization in (23) is a spe cial case of a more gene ral expres sion, w hich we give as follows. For X ∈ F n × m q , Y ∈ F N × m q and L ≥ max { n − ρ, N − σ } , let ∆ ρ,σ,L ( X, Y ) , min A ∈ F L × n q ,B ∈ F L × N q : rank A ≥ n − ρ rank B ≥ N − σ rank ( B Y − AX ) . The quantity define d above is compu ted in the follo wing lemma. Lemma 15: ∆ ρ,σ,L ( X, Y ) = h max { rank X − ρ, rank Y − σ } − dim ( h X i ∩ h Y i ) i + . Pr oof: See Appendix B. Note that ∆ ρ,σ,L ( X, Y ) is indep endent of L , for all valid L . Thus, we may d rop the s ubscript and write simply ∆ ρ,σ ( X, Y ) , ∆ ρ,σ,L ( X, Y ) . W e ca n now provide a simpler expression for ∆ ρ ( X, Y ) . Theorem 16: ∆ ρ ( X, Y ) = max { rank X − ρ, rank Y } − dim ( h X i ∩ h Y i ) . Pr oof: This follo ws immedia tely from Lemma 1 5 by noticing that ∆ ρ ( X, Y ) = ∆ ρ, 0 ( X, Y ) . From The orem 16, we ob serve that ∆ ρ ( X, Y ) depen ds on the matrices X and Y only through the ir row space s, i.e., only the trans mitted and receiv e d row spaces have a role in the decoding. Put another way , we may s ay that the cha nnel really acce pts a n inpu t subs pace h X i a nd delivers an output subspa ce h Y i . Thus, a ll the c ommunication is ma de via s ubspa ce selection. Th is ob servation provides a fund amental justification for the approach of [7]. At this point, it is useful to introdu ce the following definition. Definition 2: The injection distance b etween subspac es U and V in P q ( m ) is defined as d I ( U , V ) , max { dim U , dim V } − dim ( U ∩ V ) (25) = dim ( U + V ) − min { dim U , dim V } . 20 The injection distanc e can be interpreted as meas uring the number of error packets that an adversa ry needs to inject in order to transform an input s ubspac e h X i into an output sub space h Y i . This c an be clearly se en from the fact that d I ( h X i , h Y i ) = ∆ 0 ( X, Y ) . T hus, the injection d istance is esse ntially equal to the d iscrepancy ∆ ρ ( X, Y ) when the channe l is influenc ed only by the a dversary , i.e ., whe n the non-adversarial aspe ct of the channe l (the column-rank d eficiency of A ) is removed from the prob lem. Note that, in this c ase, the decode r (20) become s precisely a minimum-injection-distance decode r . Pr opo sition 17: The injection distance is a metric. W e de lay the p roof of Proposition 1 7 until Section V -B. W e can now use the de finition of the injection distan ce to simplify the expression for the ∆ -distanc e. Pr opo sition 18: δ ρ ( X, X ′ ) = [ d I ( h X i ,  X ′  ) − ρ ] + . Pr oof: This follo ws immedia tely after realizing that δ ρ ( X, X ′ ) = ∆ ρ,ρ ( X, X ′ ) . From Proposition 1 8, it is clear tha t δ ρ ( · , · ) is a metric if a nd o nly if ρ = 0 (in wh ich cas e it is precisely the injection metric). If ρ > 0 , then δ ρ ( · , · ) d oes not satisfy the triangle ine quality . It is worth noticing that δ ρ ( X, X ′ ) = 0 for any two matrices X and X ′ that sh are the same row space . Thus, any reas onable co de C should a void this situation. For C ⊆ F n × m q , let hC i = {h X i : X ∈ C } be the subsp ace code (i.e., a collection o f subs paces ) co nsisting of the row spaces of all matrices in C . The follo wing c orollary o f Proposition 18 is immediate. Cor ollary 19: Suppos e C is such tha t |C | = | hC i | , i.e ., no two codewords of C have the same row space . Then δ ρ ( C ) = [ d I ( hC i ) − ρ ] + . Using Corollary 19, we can restate T heorem 14 more simply in terms of the injection distance . Theorem 20: A co de C is g uaranteed to correct t pa cket errors, und er rank deficiency ρ , if and only if d I ( hC i ) > 2 t + ρ . Note that, due to e quality in Corollary 19, a con verse is indeed poss ible in Theorem 20 (contrast with Proposition 8 for