Generlized Fno nd non-fno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in rxiv:609.0585v [cs.it] 9 Sep 26 Abstrct It is known tht the Fno network hs vector liner solution if nd only if the chrcteristic of the finite field is 2; nd the non-fno network hs vector liner solution if nd only if the chrcteristic of the finite field is not 2. Using these properties of Fno nd non-fno networks it hs been shown tht liner network coding is insufficient. In this pper we generlize the properties of Fno nd non-fno networks. Specificlly, by dding more nodes nd edges to the Fno network, we construct network which hs vector liner solution for ny vector dimension if nd only if the chrcteristic of the finite field belongs to n rbitrry given set of primes {p, p 2,..., p l }. Similrly, by dding more nodes nd edges to the non-fno network, we construct network which hs vector liner solution for ny vector dimension if nd only if the chrcteristic of the finite field does not belong to n rbitrry given set of primes {p, p 2,..., p l }. I. INTRODUCTION Network coding refers to dt trnsmission scheme whereby insted of treting dt symbols s commodity, the intermedite nodes forwrd dt which re functions of incoming dt symbols. By using such trnsmission scheme it hs been shown tht the min-cut upper bound on the cpcity of multicst networks cn be chieved []. Liner network coding refers to the network coding scheme where ll the nodes re restricted to compute only liner functions. More precisely, in liner network coding, the dt symbols generted by the sources re ssumed to belong to finite field sy F q, nd nodes trnsmit F q -liner combintion of the dt symbols it receives. It is known tht the liner coding cpcity of network cn be dependent on the chrcteristic of the finite field [2]. In [2] Dougherty et l. constructed network, nmed s the Fno network, which hs sclr liner solution over ny finite field of chrcteristic 2, but hs no vector liner solution for ny vector dimension over ny finite field of odd chrcteristic. By dding more nodes nd edges to the sme network we show tht for ny set of prime numbers {p, p 2,..., p l }, the resulting network hs sclr liner solution if the chrcteristic belong to the given set of primes) vector liner solution for ny vector dimension if nd only if the chrcteristic of the finite field belong to the given set. Along with the Fno network, Dougherty et l. lso presented the non-fno network in [3] which hs sclr liner solution if the chrcteristic of the finite field is ny prime number other thn 2, but hs not vector liner solution for ny vector dimension if the chrcteristic of the finite field is 2. We use the sme network s sub-network to construct network which for ny given set of prime numbers {p, p 2,..., p l } hs vector liner solution if nd only if the chrcteristic of the finite field does not belong to the given set. Both of the Fno nd the non-fno networks hve been constructed from the Fno nd non-fno mtroid respectively. Considering the equivlence between mtroids nd liner network coding presented in [3] nd [4], our results lso show tht from the Fno nd non-fno mtroid itself, mtroids cn be constructed which re representble if nd only if the chrcteristic of the finite field belong to some rbitrry finite or co-finite set of primes. A combintion of the Fno nd non- Fno network hs been used in [2] to show tht liner network coding is insufficient. The chievble rte region for the Fno nd non-fno networks were described in reference [5]. It is to be noted tht in references [6] nd [7] it hs been lredy shown tht for ny set of prime numbers {p, p 2,..., p l }, there exists network which hs vector liner solution for ny vector dimension if nd only if the chrcteristic of the finite field belong to the given set. The uthors first proved the results for sum-networks, nd then showed tht for every sum-network there exists liner solvbly equivlent multiple-unicst network. Similrly, for ny set of prime numbers {p, p 2,..., p l }, they constructed sum-network which hs vector liner solution if nd only if the chrcteristic of the finite field does not belong to the given set. Invoking the sme liner equivlence between multipleunicst networks nd sum-networks they showed the sme properties hold for multiple-unicst networks. However, to the best of our knowledge this is the first time it is shown tht the Fno-network nd the non-fno network itself cn be used s sub-network to construct network hving the sme properties with n dded desirbility tht these networks re simpler s they hve less number of sources, terminls, nodes nd edges. The orgniztion of the pper is s follows. In Section II we reproduce the stndrd definition of vector liner network coding. In Section III, we present the generlized Fno network, nd in the following section, Section IV, we present the generlized non-fno network. We conclude the pper in Section V. II. PRELIMINARIES We represent network by grph GV, E). Some of the nodes re considered s sources nd some s terminls. It is ssumed tht the sources hve no incoming edges nd the terminls hve no outgoing edges. The set of sources is denoted by S; nd the set of terminls is denoted by T. Every
