Recursive Code Construction for Random Networks

a r X iv:086.365v2[cs.IT]28Jul28Recursive Code Construction for Random Networks Vitaly Skachek Claude Shannon Institute University College Dublin Belfield,Dublin 4,Ireland E-mail:vitaly.skachek@ucd.ie Abstract We present a modification of K¨o tter-Kschischang codes for random networks (these codes were also studied by Wang et al.in the context of authentication problems).The new codes have higher information rate,while maintaining the same error-correcting capabilities.An efficient error-correcting algorithm is presented for these codes.1Background The area of network coding has emerged since the work of Ahlswede et al.[1].It was shown that sending coded information over the network yields an ad-vantage in bandwidth utilization compared with the classical routing sce-nario.Further,the use of network coding for correction of errors in the in-formation sent over the network was suggested in [2].This approach relied,however,on the knowledge of the network topology.In [5],a new approach for error-correcting network coding was suggested.It was assumed that the network topology is not known.Further,the en-coded information was represented by vector subspaces of some vector space.So-called codes for random networks were constructed and a correspond-ing decoder was presented in that work,that uses some ideas from classical1Reed-Solomon codes and decoders.Essentially,the analogous construction was proposed several years earlier in the context of authentication codes[11].In[4]and[9],the connections between matrix codes in the rank metric[3] (see also[7])and the codes for random networks were established.In partic-ular,the code construction in[5](and[11])is a lifting of the matrix codes in[3].A construction of spread codes was presented in[6].Very recently,inde-pendently of the present of work,a generalization of codes in[9]was presented in[8].The respective codes in[8]have higher information rate than the codes in[5].In this work,we modify the construction in[5],[11].Our codes have the same error-correcting capability as the codes in[5],while the information rate of our codes is higher.This construction can be viewed as a generalization of construction in[6].On the other hand,this construction can be viewed as a special case of construction in[8].An efficient decoding algorithm is presented for a special case of our codes,it uses the decoder in[5]as a subroutine.By contrast,no efficient decoding algorithm for the codes in[8] was given.2Notations and Previous ResultsLet W be a vector space over afinitefield F q and let V,U⊆W be linear subspaces of W.We use the notation dim(V)for the dimension of V.We denote the sum of U and V as U+V={u+v:u∈U,v∈V}.If U∩V=∅, then for any w∈U+V there is a unique representation w=u+v,where u∈U and v∈V.In this case we say that U+V is a