Ring-theoretic foundation of convolutional network codingShuo-Yen Robert Li and Siu Ting Ho

Department of Information EngineeringThe Chinese University ofHong Kong, Shatin, NT, Hong Kong

Email: {bobli, stho3 } @ie.cuhk.edu.hkAbstract Convolutional network coding deals with the formula in any upstream-to-downstream order:propagation of symbol streams through a network with a linear (1) fe = Zkdefd when e c Out(T) at all nodes Ttime-invariant encoder at every node. When the symbol alphabet deln(T)is a field F, a symbol stream becomes a power series over F.

t ebnd ioPhysical implementation requires the coding/decoding kernels be with the boundary condition:restricted to finite objects. A proper domain for convolutional (2) {te e c In(S)} forms the standard basis of thenetwork coding consists of rational power series rather than vector space F'polynomials, because polynomial coding kernels do notnecessarily correspond to polynomial decoding kernels when the Let maxflow(T) and maxflow(4), respectively, denotenetwork includes a cycle. One naturally wonders what algebraic the maximum flow from the source node S to a node T andstructure makes rational power series a suitable domain for to a collection 4 of non-source nodes. From thecoding/decoding kernels. The proposed answer by this paper is Max-flow-min-cut Theorem, a prerequisite for T to receivediscrete valuation ring (DVR). A general abstract theory ofconvolutional network coding is formulated over a generic DVR a message of X symbols from S is that maxflow(T) .w.and does not confine convolutional network coding to the Based on the concept of maximum flow, the followingcombined space-time domain. Abstract generality enhances definition presents "optimal" network codes at threemathematical elegance, depth of understanding, and adaptability different levels of strength.to practical applications. Optimal convolutional network codes atvarious levels of strength are introduced and constructed for WMinition 1.3 ([3-5]). Denote VT= (f{, e E In(T)}) and V;delivering highest possible data rates. = (IVT T E 4}). Then, the network code qualifies as a

linear multicast, a linear broadcast, or a linear dispersion,I* INTRIJTII9 respectively, if the following statements hold:

The concept of network coding was formally introduced (3) dim(VT) = X for every non-source node T withby [1, 2]. In parallel, linear network coding started with [3, maxflow(T) 2 .

4] and led to wide applications including wireless and m (T) = mc

peer-to-peer communications. (4) sVT) n m o( } reA communication network is modeled as a finite directed nodim(Vr)=mimaxflow(e), c} for every collection(5) i(r= imafo()}freeycletngraph allowing multiple edges. An edge represents a of non-source nodes.

noiseless communication channel for transmitting a symbolper unit time. Assume a unique source node, denoted by S, WMinition 1.4 ([3-5]). The network code qualifies as aon every network. The symbol alphabet is a finite field generic linear network code if, for an arbitrary set of m < Xdenoted by F throughout the paper. The source S in each channels {e,, ..., em}, where each ei E Out(Ti).unit time generates a message, which consists of a fixed (6) if ({fd: d c In(Ti)}) ¢ ({fek: k . i}) for 1 < i < m, thenumber X of symbols and is represented by a c-dim row coding vectorsfel, ... ,fe. are linearly independent.vector over F. For every node T, let In(T) denote the set of Obviously a linear dispersion is a linear broadcast and aincoming channels to T and Out(T) the set of outgoing linear broadcast is linear multicast. In [4], it is shown that achannels from T. For technical convenience, the set In(S) is

regarded as cossigo.mgiaycanl.A generic linear network code iS a linear dispersion. TheNetwork Coding Theorem establishes the existence of aordered pair (d, e) of channels is called an adjacent pair generic linear network code when the base field is large

when there exists a node T with d c In(T) and e c Out(T). enough and thereby also the existence of optimal networkA. Linear network coding over an acyclic network codes of the other three types.