b c b c u u 2 u 3 u 4 u u 2 u 3 u 4 u 5 u 7 u 6 u 8 e u 5 u 7 u 6 u 8 e u 9 u 9 u 0 u 0 Fig.. solution. u u 2 u 3 c b ) The Fno network u u 2 u 3 u4 c b c b) A modified Fno network A modified Fno network which hs n l-dimensionl vector liner solution if nd only if the Fno network hs n l-dimensionl vector liner source genertes n i.i.d. rndom process uniformly distributed over finite lphbet. In liner network coding this lphbet is ssumed to be finite field F q. Any source process is independent of ll other source processes. The source process generted by source s S is denoted by X s. Ech terminl is required to retrieve the rndom process generted t some source. The symbol crried by n edge e is denoted by Y e. A node v V is clled n intermedite node if it is neither source node nor terminl node. Inv) denotes the set of edges e such tht hede) = v. Outv) denotes the set of edges e such tht tile) = v. The i th edge between nodes v nd v 2 is denoted by v, v 2, i). In cse there is only one edge between v nd v 2, we denote it by v, v 2 ). A k-dimensionl vector liner network coding is defined s follows. If e Outs) for ny s S, then Y e = A s,e X s where Y e, X s F k q nd A s,e F k k q. For ny intermedite node v, if e Outv) nd Inv) = {e, e 2,..., e n }, then Y e = A e,vy e + A e2,vy e2 + + A en,vy en, where Y e, Y e, Y e2,..., Y en F k q nd A ei,v F k k q for i n. Ech terminl t T cn compute block of k symbols from source s Y t = A e,ty e + A e2,ty e2 + + A en,ty en where Int) = {e, e 2,..., e n }, Y e, Y e2,..., Y en F k q nd A ei,t F k k q for i n. If ll terminls, by obeying the bove restrictions, cn compute block of k symbols from their respective sources in k usges of the network, then the network is sid to hve k- dimensionl vector liner solution, or nlogously, the network is sid to be vector linerly solvble for k vector dimensions. In k-dimensionl vector liner solution, k is clled s the vector dimension. When the network hs n -dimensionl vector liner solution the network is sid to be sclr linerly solvble. The mtrices indicted bove by A,b where nd b is either node or n edge re clled s locl coding mtrices. III. GENERALIZED FANO NETWORK In this section, for ny set of prime numbers {p, p 2,..., p l }, network is constructed, by dding more nodes nd edges to the Fno network, which hs vector liner solution for ny messge dimension if nd only if the chrcteristic of the finite field belong to the set {p, p 2,..., p l }. The Fno network shown in [2] ws constructed using n lgorithm given in [3] tht tkes the Fno mtroid s n input. Considering tht there is no unique lgorithm to construct network from mtroid, nd lso tht the construction lgorithm given in [3] itself leves the scope of dding more terminls, we hve found tht modified Fno network which will be shown to be liner solvbly equivlent to the Fno network) to be the one which is the specil cse of the generlized Fno network constructed in this pper.towrds this end, we present modified Fno network which hs n l-dimensionl vector liner solution if nd only if the Fno network hs n l-dimensionl vector liner solution. The Fno network reproduced from [2] is shown in Fig. nd the modified Fno network is shown in Fig. b. Ech of these two networks hs three sources which generte the rndom processes, b nd c respectively. The nodes t the bottom re the terminl nodes. Ech terminl demnds one of the source processes from, b nd c s
indicted in the figures. To note tht the modified Fno network is constructed by dding two edges nd one terminl to the Fno network. Lemm. The network shown in Fig. b hs n l-dimensionl vector liner solution if nd only if the Fno network hs n l-dimensionl vector liner solution. Proof: We first show tht if the Fno network hs n l-dimensionl vector liner solution then the modified Fno network lso hs n l-dimensionl vector liner solution. Here we give the proof for the sclr liner solution, i.e., we prove tht if the Fno network hs sclr liner solution then the modified Fno network lso hs sclr liner solution. Sy Y e = αb+βc. It we cn show tht β 0, then upon receiving b from the direct edge, node u 4 in the modified Fno network cn compute c. Sy β = 