direct sum,and denote it as U⊕V.It is easy to check that dim(U⊕V)=dim(U)+dim(V).For a vector set S⊆W,we use the notation span(S)to denote a linear span of the vectors in S.We also use the notation0m to denote all-zero vector of length m,for any integer m.Below,we recall the construction in[5].Let F=F q m be an extensionfield of F q,m>1.Then,F is a vector space over F q.Let{α1,α2,···,αℓ}⊆F be a set of linearly independent elements in F.We denoteA =span {α1,α2,···,αℓ}andW= A ⊕F={(v1,v2):v1∈ A ,v2∈F}.2For a vector v ∈W ,sometimes we may write v =(v 1,v 2),where v 1∈ A and v 2∈F (any such vector v can be viewed as an (ℓ+m )-tuple over F q ).For a vector space V ⊆W we define a projection of V on F asV |F ={v 2∈F :(v 1,v 2)∈V }.We use the notation P (W,ℓ)for the set of all subspaces of W of dimension ℓ.For U,V ∈W ,let d (U,V )=dim(U )+dim(V )−2dim(U ∩V )be a distance between U and V in the Grassmanian metric (see [5]).Let F k[x ]denote the set of linearized polynomials over F of degree at most q k −1,k ≥1.Define the mapping E :F k [x ]→P (W,ℓ)asE (f (x ))=span {(α1,f (α1)),···,(αℓ,f (αℓ))} ∈P (W,ℓ).The code K is defined in [5]asK = E (f (x )):f (x )∈F k [x ] .Below,we might sometimes use the notation K [ℓ+m,ℓ,k ]for the code K with the parameters as above.It was shown in [5]that |K|=q mk .It was also shown there,that for all U,V ∈K ,U =V ,it holds that dim(U ∩V )≤k −1,and so d (U,V )≥2(ℓ−k +1).Therefore,the minimum distance of K is at least 2(ℓ−k +1).Singleton-type upper bound on the maximum size of K ,A q (ℓ+m,ℓ,k ),was derived in [5].The bound can be written asA q (ℓ+m,ℓ,k )≤ m +k k q,(1)where m +kk q =k −1 i =0q m +k −i −1ℓk q.3Finally,the Johnson-type bound was presented in[10],as belowA q(ℓ+m,ℓ,k)≤ qℓ+m−1qℓ−1−1··· qℓ+m−k+1−1ℓ)operations over F q m.3Code ConstructionIn this section,we define a new code,based on the construction in[5],[11]. Let m≥ℓ≥k.We use the notation C[ℓ+m,ℓ,k](for a sake of simplicity, sometimes we will use the notation C instead)to define this new random net-work code defined by vector subspaces in the ambient space W of dimension ℓ+m,such that for any V∈C,dim(V)=ℓ,and for any V∈C and U∈C, dim(U∩V)≤k−1.Let N(ℓ+m,ℓ,k)= C[ℓ+m,ℓ,k] for all m,ℓ,k.We pick some code parameters m,ℓand k.Let hℓ+m be an integer, 0≤hℓ+m≤k−1.This hℓ+m is a design parameter which can be optimized later.Next,we recursively define the code C[ℓ+m,ℓ,k].•Boundary condition:m<2(ℓ−hℓ+m)orℓ−hℓ+m<k(in which case K[m,ℓ−hℓ+m,k]does not exist).We define C=K[ℓ+m,ℓ,k].•Recursive step:assume that C[m,ℓ−hℓ+m,k]={Uσ⊆F:σ=1,2,···,N(m,ℓ−hℓ+m,k)}.Let{eσ1,eσ2,···,eσℓ−hℓ+m }⊆F be a basisof Uσ.–If hℓ+m=0,then we set Sσ=∅for allσ∈N.4–Otherwise,forσ=1,2,···,⌊ℓ/hℓ+m⌋,we define sets of vectorsSσ= (αj,0m):j=(σ−1)hℓ+m+1,(σ−1)hℓ+m+2,···,σhℓ+m .In addition,forσ=1,2,···,N(m,ℓ−hℓ+m,k),we defineTσ= (0ℓ,eσj):j=1,2,···,ℓ−hℓ+m .Let.