2009 Workshop on Network Coding, Theory and Applications

Network Coding Theory via Commutative AlgebraShuo-Yen Robert Li and Qifu Tyler Sun

Department of Information Engineering

The Chinese University ofHong Kong

Shatin, NT, Hong Kong

Email: bobli@ie.cuhk.edu.hk.qfsun5@ie.cuhk.edu.hk

Abstract-The fundamental result of linear network codingasserts the existence of optimal codes over acyclic networkswhen the symbol field is sufficiently large. The restriction tojust ac~clic networks turns out to stem from the customaryalgebraic structure of data symbols as a finite field. Adopting~ata units that belong to a discrete valuation ring (DVR), thatIS, a PID with a unique maximal ideal, much of the networkcoding theory extends to cyclic networks. Being a PID with themaximal ideal 0, a field can be regarded as a degenerated DVR.Thus the field-based theory becomes a degenerated version ofthe DVR-based theory. Meanwhile, convolutional networkcoding becomes the instance when the DVR consists of rationalpower series over a field.

Besides the treatise in commutative algebra, the presentpaper also delves into the efficiency issue of code construction.Given a cyclic network, a quadratically large acyclic networkis constructed so that every optimal code on the acyclicnetwork subject to some straightforward restriction inducesan optimal code on the given network. In this way, existingconstruction algorithms over acyclic networks can be adaptedfor cyclic networks as well.

I. INTRODUCTION

The concept of network coding originated from both thenonlinear theory [1, 2] and linear theory [9, 11]. The latter,especially, carries wide applications to many establishedfields including coding theory, computer networks, computerscience, distributed data storage, information security,information theory, optimization theory, peer-to-peer (P2P)content delivery, switching theory, and wireless/satellitecommunications. It models a communication network as afmite acyclic directed multi-graph with a single source node.An edge represents a noiseless communication channel ofunit capacity. The symbol alphabet is structured as a fmitefield. The source generates a message per unit time, which isformulated as a vector over the symbol field.Upstream-to-downstream ordering of nodes enablesconcerted synchronization among all nodes so that theencoding and transmission of a message is independent ofsequential messages. This fmesses the issue of datacommunication delay and thereby allows the theory to dealwith each message individually. The main theoremguarantees the maximum data rate toward every node whenthe symbol field is sufficiently large.

Despite the theoretic restriction to acyclic networks, thetheory is applicable to cyclic networks as well because theactual transmission medium is in the combined space-timedomain and, when a time-multiplexed network is unfoldedwith respect to time, the resulting trellis network is acyclic.The coding scheme at a node on a cyclic network is timevariant, and the propagation of sequential messages mayconvolve together. On the surface, it seems that the theoreticrestriction to acyclic networks has been because of the needof upstream-to-downstream node ordering. Actually, this

978-1-4244-4724-4/09/$25.00 © 2009 IEEE 12

restriction is a much deeper mystery to be explored in thepresent paper.

Section II below algebraically structures the ensemble ofdata units as a principal ideal domain (PID) instead of afmite field. Fundamental theory of linear algebra deals withvectors over a PID and leads to a generalization of the basicconcepts of linear network coding from acyclic networks tocyclic networks. However, data propagation around a cyclevia a PID-based linear network code may be non causal. Toresolve this problem, Section III requires the PID of dataunits to possess a unique maximal ideal. In commutativealgebra, such a PID is called a discrete valuation ring (DVR).Let M denote the maximal ideal. Then, all ideals in the DVRform the infmite strictly descending chain

M => M 2 => ... => M t => ...Monotonicity of this chain is a "unidirectional attribute" ofthe DVR, which generalizes the unidirectional nature oftime.It serves to break the deadlock in cyclic transmission.Meanwhile, every field is a PID with the unique maximalideal 0 and hence can be regarded as a degenerated DVR. Inthis sense, the conventional formulation of linear networkcoding becomes a degenerated case of DVR-based networkcoding.