Over an acyclic network, the upstream-to-downstream B. Convolutional network coding over a cyclic networkpartial order among nodes allows a way of concerted Over a cyclic network, convolutional network codingsynchronization so that the encoding and transmission of r S t aeach message generated by S are independent of sequential tm-narat encode atreverod Sequentalimeagmessages. Hence linear network coding may simply dealwith each individual message instead of a whole stream. In may convolve together through cyclic transmission. For

this reason, one may as well allow feedback in the encoderthis way, the transmission medium iS purely in the space a ahnd.Tedlyi nisprbedoai whl prpgto dea isdseadd at each node. The propagation delay iS an inseparable issue,

and theransmipropagation issue,delays e-time

domai whieprpagaion dlay s diregaded.and the transmission medium iS the combined space-timeWilnition 1.1 ([3, 4]). An F-linear network code on an domain. A symbol stream is represented by a power series.acyclic communication network consists of a coding Thus the message stream generated by S becomes an w-dimcoeJfflcient kd,e E F for every adjacent pair (d, e). vector over the integral domain F[[D]] of power series,

where the variable D symbolizes a unit-time delay.Corollary 1.2. An F-linear network code on acyclicnetwork determines an w-dim column vector f. for each Wilnition 1.5 ([6]). A delay function t(d, e) over achannel e, called the coding vector, via the recursive communication network maps every adjacent pair (d, e) of

978-1-4244-1689-9/08/$25.00 ©2008 IEEE

channels to a number in {0, 1, 2, ..., oo} such that, around (7) MD M2 D ... D AD: ... D fl iMt = {O}every cycle, there is at least pair (d, e) with t(d, e) > 0. Moreover, if M = z 9, then 34k = zk.- for all k and z isA function in the form of p(D)/[1-D q(D)], where p(D) called the uniformizer of the DVR. The uniformizer is

and q(D) are polynomials in FND], is called a rational unique up to a unit factor.power series as it can be uniquely expressed as a powerseries. The set of rational power series will be denoted asF[(D)], which can be regarded as the intersection between +4F[[D]] and the quotient field F(D).A convolutional network code on a communication

network with the delay function t(d, e) is an assignment ofa coding kernel gd,e C Dt(d e) F[(D)] to every adjacent pair(d, e) of channels.

Coding kenel is the convolutional counterpart to coding 8_\coefficient in Definition 1.1. Figure 1 depicts a delayfunction over the Shuttle Network, where white arrowscorrespond to adjacent pairs.One way to define an optimal convolutional network

code, to be called a convolutional multicast, is bydecodability at all non-source nodes T with maxflow(T) >o allowing some delay. The existence of convolutionalmulticast is established in [6] and, moreover, all codingkernels can be chosen from any large enough subset of Fgure 1 Every adjacent pair (d, e) of channels in the Shuttle Network isF[(D)]. The special case for unit-delay networks also depicted by a white arrow, and the value of the delay function t(d, e) isappears in [5] and [7]. Assessing the delay per channel by prescribed next to it. On every directed cycle in the network, there is atan integer, [8] presents a construction algorithm of least one adjacent pair with non-zero delay.convolutional multicast using polynomial coding kernels. The terminology of a "discrete valuation ring" can beCoding kernels in Definition 1.5 are restricted to F[(D)] interpreted as follows. Over a DVR with the maximal ideal

instead of F[[D]] in order for implementation with finite M, the integer-valued function viamemory. On the other hand, if coding kernels were all (8) lxl =the largest k such that x c Akpolynomials, the corresponding decoding kernels (to be constitutes a natural discrete non-Archimedean valuation.defined in the sequel) could be non-polynomials. The When the DVR is F[[D]], a power series g(D) represents aessential algebraic structure that makes F[(D)] a proper symbol stream that starts at the time Jg(D)J. Thus the idealdomain of convolutional network coding/decoding kernels Dk.F[[D]] consists of those power series that representis a discrete valuation ring (DVR), which is the theme in symbol streams to appear at the time k or later.the remainder of the paper. The equality inside (7) is a direct consequence of theC. Localization, local ring, andDVR Nakayama Lemma [9]. In particular, the equalityl7inition 1.6 ([10-13]). A ring in this paper always refers nf=l(Dt .F[[D]]) = {O} reflects the physical axiom thatto a commutative ring with identity. every non-zero symbol stream must start within finite time.