0. Since c is computed t u, nd u 2, u 4 ) disconnects the source tht genertes c nd node u, the coefficient multiplying c in Y u2,u 4) cnnot be zero. Hence if β = 0, it mens Y u2,u 4) hs been multiplied by zero, nd Y e = 0. Then, from u 6, u 8 ) solely, must be retrieved by u 2. And hence the coefficient of b nd c in Y u6,u 8) is zero. So b must be retrieved t node u 2 from u 5, u 7 ) solely, nd this indictes tht the coefficient of c in Y u5,u 7) is zero. Therefore, node u cnnot compute c which is contrdiction. So, β 0. The proof for l-dimensionl vector liner solution cn be done in similr wy by tking β s l l mtrix nd showing tht mtrix β hs to be full rnk mtrix. The converse tht if the modified Fno network hs n l-dimensionl vector liner solution then the Fno network lso hs n l-dimensionl vector liner solution is immedite becuse the Fno network is sub- network of the modified Fno network. We now present the generlized Fno network in Fig. 2. The top nodes re sources, nd the sources generte the rndom processes,,,..., b p nd c respectively. The locl coding mtrices in the network re shown longside the edges in cpitl letters. To mintin the clenliness of the figure, some of the direct edges hve been depicted by edge without til node long with the nottion of the source process it is connected to. The prmeter q which determines the number of nodes nd edges in the network cn tke ny integer vlue greter thn or equl to two. Note tht the network shown in Fig 2 reduces to the modified Fno network shown in Fig. b for q = 2. Lemm 2. The network shown in Fig. 2 hs vector liner solution for ny vector dimension if nd only if the chrcteristic of the finite field divides q. Proof: We first show tht the network in Fig. 2 hs vector liner solution only if the chrcteristic of the finite field divides q. Sy, the network hs k dimensionl vector liner solution. Y u,u 3 = D + A i b i ) Y u2,u 4 = B i b i + D 2 c 2) Y u5,u 7 = M e 3 + M 2 e 24 = M D + M A i + M 2 B i )b i + M 2 D 2 c 3) Y u6,u 8 = M 3 e 3 + D 3 c = M 3 D + M 3 A i b i + D 3 c 4) Y ei = W i e 24 + = W i B i b i + U ji b j for i q W i B j + U ji )b j + W i D 2 c 5) Y u9,u 0 = M 4 e 5,7 + M 5 e 6,8 = M 4 M D + M 5 M 3 D ) + M 4 M A i + M 2 B i ) + M 5 M 3 A i )b i + M 4 M 2 D 2 + M 5 D 3 )c 6) Since t computes c, for i q, we hve: Q M D + D 4 = 0 7) Q M A i + M 2 B i ) = 0 8) Q M 2 D 2 = I 9) Since t i for 2 i q computes b i, for i, j q nd j i, we hve: K i M 4 M D + M 5 M 3 D ) = 0 0) K i M 4 M A i + M 2 B i ) + M 5 M 3 A i ) = I ) K i M 4 M A j + M 2 B j ) + M 5 M 3 A j ) + J ji = 0 2) K i M 4 M 2 D 2 + M 5 D 3 ) = 0 3) Since t q+ retrieves, for i q, we hve: Q 2 M 3 D = I 4) Q 2 M 3 A i + R i W i B i + R j W j B i + U ij ) = 0 5) Q 2 D 3 + R i W i D 2 = 0 6) Since t q++i for i q demnds c, for i, j q nd j i, we hve: V i W i B i + E i = 0 7) V i W i B j + U ji ) = 0 8) V i W i D 2 = I 9) From eqution 9), we know tht Q is invertible. Hence from eqution 8), we hve for i q : M A i + M 2 B i = 0 20) From eqution ), we know tht K i is invertible for i q. Hence from 0), we hve, M 4 M D + M 5 M 3 D = 0 2)
b c A A 2 B D B 2 D2 A B u u 2 u 3 u 4 W M M3 M 2 D 3 W W 2 b 3 b 3 U 2 U 3 b U 2 U32 b U ) U2) b q 2 u 5 u 6 U ) U )2 U q 2)) e e 2 e u 7 u 8 M 4 M 5 u 9 D 4 Q u 0 K K 2 K Q 2 R R 2 R b 3 J 2 J 3 b b3 J 2 J 32 b J ) J 2) b q 2 V V 2 V b b J ) J )2 Jq 2)) E E 2 E t t 2 t 3 t q t tq+ q+2 t q+3 t 2q c b c c c Fig. 2. Generlized Fno network: for ny integer q 2, the network is vector linerly solvble for ny vector dimension if nd only if the chrcteristic of the finite field divides q It cn be seen from eqution 4) tht D is lso invertible. Hence from eqution 2), M 4 M + M 5 M 3 = 0 22) Substituting eqution 20) in eqution ), we hve for i q : K i M 5 M 3 A i = I 23) Also from eqution 3), we hve: M 4 M 2 D 2 + M 5 D 3 = 0 24) From eqution 9), it cn be seen tht V i is invertible for i q. Hence, from eqution 8) for i, j q nd j i, W i B j + U ji = 0 25) Substituting eqution 25) in eqution 5) for i q : Q 2 M 3 A i + R i W i B i = 0 26) Since D 2 is invertible becuse of eqution 9), from eqution 6) we hve: 2 + R i W i = 0 27) Note tht since from eqution 23), M 3 A i is invertible nd from eqution 4), Q 2 is invertible, R i W i B i in eqution 26) invertible, nd hence B i nd A i must be invertible for i q. Also note tht M 2 in eqution 9) is invertible. Hence, from eqution 20), M is invertible. Also from eqution 23) M 5 is invertible. Substituting R i W i from eqution 26) in eqution 27) we hve: Q 2 M 3 A i B i = 0 28) Q 2 M 3 A i B i = 0 29)