Convolutional network coding [11, 7] is a special case ofDVR-based network coding when the DVR consists offormal power series over the symbol field IF in the variable Drepresenting a unit-time delay. In fact, the power series isnormally restricted to just rational power series for fmiteimplementability [10]. A rational power series, Le., afunction in the form p(D)/[I-D·q(D)], where p(D) and q(D)are polynomials in the polynomial ring IF[D], is invertible ifand only if it is not divisible by D. Thus, the ring IF[(D)] ofrational power series has the unique maximal ideal D·lF[(D)]and hence is also a DVR by itself.

The practical application ofconvolutional network coding,however, is hindered by the difficulty in precise inter-nodesynchronization. In general, DVR-based network coding isnot restricted to time-multiplexing or even combinedtime/space/frequency/phase/code/wavelength-multiplexingof data symbols. Generality enhances the potential ofapplicability. For example, if the uniformizer in the DVRrepresents a shift in some domain other than time, then thenetwork code is insensitive to the aforementioned hindranceof imprecise inter-node synchronization.

Section IV delves into the efficiency issue of codeconstruction. A linear network code that delivers a completemessage from the source to all eligible receivers is called alinear multicast. There has been a variety ofpolynomial-timealgorithms for the construction of a linear multicast over anacyclic network. They utilize the upstream-to-downstreamnode ordering. Given a cyclic network, we construct a

quadratically large acyclic network, on which every linearmulticast with a certain simple property immediately inducesa linear multicast on the cyclic network. In this way, efficientconstruction algorithms over acyclic networks, including [6,5] for example, become applicable to cyclic networks.

II. ABSTRACT FORMULATION OF NETWORK CODING OVER A

PRINCIPLE IDEAL DOMAIN

We adopt the following convention unless otherwisespecified• The network under consideration is represented by a

quadruple (V, E, s, OJ), where V is the set of nodes, E theset of directed edges, s the source node, and OJthe fixeddata generating rate of the source node. An edgerepresents a communication channel of the unit capacity.There may possibly be cycles in the network.

• For every node v, denote by In(v) and Out(v), respectively,the sets of its incoming and outgoing edges. In particular,In(s) consists of OJedges from nowhere to s, which willbe called data-generatingchannels.

• An ordered pair (d, e) of edges is called an adjacent pairwhen there is a node v such that d E In(v) and e E Out(v).

• Impose a linear order on edges led by data-generatingchannels. In the special case when the network is acyclic,this linear order is assumed to be in anupstream-to-downstream fashion.

• Adopt the notation Adj(·) for the adjugate ofa matrix.• For any integer k, let I k denote the kxk identity matrix.• Form the OJxlEI matrix JOJ,IEI by appending lEI-OJ columns

of zeroes to IOJ.• Let JP> denote a PID and JP>OJ the free JP>-module consisting

of OJ-dim column vectors over JP>.• Let Q denote the quotient field of JP> and regard JP> as a

subdomain of Q.

Definition 1. A JP>-linear network code C assigns a codingcoefficient kd,e in JP> to every pair (d, e) ofedges such that kd,e =owhen (d, e) is not an adjacent pair. We shall also adopt theconvention C = (kd,e). Moreover, denote by Kc the IElxlEImatrix [kd,e]d,eeE, where rows and columns are indexedaccording to the ordering of edges.

Definition 2. Given the JP>-linear network code C = (kd,e), a setof coding vectors means an assignment of an OJ-dimensionalcolumn vectorIe over JP> to each edge e such that(1) {fe, e E In(s)} forms the natural basis of JP>OJ(2) Ie = Ldeln(v)kd,efd for every node v and every e E Out(v)The two conditions can be combined in the matrix form intothe equation [f;]eeE = [f;]eeE· Kc + JOJ, lEI with the followingtwo equivalent forms:(3) [f;]eeE· (IIEI-Kc) = JOJ,IEI(4) det(IIEI-Kc) [f;]eeE = JOJ,IEI· Adj(~EI-Kc)