the coherent c ase). Theorem 20 shows that d I ( hC i ) is a fundamental pa rameter characterizing the comp lete correction capability (i.e., error correction capa bility and “rank-deficien cy correction” capab ility) of a co de in our 21 noncoh erent model. Pu t a nother way , we may s ay that a code C is go od for the model of this sub section if and only if its s ubspa ce version hC i is a good cod e in the injection metric. B. Comparison with the Metric o f K ¨ otter and Kschischang Let Ω ⊆ P q ( m ) be a subspac e cod e wh ose elements have max imum dimension n . In [7], the network is mo deled as an op erator cha nnel that takes in a subs pace V ∈ P q ( m ) and puts o ut a pos sibly diff erent subspa ce U ∈ P q ( m ) . The kind of disturban ce tha t the c hannel applies to V is cap tured b y the no tions of “insertions” an d “d eletions” of d imensions (repres ented mathema tically using op erators), an d the degree of such a dissimilarity is ca ptured by the subspac e distance d S ( V , U ) , di m ( V + U ) − di m ( V ∩ U ) = dim V + dim U − 2 dim ( V ∩ U ) (26) = dim ( V + U ) − di m V − dim U . The trans mitter selec ts some V ∈ Ω and trans mits V over the channel. The recei ver receives so me subspa ce U and, us ing a minimum sub space dis tance de coder , de cides that the subspa ce ˆ V ⊆ Ω was sen t, where ˆ V = argmin V ∈ Ω d S ( V , U ) . (27) This deco der is g uaranteed to co rrect a ll disturbanc es app lied by the ch annel if d S ( V , U ) < d S (Ω) / 2 , where d S (Ω) is the minimum subspac e d istance between all pairs of distinct codewords of Ω . First, let us po int out that this setup is indee d the sa me as that of Section V -A if we s et V = h X i , U = h Y i and Ω = hC i , whe re C is suc h that |C | = | h C i | . Also, any d isturbance a pplied by an operator channe l can be realized by a matrix mode l, a nd vice-versa. T hus, the difference b etween the approac h of this section and tha t of [7] lies in the ch oice of the d ecode r . Indeed, b y us ing Theo rem 16 a nd the defi nition of subsp ace distanc e, we ge t the following relationship: Pr opo sition 21: ∆ ρ ( X, Y ) = 1 2 d S ( h X i , h Y i ) − 1 2 ρ + 1 2 | ra nk X − ra nk Y − ρ | . Thus, we can se e tha t whe n the matrices in C do no t all have the same rank (i.e., Ω is a non-con stant- dimension code ), then the decoding rules (20) an d (27) ma y produce d if fere nt decis ions. Using ρ = 0 in the ab ove proposition (or simply using (25) a nd (26)) g i ves us another formula for the injection distance: d I ( V , U ) = 1 2 d S ( V , U ) + 1 2 | d im V − dim U | . (28) 22 V 2 V 1 V 1 ∩ V 2 V 1 + V 2 U ǫ γ ǫ increasing dimension Fig. 1. Lattice of subspaces in E xample 4. T wo spaces are joined with a dashed line if one is a subspace of the other . W e ca n now prov e a resu lt that was pos tponed in the previous se ction. Theorem 22: The injection distance is a metric. Pr oof: Since d S ( · , · ) is a metric on P q ( m ) an d | · | is a norm on R , it follo ws from (28) tha t d I ( · , · ) is also a me tric on P q ( m ) . W e now examine in more de tail an exa mple situation where the minimum-subspace -distance decode r and the minimum-discrepancy dec oder p roduce dif ferent de cisions. Example 4: For simplicity , assu me ρ = 0 . Consider a subspac e code that con tains two co dewords V 1 = h X 1 i and V 2 = h X 2 i such that γ , dim V 2 − dim V 1 satisfies d/ 3 < γ < d/ 2 , where d , d S ( V 1 , V 2 ) . Suppose the rece iv e d subspac e U = h Y i is such that V 1 ⊆ U ⊆ V 1 + V 2 and dim U = di m V 1 + γ = dim V 2 , as illustrated in Fig. 1. Then d S ( V 1 , U ) = γ an d d S ( V 2 , U ) = d − γ , while P roposition (21) gives ∆ ρ ( X 1 , Y ) = γ a nd ∆ ρ ( X 2 , Y ) = ( d − γ ) / 