Localization of an integral domain 9J at a multiplicativesubset A containing 1 but not 0 is the integral domain II.ENNRNG INTRRATII D F[(D)] ANDMconsisting of the fractions a/b, where a c 9J and b E A, We can now relate rational power series to convolutionalwith the understanding that two fractions a/b and c/d are network coding. By a convolutional encoder we shall meanequal when ad = bc. Moreover, every element x in 9i is a linear time-invariant casual convolutional encoder. A

identified with thfaciox1oht ecconvolutional encoder without feedback can be identifiedidentfied with the fraction x/1 so thato becomesansuring with its impulse response, which is represented by a powerin the localization and all elements of Aobecomeinvertile series. It is shown in [14] that a finite-state convolutionalA ring is said to be local when all non-units in it form an encoder must have the impulse response in the form of a

ideal, which would naturally be the unique maximal ideal., rational power series and can be implemented by, forWhen a principalalil domain (PID) is a local ring, it is example, linear shift registers.

called a discrete valuation ring (DVR). The convolutional encoder in Figure 2 corresponds to

Localization of an integral domain 9J at the complement p(D)1[1-D q(D)]. Every feedback signal goes through atleast one delay component and thereby ensures causality.Ofa im ideal yin integraldmain to be d n This encoder minus the feedback part is identified with the

apolynomial p(D). Thus the ensemble of finite-stateparticular if SJ =F[D] and gp =DF[D], then S1g =F[(D)]. convolutional encoders without feedback corresponds toIn fact, F[(D)] is aDVR, so is F[[D]]. the polynomial ring F[D] and the ensemble of all

Theorem 1.7. Let M denote the unique maximal ideal in a finite-state convolutional encoders corresponds to F[(D)].DVR SJ. Then JV1k for all positive integers k, are distinct The expansion from F[D] to F[(D)] is the localization ofideals~ ~ '.Moevr'ldasinteDR~1fr h F[D] at the complement of its prime ideal D.F[D]. This

descending chain ~~~~~~~~localization effectively makes all polynomials in the form

1-D.q(D), i.e., polynomials without the factor D, invertible.This is the algebraic meaning of feedback channels in a (convolutional encoder, because a feedback channel can bedevised to invert any given polynomial except for the factorof D in it. In other words, feedback channels can inverteverything except time in the combined space-time domain.Suppose that a finite-state convolutional encoder without

feedback is installed at every node of a cyclic network. Theglobal impulse response from the imaginary channels toany channels in the network is not necessarily a polynomial Fgure 2 A finite-state linear time-invariant casual sequential circuit withbut rather a rational power series. That is, encoders on a feedback that realizes the encoder with the impulse response representeddirected cycle behave as a large encoder with feedback by the rational power series p(D)/[l-D q(D)], where p(D) = po + p1D +resulting in the same effect of the aforementioned + PkDk and q(D) = qo + q1D + ... + qk 1Dk 1.localization that expands from F[D] to F[(D)]. This definition coincides with Definition 1.5 when 9J=To decode a message stream from a collection of co F[(D)]. The next theorem generalizes a result in [6].

rational power series means a form of mathematicalinversion. For the simplicity in illustration, consider the Theorem 3.2. Let J be a DVR and z the uniformizer in it.case with co = 1, which is to invert a rational power series. Given an 9J-convolutional network code, there is a uniqueThis "inversion" however cannot invert the factor D, which way to assign an c0-dim column vector over 9J as therepresents time. Suppose that the valuation of a polynomial coding vector to each channel e subject to the recursionp(D) in F[(D)] is v, that is, the highest power ofD in the (9) fe = Z gd,efd when e E Out(T)polynomial p(D) is D'. When v > 0, the inversion of a d In(T)rational power series in the form p(D)/[1-D-q(D)] must with boundary conditionincur a delay by at least v unit times. That is, we can find a (10) {fe: e E In(S)} forms the natural basis of the freerational power series of which the multiplicative product module 9SO of c0-dim column vectors over 9S.withp(D)I[1-D -q(D)] is D'. This is one way to explain why Proof Let n be the number of channels in the network.F[(D)] is the proper domain for finite-delay kernels in both Linearly order the n channels with the imaginary channelsencoding and decoding. first. Consider the nxn matrix (g d)d over 9J and anPower series over the symbol alphabet F represent e ,e