The above condition (2) amounts to a nonhomogeneoussystem of OJo(IEI-OJ) linear equations over P for the OJo(IEI-OJ)variables that are entries to the vectors Ie, e E E\In(s). Thediscriminant of this linear system is det(~EI-Kc). When it iszero, none or multiple solutions exist. On the other handwhen it is nonzero, C = (kd,e) can be regarded as a Q-linearnetwork code and thereby determines a unique set of codingvectorsIe E QOJ by the formula(5) [f;]eeE = det(IIEI-Kc)-1 JOJ,IEI· Adj(IIEI-Kc)

2009 Workshop on Network Coding, Theory and ApplicationsThus, when det(IIEI-Kc) * 0, the JP>-linear network code Cdetermines a unique set ofcoding vectors or none dependingwhether det(IIEI-Kc) E JP> divides all entries in the matrix JOJ, lEI

·Adj(IIEI-Kc) or not.

Definition 3. A JP>-linear network code C is said to benonsingular when det(IIEI-Kc) * O. A nonsingular JP>-linearnetwork code is said to be normal when it determines aunique set of coding vectors.

Represent the message from the source by a row vector mT,

where m E JP>OJ. For a normal JP>-linear network code withcoding vectors Ie, the intended data unit for the transmissionover an edge e is mT1e.According to (2), mT1e = Ldeln(v)kd,emT-fd for an edge e E Out(v). That is, an outgoing data unitfrom a node v is a linear combination of incoming data units,where the "linear gains" are the coding coefficients.Normality of a JP>-linear network code identifies the codingvectors unambiguously and hence is a prerequisite to thenotion of data propagation via the code. The remainder ofthis section deals with normality and optimality of a code,and the next section will be concerned with causality in datapropagation via a code.

Corollary 4. A nonsingular JP>-linear network code C isnormal if and only if det(IIEI-Kc) divides all entries in thematrix JOJ, lEI •Adj(~EI-Kc). This, in particular, is the case whendet(~EI-Kc) is a unit in JP>. Thus, a nonsingular JP>-linearnetwork code may be regarded as a normal Q-linear networkcode.

Corollary 5. Assume that the network is acyclic. Then,det(~EI-Kc) = 1 for every JP>-linear network code C and,consequently, C is normal.

Definition 6. The normalization of a nonsingular JP>-linearnetwork code C = (kd,e) means the JP>-linear network code(It d,e), where• It d,e = det(IIEI-Kc)·kd,e for d E In(s)• It d,e = kd,e for d E E\In(s)

Corollary 7. The normalization of a nonsingular JP>-linearnetwork code is normal. Moreover, for a nonsingularJP>-linear network code C with coding vectors Ie E QOJ, thecoding vectors of the normalization are given by:(6)f~ = det(~EI-Kc)1e for e ~ In(s), whilej"; for e E In(s) still

abide by the rule (1).Proof Skipped. -

In a normal JP>-linear network code, coding vectors ofdata-generating channels generate the free module JP>OJ, whichcontains all coding vectors. Every submodule of JP>OJ is a freeJP>-module according to the invariantfactor theorem offreesubmodule (See Chapter 12 of[4], for example.) In particular,coding vectors of incoming edges to a node v generates a freesubmodule, of which the rank represents the data receptionrate ofv from the source s.

Convention. Denote the maximumflow from the source s toa node v by maxflow(v). It is equal to the minimum cutbetween sand v.

Definition 8. A sink means a node v with maxflow(v) ~ OJ. Anormal JP>-linear network code with the coding vectors Ie iscalled a JP>-linear multicast when(7) rankp«f;: eEIn(v)) = OJ for every sink v

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The point-to-point rate of information flow, as well ascommodity flow, from s to v is bounded by maxflow(v).Thus a sink means node eligible for receiving data from s atthe full rate co. A linear multicast is an optimal network codein the sense of enabling every eligible node to receive at thefull rate. It is the weakest sense of optimality in theconventional theory of linear network coding [14] . In itssimplest form, the fundamental theorem of network codingasserts the existence of an F-linear multicast on an acyclicnetwork for every sufficiently large field F . The followinglemma generalizes this theorem from acyclic networks to allnetworks.