2 , ǫ . Sinc e, by assu mption, d − γ > γ an d ǫ < γ , it follo ws that d S ( V 1 , U ) < d S ( V 2 , U ) but ∆ ρ ( X 1 , Y ) > ∆ ρ ( X 2 , Y ) , i.e., the decod ers (27) and (20) w ill prod uce dif fe rent decisions. This situation can be intuitively explained as follows. The decod er (27) fa vors the s ubspa ce V 1 , which is c loser in subspace distance to U than V 2 . However , since V 1 is low-dimensional, U can only be produced from V 1 by the inser tion of γ dimens ions. The decode r (20), on the other ha nd, fav ors V 2 , which, a lthough farther in subsp ace dis tance, ca n produce U after the replacement o f ǫ < γ dimens ions. Since on e p acket error mu st occur for each inse rted or rep laced dimension , we conclud e tha t the d ecode r (20) finds the solution that minimizes the n umber of packet errors obse rved. Remark: Th e subs pace metric of [7] treats inse rtions and de letions of dimens ions (called in [7] “errors” and “e rasures”, res pectively) symmetrically . However , depe nding u pon the p osition of the adversary in 23 the network (namely , if there is a source-destination min-cut between the adversa ry and the des tination) then a single error packet may c ause the replaceme nt o f a dimension (i.e., a simultaneous “e rror” and “erasure” in the terminology of [7]). The injection distanc e, which is des igned to “explain” a received subspa ce with a s few error -packet injections as poss ible, properly ac counts for this phe nomenon , and hence the corresponding d ecode r produ ces a diff erent result tha n a minimum subspac e distanc e d ecode r . If it we re possible to restrict the adversary so that e ach error- packet injec tion would only cau se either an inse rtion or a deletion of a dimension (but not both), then the subsp ace distance o f [7] would indeed be appropriate. Howe ver , this is n ot the model c onsidered here. Let us n ow discuss an important fact abo ut the subsp ace distance for g eneral subsp ace codes (ass uming for simplicity that ρ = 0 ). The packet error correction capab ility of a minimum-subspa ce-distance de coder , t S , is not neces sarily equ al to ⌊ ( d S ( C ) − 1) / 2 ⌋ o r ⌊ ( d S ( C ) − 2) / 4 ⌋ , but lies somewhere in be tween. For instance, in the case of a co nstant-dimension code Ω , we h ave d I ( V , V ′ ) = 1 2 d S ( V , V ′ ) , ∀V , V ′ ∈ Ω , d I (Ω) = 1 2 d S (Ω) . Thus, Theorem 20 implies that t S = ⌊ ( d S ( C ) − 2) / 4 ⌋ exa ctly . In other words, in this s pecial case , the approach in [7] coincides with that of this p aper , and Theorem 20 provides a conv e rse that was missing in [7]. On the o ther hand , s uppose Ω is a su bspac e code cons isting of just two codewords, one o f which is a subsp ace o f the other . The n we h ave precise ly t S = ⌊ ( d S ( C ) − 1) / 2 ⌋ , since t S + 1 packet-injections are needed to get past halfway betwee n the co dewords. Since no s ingle quantity is known that perfectly d escribes the pac ket error co rrection capability of the minimum-subspac e-distance de coder (27) for gen eral subsp ace code s, we cannot provide a definitiv e comparison betwee n deco ders (27) and (20). Howev er , we ca n s till compu te bou nds for codes that fit into Example 4. Example 5: Let us c ontinue with Ex ample 4 . No w , we a djoin another codew ord V 3 = h X 3 i such that d S ( V 1 , V 3 ) = d and wh ere γ ′ , dim V 3 − dim V 1 satisfies d/ 3 < γ ′ < d/ 2 . Also we assu me that d S ( V 2 , V 3 ) is suf ficiently lar ge so a s not to interfere w ith the prob lem (e.g., d S ( V 2 , V 3 ) > 3 d/ 2 ). Let t S and t M denote the packet e rror c orrection cap abilities of the de coders (27) and (20), res pectiv ely . From the ar gument of Ex ample 4, we g et t M ≥ max { ǫ, ǫ ′ } , while t S < min { ǫ, ǫ ′ } , where ǫ ′ = ( d − γ ′ ) / 2 . By cho osing γ ≈ d/ 3 and γ ′ ≈ d/ 2 , we get ǫ ≈ d/ 3 and ǫ ′ ≈ d/ 4 . Th us, t M ≥ (4 / 3) t S , i.e., we obtain a 1/3 increase in error c orrection capability by using the d ecoder (20). 