symbol streams either in transmitted data or used as kernels wixn matrix J formed by anzrxos identity matrix appendedin linear coders/decoders. F[[D]] represents the combined with n-ob columns of zeroes. The two conditions (9) anddomain of space and discretized time. The subset F[(D)] (10) can be combined in the matrix form:represents data as well as coding/decoding kernels that are (11) (fe )e (fe )e (g d,e )d,e + Jfinitely expressible and hence implementable. Both F[[D]] (12) (fe)e *ln - (gd,e)d,e = Jand F[(D)] are DVRs. In the above discussion oncoding/decoding kernels over F[(D)], we can substitute ClaimthatF[(D)] by F[[D]] mathematically, if not for the sake of (13) det 1In- (g de)de 1 is divisible by z.physical implementability. Represent the combined Let a chain of distinct channels (el, e2, e3, ..., ek) be called aspace-time domain by either F[[D]] or F[(D)] depending on loop when (el, e2), (e2, e3), ..., (ek 1, ek), (ek, el) are adjacentwhether implementability is required. Let M denote the pairs. Moreover, define the value over this loop to beideal generated by D. Then, M consists of all non-units and (_)kgg( which is divisible by zis therefore the unique maximal ideal. The property (7) (-)el,e2ge2,e3 gekl,ekgek,elbcharacterizes the unidirectional nature of time. If we ask because of the casual assumption on the delay function t(d,what algebraic structure of F[[D]] and F[(D)] makes them e). Denote by md,. the (d, e) entry in the matrix.mathematically suitable domains for convolutional network Then, det [In - (g d,e)d,e ]is a sum of n! summands, each incoding, the answer seems to be DVR. In the next section, the form of ±J7J mdG(d) for some permutation a. Note thatthe theory of convolutional network coding is formulated dover any DVR instead of F[(D)], while the unit-time D is md,,(d) is 1 when cy(d) = d, is -gd,p(d) when (d, c(d)) is angeneralized to the uniformizer of the DVR. In this way, the adjacent pair, and is 0 otherwise. Thus when ± n7d MdG(d) istheory is not confined to the space-time domain. Abstractgenerality enhances mathematical elegance and non-zero for some cy, either it is equal to the product of allendertanditengas els aapthabiit to a lcaios values over a number of disjoint loops or ca is the identityuncderstanding as wel as acdaptability tO applications.

mapping, which yields ±l7ld md,<7(d) 1. Thus allIII. GNRRAL ANRACT FomII9 0

CMII9AL NEWOK CING summands in det [In - (g de)de] are divisible by z except

Wilnition 3.1 Let SJ be a DVR and z be the uniformizer. the summand 1 corresponding to the identity mapping. ThisAn ~Y-convolutional networ-k code on a communication justifies the claim (13).network with the delay function t(d, e) is an assignment of Because z. J is the unique maximal ideal in ~J, thea coding ker-nel gd,e E zt(de).-y to every adjacent pair (d, e) inverse of det [In - (g de)d,e] exists within SJ. Denote theof channels.

adjoint matrix of In. - (gd,e)d,C by A. From (12),

r , n-.1 of non-source node.(14) (fe)e = det[In -(gd,e)d,ej1 *'J A is a matrix over S.This provides a close-form expression for coding vectors of At every node T, let (fe )eeIn(T) denote the coxlln(T) matrix

the 9J-convolutional network code in terms of coding that juxtaposes coding vectors fe, e E In(T). A decodingkernels. E matrix at the node T means an Iln(T)lx co matrix MT over 9J

such thatEample. Figure 3 depicts an 9i-convolutional network (18) (fe )enT) 'MT = zt(T)I. for some integer t(T) . 0code on the Shuttle Network, where 9J= F[(D)] and every e in T cacoding kernel is 0, 1 or D. An entry in a decoding matrix is called a decoding kernel.

Eample. In Figure 3, a decoding matrices at nodes R andAL (-D -1 I(-D O0Y are K ID and ji

I ,respectively.