Lemma 9. Let F be a field . There exists an F-linear multicastwith all coding coefficients belonging to any sufficientlylarge subset Fin F.

Proof Analogous to the approach in [7]. _

Theorem 10. A P-linear multicast exists.Proof Because of Lemma 9, we need consider only the

case when P is not a field . Then, IPI is infinite since everyfinite integral domain is a field . By Lemma 9, there exists aQ-linear multicast with coding coefficients in P , which maybe regarded as a nonsingular P-linear network code . Thelemma below shows that the normalization of thisnonsingular P-linear network code is a P-linear multicast. _

Lemma n. Let C be a nonsingular P-linear network code.Then, the normalization of C is a P-linear multicast if andonly ifC is a Q-linear multicast.

Proof Skipped. _

Example. Figure 1 depicts a GF(3)-linear multicast on theShuttle Network, which contains cycles.

Figure 1. A GF(3)-linear multicast on the Shuttle Network, which containscycles, is given in terms of coding coefficient s and coding vectors. Anadjacent pair is depicted by a white arrow inside a node.

The next theorem "normalizes" the inverse homomorphicimage of a linear multicast into a linear multicast. It is a toolto be employed in Theorem 21 of Section IV.

Theorem 12. Let J.1 be a homomorphism from P to anotherPID P', and C = (kd.e) a P-linear network code such that thehomomorphic image (ji.,kd.e) ) is a P'-linear multicast. Then,the normalization of C is a P-linear multicast.

Proof Applying componentwise, the homomorphism J.1extends to a mapping from P'" to (P')'"and also to a mappingfrom matrices over P to matrices over Pl .

Write C' = (ji.,kd.e))' Since det(IIEI-Kc) *" 0 and

2009 Workshop on Network Coding, Theory and Applications,u(det(IIEI-Kc)) = det(IIEI-Kc) , we find that det(IIErKc) *" O.Thus Cis nonsingular. Letf~E(p')'" andfeEQ"', respectively,denote the coding vectors of C' and C. Applying (4) to C andthen to C', one can show

,u(det(IIErKc) [fe]ee E) = det(IIErKc) [f~]eeE

Let ge denote the coding vectors of the normalization of C.Then, ge= det(IIErKc)fe for e~In(s) according to Corollary 7.For every sink v,

rankll'«ge: eEIn(v))= rankll'«det(IIErKc)fe: eEIn(v));;::: rankll"«,u(det(IIEl-Kc)fe): eEIn(v))= rankll"«det(IIErKc}r~: eEIn(v))=co _

III. CAUSAL DATA PROPAGATION By NETWORK CODING

Convention. Hereafter let lID denote a DVR and z theuniformizer, that is, the generator of the maximal ideal in lID.The unifonnizer is unique up to a unit factor . In the particularinstance when lID = F[(D)], the unifonnizer z is D.

Represent the message from the source by a row vector mT,

where m E P "'. If there is a sensible way ofdata propagationvia a normal P-linear network code with coding vectors fe,then each edge e must carry the data unit mTJe. This makesevery outgoing data unit from a node a linear combination ofincoming data units to that node . However, there is thequestion on how do edges around a cycle acquire theirrespective data units for transmission without a deadlock.The question pertains to the algebraic structure of P . Thealgebraic structure of a DVR incorporates a unidirectionalattribute similar to time . In fact, all ideals in IDl form theinfmite strictly descending chain

(z) ::::> (:l) ::::> ••• ::::> (z~ ::::> •• •

This makes IDl a suitable domain of data units . Causaltransmission ofdata units via a IDl-linear linear network coderequires the coding coefficient for at least one adjacent pairalong every cycle to be divisible by z.

Definition 13. A delay function on the network is anonnegative integer function t, defined over the set ofadjacent pairs such that, along every cycle, there is at leastone pair (d, e) with t(d, e) > O. A D-Iinear network code issaid to be t-causal ifthe coding coefficient for every adjacentpair (d, e) is divisible by z/(d. e). A causalIDl-linear networkcode means one that is t-causal for some t.