24 V I . C O N C L U S I O N W e have address ed the problem of error correction in network coding un der a worst-case adversarial model. W e show tha t certain metrics n aturally arise as the fundame ntal p arameter d escribing the error correction ca pability of a code; name ly , the rank metric for cohe rent network c oding, and the injection metric for noncohe rent network c oding. For coh erent ne twork c oding, the frame work bas ed o n the rank metric e ssen tially s ubsumes p revious a nalyses an d c onstructions, with the advantage of providing a clear separation between the problems of des igning a feasible network code a nd an error-correcting outer code. For nonc oherent ne twork coding, the injection metric provides a measure of code performanc e tha t is more p recise, when a non-cons tant-dimension co de is use d, tha n the so-called subs pace metric. The design of general subspace codes for the injec tion metric, as well as the deri vation of bounds, is left as an open problem for future research. A P P E N D I X A D E T E C T I O N C A P A B I L I T Y When dealing with communication over an adversarial cha nnel, the re is little justification to consider the pos sibility of error detection. In p rinciple, a co de shou ld be designe d to be u nambiguou s (in which case e rror de tection is not need ed); otherwise, if the re is any p ossibility for a mbiguity at the receiver , then the adversary will c ertainly exploit this possibility , leading to a high proba bility of decoding failure (detected error). Still, if a system is su ch that (a) s equen tial tr ansmission s are made over the same channel, (b) there exists a fee dback link from the rec eiv er to the transmitter , and (c) the a dversary is not able to fully exploit the chann el at all times, then it might be worth using a code with a lower correction capability (b ut higher rate) that has some ab ility to detect errors. Follo wing c lassical co ding the ory , we cons ider error detection in the presen ce of a bo unded error- correcting d ecode r . More prec isely , define a bou nded-discrepancy decoder with co rrection radius t , or simply a t -discr e pancy -correcting dec oder , by ˆ x ( y ) =      x if ∆( x, y ) ≤ t a nd ∆( x ′ , y ) > t for all x ′ 6 = x , x ′ ∈ C f otherwise . Of course, whe n using a t -discrepancy-correcting decoder , we implicitly assume that the code is t - discrepancy-co rrecting. Th e d iscrepancy detection capability of a code (under a t -disc repancy-correcting decode r) is the maximum value of discrepancy for which the decoder is gua ranteed not to make an undetected error , i.e., it must return either the c orrect codeword or the failure symb ol f . 25 For t ∈ N , let the function σ t : X × X → N be given by σ t ( x, x ′ ) , min y ∈Y : ∆( x ′ ,y ) ∆( x, y ) − 1 . (29) Pr opo sition 23: The discrepa ncy-detection c apability of a c ode C is given exac tly by σ t ( C ) . T hat is, under a t -discrepancy-correction de coder , any discrep ancy o f ma gnitude s can be detected if and only if s ≤ σ t ( C ) . Pr oof: Let t < s ≤ σ t ( C ) . Suppose that x ∈ X is tr ansmitted an d y ∈ Y is recei ved, where t < ∆( x, y ) ≤ s . W e will show that ∆( x ′ , y ) > t , for all x ′ ∈ C . Suppos e, by way o f contradiction, tha t ∆( x ′ , y ) ≤ t , for s ome x ′ ∈ C , x ′ 6 = x . Then σ t ( C ) ≤ ∆( x, y ) − 1 ≤ s − 1 < s ≤ σ t ( C ) , which is a contradiction. Con versely , ass ume that σ t ( C ) < s , i.e. , σ t ( C ) ≤ s − 1 . W e