(2) tℓ+m= min{⌊ℓ/hℓ+m⌋,N(m,ℓ−hℓ+m,k)}if hℓ+m>0N(m,ℓ,k)if hℓ+m=0We define vector spaces Vσ∈P(W,ℓ),forσ=1,2,···,tℓ+m,as Vσ=.Finally,we span(Sσ∪Tσ).We also define a set B={Vσ}σ=1,2,···,tℓ+m define a code C as C=K[ℓ+m,ℓ,k]∪B.4Code Parameters4.1Recursive Formula for the Number of Codewords The code C as above is obviously a set of subspaces of dimensionℓin the space W of dimensionℓ+m.The number of codewords in C is given by the recursive relationN(ℓ+m,ℓ,k)=q mk+tℓ+m,(3) where tℓ+m is given by(2).The boundary conditions areN(ℓ+m,ℓ,k)= K[ℓ+m,ℓ,k] =q mk if m<2(ℓ−hℓ+m)orℓ−hℓ+m<k.(4)Please note that if K[m,ℓ−hℓ+m,k]=∅,we get N(m+ℓ,ℓ,k)>q mk,thus having more codewords compared with the construction in[5].Otherwise, we have N(m+ℓ,ℓ,k)=q mk.54.2Optimization of hℓ+mNext,let us discuss the choice of the parameter hℓ+m.We are interested in hℓ+m that maximizes tℓ+m in(2).Note,however,thatN(m,ℓ−hℓ+m,k)≥q(m−ℓ+hℓ+m)·k,where the right-hand side is an increasing function of hℓ+m.Forfixed m,ℓand k,the function⌊ℓ/hℓ+m⌋does not increase with hℓ+m.Therefore,we have the following lemma.Lemma4.1If for some h=0,1,···,k−2,ℓ⌋.The number of codewords corresponding to the selec-h+1tion hℓ+m=h is either⌊ℓ⌋.h+1Theorem4.2Letℓ+m,ℓand k be the parameters of the code C,and let hℓ+m=h be the smallest value that maximizes the number of words in C. In addition,assume that one of the following holds:(C1)ℓ≥4,k=1;(C2)ℓ≥3,k≥2.If h>0,then m<2(ℓ−h),and so C[m,ℓ−h,k]does not exist.Proof.Consider the code C[ℓ+m,ℓ,k].Assume that it is constructed recursively from the code C[m,ℓ−h,k]for some0≤h≤k−1,as it was described in the previous section.The later code exists only ifm≥2(ℓ−h).(5)6Next,assume also that the smallest optimal value of hℓ+m is h>0.Then, hℓ+m=h−1is not an optimal value,and so we have⌊ℓ/h⌋>q(m−ℓ+h−1)·k,due to Lemma4.1(with respect to h−1instead of h).This can be rewritten aslog q(⌊ℓ/h⌋)>(m−ℓ+h−1)·k.We substitute(5)to obtain thatlog q(⌊ℓ/h⌋)>(ℓ−h−1)·k.(6) Next,we obtain a contradiction,by showing that(6)does not hold for any h,1≤h≤ℓ−1.Indeed,consider the following four cases.•Case h>ℓ/2.The LHS of(6)is zero,while the RHS is non-negative.•Case h=1.Then,(6)is rewritten aslog q(ℓ)>(ℓ−2)·k.This is false for allℓand k satisfying either(C1)or(C2).•Case h=ℓ/2.Then,(6)is rewritten aslog q(2)>(ℓ/2−1)·k.This is also false forℓand k satisfying either(C1)or(C2).•Case1<h<ℓ/2.The second derivative of the function k(ℓ−h−1)(ina variable h)is0,while the second variable of the function log q(ℓ/h)(ina variable h)is positive for1<h<ℓ/2.In addition,for the endpointsof that interval,k(ℓ−h−1)≥log q(ℓ/h).Therefore,this relation is also true for the whole interval.We obtain a contradiction.The contradiction follows from the assump-tion(5).Therefore,the only possible situation is that m<2(ℓ−h).74.3Explicit Formula for the Number of CodewordsIf all hℓ+m’s in the construction are zeros for