When 91 = F[(D)], a convolutional multicast has beendefined in [6] as a convolutional network code such that(19) Every node T with maxflow(T) . c0 possesses a

1 decoding matrix over F[(D)].We now prove the consistence of this with Definition 3.3.Let T be a node with a decoding matrix MT. Since

t 1 si ,^,| \ / X ~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~t(T) ITfeIen(T) 'MT= z II the rank of the matrix (fe E(T)( eln(T) I Iln(S) (fe'nT

__Il\ 110 0 must be exactly co. Equivalently, rank(VT) = . Conversely,claim the existence of a decoding matrix at every node Twith rank( VT) = ). Thus, letfei,f2, J,f be a basis of the

1 ~ ~~Jj-module VT. Juxtapose these co column vectors into a

non-singular co~x0o matrix over 9J1. Then, the adjoint matrixcan be expanded into a decoding matrix by appropriately

Fgure 3 A convolutional multicast is prescribed over the Shuttle Network inserting |In(T) - co rows of zeroes.with co 2 and the same delay function as in Figure 1. A coding kernel is Winition 3.4 A set 4 of exactly o channels, includingindicated next to every adjacent pair of channels represented by a whitearrow. The convolutional network code determines a coding vector fe, possibly imaginary channels, will be called a basis of thewhich is an co-dim column vector over F[(D)], for each channel e. network when there are co channel-disjoint paths starting

In an 9J-convolutional network code, the coding vectors from imaginary channels and ending at channels in 4. Letof imaginary channels form the natural basis of an finitely V4 denote the 9J-module generated by {fe e E 4,}. Angenerated free module 9SO over 9S. The coding vector of 9J-convolutional network code is said to qualify as a basicevery other channel is a vector in this module according to 9i-convolutional network code ifthe close form formula (14). With regard to an (20) rank(V4) = co for every basis 4 of the network9J-convolutional network code, denote by VT the 9J-module A similar condition to (20) on a set . of channels in angenerated by {fe e E In(T)} and V; the 9J-module acyclic network will appear in [16] regarding generic lineargenerated by { VT: T E 4}. These are all submodules of the network codes.free module 9SO and hence are free modules. By theInvariant Factor Theorem of Free Submodule over a PID Theorem 3.5. On a communication network with any delay([15]), the invariant factors of this submodule can be taken function, a basic 9J-convolutional network code on theto be non-decreasing powers of the uniformizer z. network is an 9J-convolutional dispersion.

The data rate delivered from the source to any set of Proof. Let 4 be an arbitrarily given collection of non-sourcenon-source nodes is bounded by the maximum flow in the nodes. Given a basic 9J-convolutional network code withnetwork flow theory. We are interested in convolutional.... ~~~any delay function, we want to show that rank(V;)=network codes that achieve this intrinsic bound. Here in anetwork with a prescribed delay function, a path means a mmonmaxfrow there are maxflowsi) chanlchain of distinct channels such that every two consecutive disjointptsfmatelource cnoeSodesjint pt thuschannels make an adjacent pair with afinite delay. there exist min{maxflow(r),cdchannel-disjoint paths that

start from imaginary channels and end at channels in the setWMinition 3.3 An 9J-convolutional network code is said to {e : e E In(T) for some T E 4}. Let 4 denote the set of thequalify as an 9Y-convolutional multicast, 9Y-convolutional ending channels on these paths. If 1,1 <c0, expand the set 4broadcast, or 9Y-convolutional dispersion, respectively, if into the set ,' by the co - 1,1 imaginary channels not on thethe following statements hold: said paths. Else, let 4,' = 4. Clearly, 4,' is a basis of the(15) rank(VT) =0) for every non-source node T with network. With regard to the given basic SJ-convolutional

maxflow(T) .0). network code, we have rank(Ve,) =0). Thus the coding(16) rank(VT) = min{maxflow(T), 0)} for every vectors fe are linearly independent for all e E 4,',and, in

non-source node T. particular, for all e E 4,. Therefore, rank(V;) 2 rank(Ve,) =1(17) rank(V;) =min{maxflow(4), 0)} for every collection =min{maxflow(4), oz4. That is rank(V~) attains its obvious

upper bound min{maxflow(4), coz. E vectors fe, ee4, into the coxco matrix (fe)ee . Claim that