2 + Q 2 M 3 M M 2) = 0 [ from 20)] 30) Q M 3 M )M 2 = 0 3) Q 2 M 5 M 4M 2 = 0 [ from 22)] 32) Q D 3 D ) = 0 [ from 24)] 33) 2 = 0 34) q) 2 = 0 35) Since Q 2, D 3 nd D 2 re ll full rnk mtrices, it must be tht q = 0. Now note tht over ny finite field of certin chrcteristic, q is zero if nd only if the chrcteristic divides q. We now show tht the network hs sclr liner solution over chrcteristic which divides q. Consider the following messges to be crried by the edges. p Y u,u 3 = + b i Y u2,u 4 = b i + c Y u5,u 7 = Y u,u 3 Y u2,u 4 = c Y u6,u 8 = Y u,u 3 c = + b i c for i q : Y ei = b i + c Y u9,u 0 = Y u6,u 8 Y u5,u 7 = Now, we show tht the terminls cn decode their desired rndom vribles s follows. At terminl t, with the opertion Y u5,u 7, rndom vrible c cn be determined. For i q, the terminl t +i decodes b i s m m= b m p j=,j m b j = b i. Since ll opertions re over the finite field of chrcteristic which divides q, terminl t q+ decodes s Y u6,u 8 p Y e i = qc =. For i q, the terminl t q++i performs the opertion Y ei b i to derive c. Theorem 3. For ny set of prime numbers {p, p 2,..., p l }, there exists network, constructed by dding more nodes nd edges to the Fno network, which hs vector liner solution if nd only if the chrcteristic of the finite field belongs to the given set. Proof: The network in Fig. 2 constructed for q = p r.pr2 2....pr l l, where r, r 2,..., r l Z + is such network. b i IV. GENERALIZED NON-FANO NETWORK In this section, for ny set of prime numbers {p, p 2,..., p l }, we dd more nodes nd edges to the non-fno network, to construct network which hs vector liner solution for ny vector dimension if only if the chrcteristic of the finite field does not belong to the given set. For this purpose, we first construct modified non-fno network, shown in Fig. 3b, which is liner solvbly equivlent to the non-fno network shown in [3] nd reproduced here in Fig 3. The nodes t the top which hve no incoming edges re the sources, nd they not lbelled to reduce clumsiness. The source process generted by source node is indicted bove the node. For ny terminl, the source processes demnded by the terminl re indicted below the terminl. Lemm 4. The modified non-fno network in Fig. 3b hs n l-dimensionl vector liner solution if nd only if the network in Fig. 3 hs n l-dimensionl vector liner solution. Proof: As the terminl t 4 in the modified non-fno network hs demnds which is subset of wht t 4 demnds in the non-fno network, it is self-evident tht if the non-fno network hs n l-dimensionl vector liner solution then the modified non-fno network too hs n l-dimensionl vector liner solution. We now show tht if the modified non-fno network hs sclr liner solution then the non-fno network too hs sclr liner solution. Assume sclr liner solution of modified non-fno network. It cn be seen tht if the coefficient of in Y e is non-zero then cn be retrieved from Y e by the node t 4 in Fig. 3b s it lredy knows. If however, the coefficient of in Y e hd been zero, then the coefficient of in Y e hd lso to be zero, s Y e cnnot be multiplied by zero t node t 2 since t 2 needs to use the informtion in Y e to compute. However, if the coefficient of in Y e is zero, then the node t 3 in Fig. 3b won t be ble to compute. Similr rgument cn be used to derive tht the coefficient of in Y e 2 is non-zero. And hence the node t 4 in Fig. 3b cn compute ll of, nd. The proof for l-dimensionl vector liner solution cn be done in similr wy. We now present the generlized non-fno network in Fig. IV. The source nodes re not lbelled in the figure for the ske of clenliness in the digrm.,,,..., b q re the rndom processes generted by the sources. Note tht for i q, there exists no pth between tile i ) nd the source tht genertes messge b i. Here lso, the prmeter q cn tke ny integer vlue greter thn or equl to two. It cn be verified tht the network shown in Fig. 3b reduces to the modified non-fno network shown in Fig. 3b for q = 2. Lemm 5. The network shown in Fig. IV hs vector liner solution for ny vector dimension if nd only if the chrcteristic of the finite field does not divide q. Proof: We first list the messges crried over by the
00 e e e 2 e b e e e 2 e b t t 2 t 3 t 4,, t t 2 t 3 t 4 ) The non-fno network b) A modified non-fno network Fig. 3. A modified non-fno network which hs n l-dimensionl vector liner solution if nd only if the Fno network hs n l-dimensionl vector liner solution. b q D q D 2 D M C C q C 2 B 2 B 2 A B q A2 B q2 B q A q B 2q e e e 2 e q e b M 2 M3 000 000 000 00 000 000 000 U 2 U R K K R K 2 2 q 000 000 000 R q 0 b q U q M 4 t t 2 t 3 t q+ t q+2 Fig. 4. Generlized non-fno network: for ny integer q 2, the network is vector linerly solvble for ny vector dimension if nd only if the chrcteristic of the finite field does not divide q.