will s how that an unde tected e rror may occur . Since σ t ( C ) ≤ s − 1 , the re exist x, x ′ ∈ C suc h that σ t ( x, x ′ ) ≤ s − 1 . Th is implies that there exists some y ∈ Y s uch that ∆( x ′ , y ) ≤ t an d ∆( x, y ) − 1 ≤ s − 1 . By ass umption, C is t -disc repancy-correcting, so ˆ x ( y ) = x ′ . Thu s, if x is transmitted a nd y is received, an undetec ted error will oc cur , even tho ugh ∆( x, y ) ≤ s . The result ab ove has also bee n obtained in [12], a lthough with a different n otation (in pa rticular , treating σ 0 ( x, x ′ ) + 1 as a “distanc e” func tion). Below , we characterize the detec tion capab ility of a code in terms of the ∆ -distan ce. Pr opo sition 24: For any cod e C , we have σ t ( C ) ≥ δ ( C ) − t − 1 . Pr oof: For any x, x ′ ∈ X , let y ∈ Y be a solution to the minimization in (29), i.e., y is suc h that ∆( x ′ , y ) ≤ t a nd ∆( x, y ) = 1 + σ t ( x, x ′ ) . Then δ ( x, x ′ ) ≤ ∆( x, y ) + ∆( x ′ , y ) ≤ 1 + σ t ( x, x ′ ) + t , which implies that σ t ( x, x ′ ) ≤ δ ( x, x ′ ) − t − 1 . Theorem 25: Suppose tha t ∆( · , · ) is normal. For every code C ⊆ X , w e have σ t ( C ) = δ ( C ) − t − 1 . Pr oof: W e just nee d to show that σ t ( C ) ≤ δ ( C ) − t − 1 . T ake any x, x ′ ∈ X . Sinc e ∆( · , · ) is normal, there exists s ome y ∈ Y suc h that ∆( x ′ , y ) = t and ∆( x, y ) = δ ( x, x ′ ) − t . Thu s, σ t ( x, x ′ ) ≤ ∆( x, y ) − 1 = δ ( x, x ′ ) − t − 1 . A P P E N D I X B P RO O F O F L E M M A 1 5 First, we recall the follo wing us eful res ult shown in [8, Proposition 2]. Let X, Y ∈ F N × M q . Then rank ( X − Y ) ≥ max { rank X , rank Y } − dim ( h X i ∩ h Y i ) . (30) 26 Pr oof of Lemma 1 5: Us ing (30) and (3), we h ave rank ( AX − B Y ) ≥ max { rank AX , rank B Y } − dim ( h AX i ∩ h B Y i ) ≥ max { rank X − ρ, rank Y − σ } − dim ( h X i ∩ h Y i ) . W e will now sh ow that this lo we r bound is achiev able. Ou r approach will be to c onstruct A as A = A 1 A 2 , where A 1 ∈ F L × ( L + ρ ) q and A 2 ∈ F ( L + ρ ) × n q are both full-rank matrices. Then (3) guarantees that rank A ≥ n − ρ . The matrix B will be constructed similarly: B = B 1 B 2 , whe re B 1 ∈ F L × ( L + σ ) q and B 2 ∈ F ( L + σ ) × N q are both full-rank. Let k = rank X , s = rank Y , and w = dim ( h X i ∩ h Y i ) . Let W ∈ F w × m q be such that h W i = h X i ∩ h Y i , let ˜ X ∈ F ( k − w ) × m q be s uch that h W i + h ˜ X i = h X i and let ˜ Y ∈ F ( s − w ) × m q be su ch tha t h W i + h ˜ Y i = h Y i . Then, let A 2 and B 2 be such tha t A 2 X =      W ˜ X 0      and B 2 Y =      W ˜ Y 0      . Now , c hoose any ¯ A ∈ F i × ( k − w ) q and ¯ B ∈ F j × ( s − w ) q that have full row rank, wh ere i = [ k − w − ρ ] + and j = [ s − w − σ ] + . For instanc e, we ma y pick ¯ A = h I 0 i and ¯ B = h I 0 i . Finally , let A 1 =      I 0 0 0 0 ¯ A 0 0 0 0 I 0      and B 1 =      I 0 0 0 0 ¯ B 0 0 0 0 I 0      where, in both cases, the up per identity matrix is w × w . W e have rank ( AX − B Y ) = rank ( A 1 A 2 X − B 1 B 2 Y ) = rank (      W ¯ A ˜ X 0      −      W ¯ B ˜ Y 0      ) = max { i, j } = max { k − w − ρ, s − w − σ, 0 } = [max { k − ρ, s − σ } − w ] + . 27 A C K N O W L E D G E M E N T The authors would like to thank the anonymous reviewers for the ir helpful comments. R E F E R E N C E S [1] N. Cai and R. W . Y eung, “Network coding and error correction, ” in P r oc. 2002 IEEE Inform. Theory W orkshop , Bangalore, India, Oct. 20–25, 2002, pp. 119–122 . [2] R. W . Y eung and N. Cai , “Network error correction, part I: Basic concepts and upper bounds, ” Commun. Inform. Syst. , vol. 6, no. 1, pp. 19–36, 2006. [3] N. Cai and R. W . Y eung, “Network error correction, part II: L owe r bounds, ” Commun. Inform. Syst. , vol. 6, no. 1, pp. 37–54, 2006. [4] S. Y ang