Theorem 14. A causalIDl-linear network code Cis normal. Infact, det(IIErKc) is a unit in IDl.

Proof Skipped. _

Theorem 15. Given a delay function t, there exists at-causalIDl-linear multicast.

Proof Skipped. _

Example. Figure 2 depicts a causalIDl-linear multicast on theShuttle Network of Figure 1.

Given a causal IDl-linear network code, juxtapose theincoming coding vectors fe to a node v into an coxIIn(v)1matrix [fe]eeIDf)' Represent the message from the source by arow vector m ,where m E IDl"'. Thus the data received by thenode v is represented by the row vector mT·[fe]eeID(v). Anexplicit form ofdecoding the message mT is to calculate Z8mT,

for some integer t5;;::: 0, from mT·[fe]eeID(v). Here t5 means a

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"delay measured by the exponent of z" in reflection of thedefinition of a causal lID-linear network code. A prerequisiteis the full rank OJ ofthe matrix [fe]eeln(v), because the messagevector mT is arbitrarily given.

Definition 16. For a causal lID-linear network code with thecoding vectors fe, a decoding matrix with delay 0 ~ 0 at anode v means an IIn(v)lxOJ matrix M over lID such that[fe]eeln(vr M = ZOJar

Theorem 17. For a lID-linear multicast, the submodule S, oflID'" spanned by incoming coding vectors to a node v is free.Assume that v is a sink so rankll(§v) = OJ. Denote the invariantfactors of'S, in lID'"by Zi" l' ,... ,Zi@, where i I :::; . . . :::;iar Let 0~ iar Then, a decoding matrix with delay 0 exists at v.

Proof According to the invariantfactor theorem offreesubmodule, §v is a free submodule of lID'" and there exists abasis {u" ... , u",} of lID'" such that {z''u1' ••• , Zi@U",} is a basisof S; Thus , the matrix [Uj]I';;S'" is invertible and the vectorszOu" ... , zOu", are all lID-linear combinations of the codingvectors fe, eEIn(v). Translating into the matrix form, thereexists an IIn(v)lxOJ matrix M over lID such that zO[ujhsjs", =

[fe]eeln(v)' M. Hence M([ujh,;;s",r I is a decoding matrix withdelay oat v. •

Figure 2. A IIJi-linear multicast on the Shuttle Network is given. Let t be anydelay function subject to the specification of t(d, e) = n when the codingcoefficient kd•e = Z'. Then, the given IIJi-linear multicast is r-causal,

IV. CONSTRUCTION OF OPTIMAL NETWORK CODES BY

NETWORK DE-CYCLING

Hereafter let N denote the network (V, E, s, OJ). Asdepicted in figure 3, we shall associate N with an acyclicnetwork N' according to Algorithm 18 below. The mostdownstream nodes inN'all turn out to be sinks. One ofthem ,denoted as V4, corresponds to each sink v in N. An extra one,denoted as S4, corresponds to the source node s in N. Let C'be a lP'-linear multicast on N'subject to a simple restriction(8) in Theorem 19 below. By the formula (9), a lP'-linearnetwork code C is induced on N. The full data reception rateof S4 via the lP'-linear multicast C' then implies thenonsingularity of C. Thus, C can be viewed as a normalQ-linear network code. Meanwhile, the full data receptionrate ofV4 via C' can be proven to imply the full data receptionrate of v via the Q-linear network code C. The key to thisproof is in transforming the problem into a form suitable forinvoking the Duality Theorem of Vector Matroids (See , for

2009 Workshop on Network Coding , Theory and Applicationsinstance, Theorem 2.2.8 of[I2].) The full data reception rateof all sinks v then qualifies C as a Q-linear multicast.According to Lemma 11, the normalization of C is a lP'-linearmulticast. This construction of a lP'-linear multicast on N issummarized by Theorem 19. Through this theorem, efficientalgorithms in the literature for the construction of IF-linearmulticast on an acyclic network can be adapted for a cyclicnetwork.