The three types of optimal 9J-convolutional network this determinant is non-zero in St.code in Definition 3.3 are obviously at hierarchical levels To justify the claim, it suffices find a way of evaluatingof strength. Together with Theorem 3.5, we have the indeterminates over 9J so that det(fe)ee- gets a

Basic SJ-convolutional network codenon-zero value in 9S. For this purpose, we may assume

=> 9J-convolutional dispersion without loss of generality that the ordering among the c)=> 9J-convolutional broadcast channels in 4 is consistent with the ordering among the=> 9J-convolutional multicast imaginary channels that are respectively linked to them

Eample The F[(D)]-convolutional network code on the through the said co channel-disjoint paths. Thus we set theShuttle Network in Figure 3 is a basic one. Adopt the indeterminate corresponding to each adjacent pair (d, e) togeneric notation eAB for the channel from a node A to a zt(d,e) E 9t when (d, e) is along one of these co paths andnode B and let one of the imaginary channels by eos. otherwise to 0 E 9S. Through this particular evaluation, theConsider the collection = {U, V} of two nodes. Clearly matrix (f)

e

becomes diagonal with all entries on themaxflow(4) = 2 = co. The collection {eRv, exv} of two ee*incoming channels to 4 forms a basis of the network. The main diagonal being powers of z and hence its determinantchannel evu can be coupled by an imaginary channel to also becomes a power of z. This justifies the claim.form another basis {evu, eos}. And {evu, euw} is not basis. Consider the product 1l7det(fe)ec4 taken over all

tV E sliCE D OFIAL 9S-CM)I9AL possible bases 4 of the network. Since H det(fe)ee; # ONtwoK CS

The existence of F[(D)]-convolutional multicast has in Sti, it is in the form of p0(XI .Xk) wherebeen shown in [6], where the proof borrows some I-z-qo(xl,...,xk)technique from [7]. In this section, we generalizes the proof po(x1, ..., Xk) # 0 E 9J[x1, ..., Xk]. The size of a DVR isfor the existence of a basic 9J-convolutional network code always infinite because the only finite integral domains arefor any DVR 9J, which implies the existence of all the other fields. Let E be an arbitrary subset of 9J with a sizethree types of optimal 9J-convolutional network codes. exceeding the degree of every indeterminate xj in the

polynomial po(x1, ..., Xk). By definition, the DVR 9J is aTheorem 4.1. For every DVR ~J, a basic SJ-convolutional PID. Hence there is a way of evaluating all thenetwork code exists on every network with any delay i enae overe is s tato(eat, al)re e

function t(d, e). Moreover, the coding kernels can bechosen from any large enough subset of 93. a non-zero value in 9J and consequently det(fe )ee= also

receives a non-zero value for every basis 4 of the network.ProofI On a communication network with the delay Thogtiseauinofndernts,vryg,function t(d, e), an 9J-convolutional network code consists

t

ofacol g (de).j for every ad pair assumes a value in the form ofpl(1-z.q), where p, q E 9S.of codingkernel,e djacent (d,~ ~~~~~~~Since 1-zq is invertible in SJ the ratio p/(1-z.qc E S. Thus

e), where z is the uniformizer of 9J. The associated coding theevluatio indertieinathe re thepassignmen ofThus

vectorfe for each channel e is calculated by the close-form .. .. r

formula (14). We now construct a basic 9Ji-convolutional gd,e to adjacent pairs (d, e) into a basic 9J-convolutionalnetwork code. Enetwork code, in two steps.

At the first step, for every adjacent pair (d, e) with t(d, e) kample Figure 4 illustrates the above proof by an< oo, create an indeterminate that ranges over 9S. Let these assignment of gd,e to adjacent pairs (d, e) in the Shuttleindeterminates be xl, ..., Xk. Denote by 9t* the integral Network. Here the indeterminates xl, ..., xk are denoted as r,domain consisting of rational functions in the form of SI, S2, ul, u2, vI, v2, wI, w2, xI, x2, y. When the

x, ...IXk) indeterminates are evaluated over 9J, the assignment of gd,e1z ;'I} x) where p(x1' Xk),q(xl, ..., Xk) E to adjacent pairs (d, e) reduces to an 9J-convolutional