and R. W . Y eung , “Characterizations of network err or correction/detection and erasure correction, ” i n Proc . NetCod 2007 , San Diego, CA, Jan. 2007. [5] Z. Z hang, “Linear netwo rk error correction codes in packet networks, ” IE EE T rans. Inf. Theory , vol. 54, no. 1, pp. 209–21 8, 2008. [6] S. Jaggi, M. Langberg, S . Katti, T . Ho, D. Katabi, M. M ´ edard, and M. Effros, “Resili ent network coding in t he presence of Byzantine adversaries, ” IEEE T rans. Inf. T heory , vol. 54, no. 6, pp. 2596–2603, Jun. 2008. [7] R. K ¨ otter and F . R. Kschischang, “Coding for errors and erasures in random network coding, ” IE EE T rans. Inf. Theory , vol. 54, no. 8, pp. 3579–359 1, Aug. 2008. [8] D. Silva, F . R. Kschischang, and R. K ¨ otter , “ A rank-metric approach to error control in random network coding, ” I EEE T rans. Inf. T heory , vol. 54, no. 9, pp. 3951–3967, 2008. [9] S. Y ang and R. W . Y eung, “Refined coding bounds for network error correction, ” i n Proc. 2007 IEEE Information T heory W orkshop , Bergen, Norway , Jul. 1–6, 2007, pp. 1–5. [10] S. Y ang, C. K. Ngai, and R. W . Y eung, “Construction of linear network codes that achie ve a refined Singleton bound, ” in Pr oc. IEEE Int. Symp. Information Theory , Nice, France, Jun. 24–29, 2007, pp. 1576–1580. [11] R. Matsumoto, “Construction algorithm for network error-correcting codes attaining the S ingleton bound , ” IEI CE T ransactions on Fundamentals , vol. E90-A, no. 9, pp. 1729–173 5, Sep. 2007. [12] S. Y ang, R. W . Y eung, and Z. Z hang, “W eigh t properties of network codes, ” Eur opean T ransa ctions on T elecommunica tions , vol. 19, no. 4, pp. 371–383, 2008. [13] A. R. Rao and P . B himasankaram, Linear Algebra , 2nd ed. Ne w Delhi, India: Hindustan Book Agency , 2000. [14] E. M. Gabidulin, “Theory of codes with maximum rank distance, ” Pr obl. Inform. Tr ansm. , vol. 21, no. 1, pp. 1–12, 1985. [15] D. Si lv a, “Err or control for network coding, ” P h.D. dissertation, Univ ersi ty of T oronto, T oronto, Canada, 2009. 28 Danilo Silva (S’06) receive d t he B.Sc. degree from the Federal University of Pernambuco , Recife, Brazil, i n 2002, the M.S c. degree from the Pontifical Catholic University of Rio de Janeiro (P UC-Rio), R io de Janeiro, Brazil, in 2005, and the P h.D. degree from the University of T oronto, T oronto, Canada, in 2009, all in electrical engineering. He is currently a Postdoctoral Fellow at the Univ ersit y of T oronto. His research i nterests include channel coding, information theory , and network coding. Frank R. Kschisch ang (S’83–M’91–SM’0 0–F’06) receiv ed the B. A.Sc. degree (with honors) from the Uni versity of British Columbia, V anc ouve r , BC, Canada, in 1985 and the M.A.S c. and Ph.D. degrees from the University of T oronto, T oronto, ON, Canada, in 1988 and 1991, respective ly , all i n electrical engineering. He is a Professor of Electrical and Computer E ngineering and Canada Research Chair in Communication Algorithms at the Univ ersity of T oronto , where he has been a faculty member since 1991. D uring 1997–1998, he was a V isiting Scientist at the Massachusetts Instit ute of T echnology , Cambridge, and in 2005 he was a V isiting Professor at the ETH, Z ¨ urich, Swi tzerland. His research interests are focused on the area of channel coding techniques. Prof. Kschischang was the recipient of the Ontario Premier’ s R esearch Excellence A ward. From 1997 to 2000, he served as an Associate Editor for Coding Theory for the I E E E T R A N S AC T I O N S O N I N F O R M A T I O N T H E O RY . He also served as T echnical Program Co-Chair for the 2004 IE EE International Symposium on Information Theory (IS IT), Chicago, IL, and as General Co-Chair for IS IT 2008, T oronto.