We now present the algorithm for associating N with anacyclic network N~

Algorithm 18. Nodes ofN'are on five layers, labeled 0 to 4from upstream to downstream, as exemplified by Figure 3:• Layer 0 consists of the source node to be denoted as So.

• On layer 1, there is a node eI corresponding to every edgee E E\In(s).

• On layers 2 and 3, there are nodes e2 and e3, respectively,corresponding to every edge e E E.

• Layer 4 consists ofa node V4 corresponding to each node vin N that is either the source or a sink.

There are lEI-OJ data-generating channels terminating at thesource node So. All other edges are between adjacent layers.Corresponding to every adjacent pair (d, e) in N, there is theedge de from e, to di. Corresponding to every e E E\In(s),there are the edge e(l) from So to e, and the edge e(2) from e, toe2. Corresponding to every e E E, there is the edge e(3) frome2 to e3. Finally, incoming edges to a layer-4 node V4 areprescribed in two steps:(i) Arbitrarily take OJ edge-disjoint paths in N that lead

from data-generating channels to the node v.(ii) For every e E E, install an edge from e3 to V4 unlesse E

In(v) and is an edge on these paths.Altogether, the number of edges in N' is bounded byIEI+IEI+(I£I+IEI

2)+IEI+I£I'I11 ;:z 1£12

•

(a) (b)Figure 3. (a) The given network N contains a cycle. (b) The associatedacyclic network N' consists of five layers of nodes. The two networks areso related that a linear multicast on N ' subject to a straightforwardcondition induces a linear multicast on N.

Before presenting Theorem 19 as well as two applicationtheorems, we shall break down the construction ofN' fromN into three steps for enhanced transparency. Thisnecessitates the notion of a bipartite network, which is afinite directed graph with the following attributes.• Nodes are partitioned into squares and triangles.

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Figure 4. Through the "node dilation" process, the network in Figure 3(a)is converted into the bipartite network of (a). The acyclic counterpart ofthis bipartite network appears in (b), which is the network in Figure 3(b)minus nodes and edges at the top and bottom layers.

There is an acyclic counterpart to every bipartite network.The acyclic version of NB, to be denoted by N'e, isconstructed by the following "de-cycle" process .• All squares become triangles and vice versa. The notation

e) becomes for a square and e2 for a triangle in N's.• The ro data-generating channels are removed. In

replacement, IEI-rodata-generating channels are created,each toward a different square.

• As a result ofthe square/triangle inversion, an edge e frome) to e2 in NB becomes an arrow in N's, which will bedenoted as e(2).

• Meanwhile, we reverse the orientation of arrows in NB sothat they remain arrows in N'e-Thus an arrow de from dito e) in NB becomes an arrow from e) to d2 in N's, whichwill be denoted by de .

• Corresponding to every triangle e2 in NB, a new square e3and a new edge e(3) from e2 to e3 are created for N's.

• There are two kinds of directed links between nodes: Anarrow connects from a square to a triangle, and an edgeconnects from a triangle to a square. There is exactly oneincoming edge to each square and exactly one outgoingedge from each triangle.

• A data-generating channel is an edge without anoriginating triangle, and its terminating square is called asource. There may be multiple sources.

The formulation of a bipartite network does not totallyconform to the formulation of a network in Section II. Thereis a bipartite version of every network. The bipartite versionof N, to be denoted by NB, is constructed by the following"node dilation" process.• Corresponding to every data-generating channel e in N,

there are the square e2 and the data-generating channel etoward e2 in NB•

• Corresponding to every e e E\In(s), there are the trianglee" the square e2, and the edge e from e) to e2 in NB•

• Corresponding to every adjacent pair (d, e) in N, there isan arrow de from dz to e).

In summary, every edge e in N remains an edge in NB whileevery node v in N is dilated into a subnetwork in NB• Thebipartite version of the network in Figure 3(a) is depicted byFigure 4(a) .