network code. In order for the SJ-convolutional network93[xl, ..., Xk]. For every adjacent pair (d, e) of channels, let

newrco.Inrdrfrte9cnvlinantok

t(d,e) code to be basic, we need to achieve rank(V4) = 2 for allgd,e = z *xj where xj is the indeterminate that correspondsto the adjacent pair (d, e). We want to solve the recursive bases 4 of the network, where V4 denote the 9J-moduleformula (9) with the boundary condition (10) for an co-dim generated by {fe: e E 4,.We now illustrate how to achieve this optimal rank

derivation from (9) and (10) to (14) in the proof of simultaneously for just three exemplifying bases: {evu,Theorem 3.2 is still valid here. The formula (14) shows that eos}, {euw, ewx} and {exv, eyw}. Without loss of generality,fe indeed exists uniquely and is a vector over 9S. The let feos = (0 1) .Write h= 1/(1-z u2v2w/x). Then,remainder of this proof is to convert the assignment of gd,e det[(fe)ee{e es} ] = hrs1v1to adjacent pairs (d, e) into a basic SJ-convolutional vonetwork code. det[(fe )ee{euw,ewx } ] = h2 (rs1s2u2v1w2y - z. rs1s2u2v1v2w1w2x1y)

Let 4, be an arbitrary basis of the network. That is, there dt(ee{~y} r122111are 0) channel-disjoint paths starting from imaginaryee X Ychannels and ending at channels in 0,. Juxtapose the column The numerator of the product of these three determinants is

rsiv, .(rs, s2u2v1w2y-zrs,s2u22vIv2w1w2x1y).rsIs2u2v1w1X1y- time, frequency, or code. One natural extension of theIn this polynomial, the maximum degree of any present work would be a continuous-time theory based on aindeterminate is 2, i.e., the degree of u2. If E is any subset "continuous version of the notion of a DVR."of 9J with a size greater than 2, the above product assumes Another direction for future research is to improve thesome non-zero value in 9J when all the indeterminates construction algorithm for optimal convolutional networkrange over E. Thus the indeterminates can be evaluated so codes at the four hierarchical levels of strength. Thethat rank(V4) = 2 when 4 is {evu, eos}, {euw, ewx}, or {exv, improvement can be the minimization of the computationaleyw}. It turns out that the evaluation of r = si = S2 = ul = u2 complexity or the delay incurred in decoding. The amount= VI = V2 =WI = W2 = xI x2 = Yy = 1 suffices to achieve of delay directly pertains to the packet buffer size in therank(V4) 2 for all bases , of the network in the present implementation. The issue of computational complexityinstance. here is quite different from that in the construction of linear

'1 'il: network codes (See, for example [17].) In linear networkcoding one needs only construct the coding kernels and thedecoding kernels are automatically elements of the base

- g | | Xlfield. In convolutional network decoding, if the codingkernels are obtained through a computation algorithmrather than a close-form formula, then the most criticalissue of computational complexity is in decoding.

Acknowledgement. This work was supported in part bygrants No. CUHK4231/04E, 414005, 413806, 414307,and N CUHK411/07 from the Research Grants Councilof the Hong Kong Special Administrative Region, China.The authors would also like to acknowledge the helpful

Cjl - w [:,, ] l i ~~~~criticism of X. Jonathan Tan and Jennifer X. Wu on theoriginal manuscript.

RRNCEFigure 4. Let 9i be a DVR with the uniformizer z. Let 9* denote thepolynomial ring 93[r, sI, s2, ul, u2, vI, v2, wI, w2, xI, x2, y]. The figure [1] S.-Y. R. Li and R. W Yeung, "Single-source network informationshows an assignment of gd,e to adjacent pairs (d, e) in the Shuttle Network flow," Proceedings of the 1999 IEEE Information Theory Workshop,with respect to the same delay function as prescribed in Figure 1. Each gd,,e Metsovo, Greece, p. 25, June-July, 1999.shown in white, corresponds to one of the indeterminates in 93*. The [2] R. Alshwede, N. Cai, S.-Y. R. Li and R. W Yeung, "Network

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