edges. Y e = M + C i b i for i q: Y ei = A i + Y eb = D i b i B ji b j Since, node t computes, for i q we hve: M 2 M = I 36) M 2 C i + M 3 D i = 0 37) Since, node t i+ for i q computes b i, for i, j q, j i we hve: K i M + R i A i = 0 38) K i C i = I 39) K i C j + R i B ji = 0 40) Since, node t q+2 computes, for i q we hve, U i A i = I 4) U j B ij ) + M4 D i = 0 42) Since, from eqution 39), for i q, C i is invertible, nd M 2 is invertible from eqution 36), M 3 D i for i q is invertible from eqution 37), nd hence M 3 is invertible. Also, since both of K i nd C i re invertible for i q becuse of eqution 39), R i nd B ji for i, j q, j i re invertible from eqution 40). Moreover, note tht M is invertible from eqution 36). Now substituting eqution 37) in eqution 42) we get for i q: U j B ij ) M4 M 3 M 2C i = 0 U j R ) j K j C i M4 M3 M 2C i = 0 [from 40] U j A j M C ) i M4 M3 M 2C i = 0 [from 38] U j A j M 2 C i ) M4 M 3 M 2C i = 0 [from 36] U j A j M 4 M3 ) M2 C i = 0 I Ui A i M 4 M3 ) M2 C i = 0 [from 4] I U i A i M 4 M3 = 0 U i A i + M 4 M3 = I U i A i = I M 4 M3 43) Now, substituting eqution 43) in eqution 4) we get: I M 4 M3 ) = I qi qm 4 M 3 = I qm 4 M 3 = q )I 44) Now, if the chrcteristic of the finite field divides q, then q = 0, nd eqution 44) results into 0 = I, which is contrdiction. We now show tht the network in Fig. IV hs sclr liner solution if the chrcteristic of the finite field is does not divides q. Note tht n element in finite field hs n inverse if nd only if the chrcteristics of the finite field does not divide tht element. Consider the following messges to be trnsmitted by the edges: Y e = + b i for i q : Y ei = + Y eb = b i We now show tht the terminls cn compute their respective demnds. The terminl t computes s Y e Y eb =. For i q, the terminl t +i decodes b i by the opertion Y e Y ei. At terminl t q+2, since q hs n inverse in the finite field, q q Y e i q )Y eb ) =. Theorem 6. For ny set of prime numbers {p, p 2,..., p l }, there exists network, constructed by dding more nodes nd edges to the non-fno network, which hs vector liner solution if nd only if the chrcteristic of the finite field does not belong to the given set. Proof: The network in Fig. IV constructed for q = p r.pr2 2....pr l l, where r, r 2,..., r l Z + is such network. V. CONCLUSION The Fno nd non-fno networks hve been used in the literture to show the insufficiency of liner network coding. In this pper, we hve first constructed network, nmed s the generlized Fno network, which for ny set of primes {p, p 2,..., p l }, hs vector liner solution if nd only if the chrcteristic of the finite field belongs to {p, p 2,..., p l }. This network reduces to the known Fno network s specil cse. We hve then constructed network which for ny set of primes {p, p 2,..., p l }, hs vector liner solution if nd only if the chrcteristic of the finite field does not belong to {p, p 2,..., p l }. This network reduces to the non-fno network s specil cse. REFERENCES [] S. R. Li, R. W. Yeung, nd N. Ci, Liner network coding, IEEE Trnsctions on Informtion Theory, 2003. b j
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