2009 Workshop on Network Coding, Theory and ApplicationsThe acyclic counterpart to the bipartite network in Figure

4(a) is depicted by Figure 4(b). When the square/triangle andarrow/edge distinctions are ignored, the acyclic bipartitenetwork N'« becomes a subnetwork ofN~ In fact, N» is N'minus nodes and edges at the top and bottom layers. Insummary, the construction ofN'is in three steps: first fromN to the bipartite version NB, then to the acyclic counterpartN's, and fmally to N'by the appendage ofthe top and bottomlayers.

Theorem 19. Let C' be a lP'-linear multicast on N'subject tothe constraint that:(8) The coding coefficient Kx,y = 1 when (x,y) = (e() , e(2») or

(e(2), e(3)) for some e e E\In(s) .Let C be the induced lP'-linear network code (kd,e) on N via

{

k' - ·k'- , when de In(s)(9) k

d e= e(l),de de,d(3)

, -k' d-·k'd-d ,whendeE\In(s)e(l) ' e e, (3 )

Then, the normalization of C is a lP'-linear multicast.

We now present two applications to the construction ofcausal linear multicast on a cyclic network.

Theorem 20. Given a delay function t on N, define a delayfunction t' on the N'via(10) I'(x,y) = t(d, e) ifx = de andy = d(3) for some adjacent

pair (d, e) in N. Else, I'(x, y) = O.Let C' be a r'-causal B-linear multicast on N'subject to (8)and C the corresponding D-linear network code via (9). Then,C is a r-causal D-linear multicast.

Proof The normalization ofC is a II])-linear multicast on Nby Theorem 19. Then, C is a Q-linear multicast on NbyLemma II , where Q in this case is the quotient field of 11]).

Moreover, the l' -causality of C' implies the r-causality of C.Thus, C is also a normal II])-linear network code by Theorem14. Denote the coding vectors ofC byIe e 11])"'. For every sinkv, rankIl«fe: eeIn(v)) = rankl/(fe : eeIn(v)) = ro. _

Theorem 21. Let IF be a field. Let C' be an IF-Linearmulticast on N'subject to (8) and C = (kd,e)the correspondingIF-linearnetwork code via (9). Given a delay function ton N,at-causal IF[(D)]-linear multicast H = (hd,e) can beconstructed by:( II ) h = k ·D1(d,e)d,e d,e

Proof Applying Theorem 19 and then Lemma 11, wefind C itself an IF-linear multicast. Let J1 be thehomomorphism from IF[(D)] to IF that preserves IF and mapsD to 1. Thus, J1(hd,e) = kd,e' Substituting P, P', and C inTheorem 12 with IF [(D)] , IF, and H, respectively, we find thenormalization of Han IF[(D)]-linear multicast. Clearly, Hist-causal , and thus normal by Theorem 14. Hence, H can alsobe shown an IF[(D)]-linear multicast. _

An IF-convolutional network code means an IF[(D)]-linearnetwork code, and an IF-convolutional multicast means anIF[(D)]-linear multicast. Given a delay function t on N,Theorem 21 adapts an IF-linear multicast on N'under theconstraint (8) into a t-causallF-convolutional multicast on Nthrough the simple formula (11). The adaptation algorithm isindependent of the delay function t except for the plainappearance of t in (11). The remainder of this section dealswith construction of IF-linear multicast on N ' under theconstraint (8) for sufficiently large IF as well as the issue of

(b)(a)

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computational complexity. In fact, we shall impose twofurther constraints on such construction:• The coding vectors for edges out of So form the natural

basis oflF1EI- m•

• The coding coefficients are fIXed to 1 for (x,y), where X isthe incoming edge to some layer-lor -3 node in N~

These two constraints and (8) on the construction oflF-linearmulticast on N'can be combined into:(12) Only coding coefficients for adjacent pairs in the form of

( de, d(3)) need be calculated. Such adjacent pairs in N'are in one-to-one correspondence with the ones in N.

Below we adapt two of the most popular approaches in theliterature for constructing IF-Linear multicast on acyclicnetworks into the construction of an IF-linear multicastsubject to (12) on the special topology ofN~

The flow path approach exemplified by [61. For every sink vofa given acyclic network, a set of IIn(s) Iedge-disjoint pathsleading from In(s) to v are identified. Delete from thenetwork all edges that are not on any ofthese paths. Then, theassignment process of coding coefficients is by dealing withone node at a time from upstream to downstream. In dealingwith a node u, the involved adjacent pairs are (d, e) with d E

In(u) and e E Out(u). Through normalization, we maychoose to set one of these coding coefficients to be 1.Throughout the assignment process, we keep the accruedcoding vectors for edges on different paths in a set linearindependent. The complexity is O(1]Joi...t5+1])), where J isthe number of sinks in the given acyclic network and 1] isthe number ofedges.

Over the special topology ofN~ all lEI-OJ edge-disjointpaths leading from data-generating channels to the sink S4

can be chosen to be in the form of (e(1), e(2), e(3), e3s4), whereeEE\In(s). In dealing with the node el during the assignmentprocess, the adjacent pair (e(l), e(2)) is assigned the codingcoefficient 1. In dealing with the node e2, where e E E\In(s),the adjacent pair (e(2), e(3)) is assigned 1. With thesespecifications, this approach constructs an IF-linearmulticast subject to (12) on N' at the complexity of0(IEI·(t5+1)·(IEI-OJ)·(t5+1+IEI-OJ)) = 0(IEI3.b), where Jis thenumber of sinks in N.

The matrix completion approach exemplified by [51.Associate each adjacent pair (d, e) in a given acyclic networkwith an indeterminate Xd,e. Let IF[*] denote the polynomialring in these indeterminates over the field IF. Let 1] denote thenumber of edges. Also associate every sink with anappropriate nonsingular 1]X 1] matrix over IF[*] such thatevery entry is one of the indeterminates or a scalar. Then, inan iterative process, every step replaces an indeterminate bya scalar in such a way that preserves the full rank of thematrix associated with every sink. After all indeterminatesare replaced by scalars, the full rank of a matrix implies thefull rate of data transmission from the source to a sink. Thecomplexity is 0(J1]3Iog1]), where Jis the number of sinks inthe given acyclic network.

We now adapt this approach to the construction of anIF-Iinear multicast subject to (12) on the special topology ofN~ for which the existence is guaranteed by the abovediscussion on the flow path approach. Thus associate eachadjacent pair in the form ( de , d(3)) with an indeterminate Xd,e.

These indeterminates can be used as coding coefficients,

2009 Workshop on Network Coding, Theory and Applicationstogether with pre-assigned 0-1 coefficients, for an IF[*]­linear network code C' subject to (12) on N~ Denote by f';the corresponding coding vector for the channel e(3)

corresponding to each channel e in N. For every sink V4 ofN~juxtapose the coding vectors f~, where e3 is adjacent to V4,

into an (IEI-OJ)x(IEI-OJ) matrix over IF[*], which is the matrixassociated with the sink V4 in the current adaptation of thematrix completion approach. The matrix is nonsingular,because the guaranteed existence of an IF-Iinear multicastsubject to (12) on N' says that there is a way to replace eachindeterminate Xd,e by a scalar kd,e such that each resulting(IEI-OJ)x(IEI-OJ) matrix over IF is nonsingular. The currentadaptation ofthe matrix completion approach then starts withthe t5+ 1 (IEI-OJ)x(IEI-OJ) matrices over IF[*], where J is thenumber of sinks in N. In this way, an IF-Iinear multicastsubject to (12) on N' is constructed at the complexity of0« t5+1)·(IEI-OJ)310g(IEI-OJ)) = O(J·IE1310gIE!).

ACKNOWLEDGMENT

The authors appreciate helpful comments on Section IVby Sidharth Jaggi and Siu Ting Ho, as well as beneficialsuggestions by the reviewers. Work of the first author wassupported in part by GRF grant 413806, GRF grant 414307,CRF grant CUHK2/06C, and NSFC-RGC joint researchgrant N_CUHK411/07 from the Research Grants Council ofthe HKSAR, China.

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