allℓand m,then tℓ+m= N(m,ℓ,k).Therefore,(3)becomesN(ℓ+m,ℓ,k)=q mk+N(m,ℓ,k),and thusN(ℓ+m,ℓ,k)=q mk+q(m−ℓ)k+···+q(r+ℓ)kq(ℓ+m)k−q(r+ℓ)k=Since v∈V,we may write for someσ,1≤σ≤tℓ+m,for some b i∈F q (for i=1,2,···,hℓ+m)and c i∈F q(for i=1,2,···,ℓ−hℓ+m):v=hℓ+mi=1b i·(α(σ−1)hℓ+m+i,0m)+ℓ−hℓ+m i=1c i·(0ℓ,eσi)= hℓ+m i=1b iα(σ−1)hℓ+m+i,ℓ−hℓ+m i=1c i eσi .In particular,we haveℓi=1a iαi=hℓ+m i=1b iα(σ−1)hℓ+m+i.Since allαi’s are linearly independent,it follows thata i=b i−(σ−1)hℓ+m if(σ−1)hℓ+m+1≤i≤σhℓ+m0otherwise. Finally,we obtain that every v∈U∩V can be written asv=hℓ+mi=1b i·(α(σ−1)hℓ+m+i,f(α(σ−1)hℓ+m+i)).Therefore,dim(U∩V)≤q hℓ+m≤q k−1.3.U,V∈B.Take any v∈U∩V,v=0.Since v∈U,we may write forsomeσ,1≤σ≤tℓ+m,for some a i∈F q(for i=1,2,···,hℓ+m)and for some d i∈F q(for i=1,2,···,ℓ−hℓ+m):v=hℓ+mi=1a i·(α(σ−1)hℓ+m+i,0ℓ)+ℓ−hℓ+m i=1d i·(0m,eσi)= hℓ+m i=1a iα(σ−1)hℓ+m+i,ℓ−hℓ+m i=1d i eσi .(7)9Since v∈V,we may write for someτ,1≤τ≤tℓ+m,τ=σ,for some b i∈F q(for i=1,2,···,hℓ+m)and for some c i∈F q(for i= 1,2,···,ℓ−hℓ+m):v=hℓ+mi=1b i·(α(τ−1)hℓ+m+i,0ℓ)+ℓ−hℓ+m i=1c i·(0m,eτi)= hℓ+m i=1b iα(τ−1)hℓ+m+i,ℓ−hℓ+m i=1c i eτi .(8) By comparing thefirst part of v in(7)and(8),we obtainhℓ+mi=1a iα(σ−1)hℓ+m+i=hℓ+m i=1b iα(τ−1)hℓ+m+i.However,sinceα(σ−1)hℓ+m+i ’s andα(τ−1)hℓ+m+i’s are all different andlinearly independent,we obtain that all a i=b i=0for i=1,2,···,hℓ+m.From the second part of v in(7)and(8),we obtainℓ−hℓ+mi=1d i·eσi=ℓ−hℓ+m i=1c i·eτi.Therefore,v=(0ℓ,u),where u∈Uσ∩Uτ.Since,for allσandτ, dim(Uσ∩Uτ)≤k−1,it follows that dim(U∩V)≤k−1.5Decoding5.1Simple CaseBelow,we present a recursive decoding algorithm for the code C,when h i=0 (for all i).A decoding algorithm for the code K was presented in[5].Suppose that V∈K is transmitted over the operator channel(see[5]for details).Suppose also that an(ℓ−κ+γ)-dimensional subspace U of W is received,where dim(U∩V)=ℓ−κ.We use a modification of the decoding algorithm in[5]10as follows.Given a received vector space U of W,the decoder is able to recover a single V∈K wheneverκ+γ<ℓ−k+1.If the decoding fails,the decoder returns a special error message‘?’(such a modification is straight-forward).We will denote this decoder D Kℓ+m,ℓ,k:W→K[ℓ+m,ℓ,k]∪{‘?’}.Now,suppose that V∈C is transmitted over the operator channel,and an (ℓ−κ+γ)-dimensional subspace U of W is received,and dim(U∩V)=ℓ−κ. We will denote the decoder for the code C as Dℓ+m,ℓ,k:W→C[ℓ+m,ℓ,k]∪{‘?’}.The decoder is summarized in Figure1.If the decoding fails,the decoder returns an error message‘?’.As we show in the sequel,the decoder Dℓ+m,ℓ,k is able to recover V∈C from U∈W given thatκ+γ<ℓ−k+1.Figure1:Decoder Dℓ+m,ℓ,k for the code C.Theorem5.1Let V be transmitted over the operator channel and let U be received.In addition,let dim(V)=ℓ,dim(U)=ℓ−κ+γ,dim(U∩V)=ℓ−κ.Then,the decoder in Figure1is able to recover the original codeword V∈C from U given thatκ+γ<ℓ−k+1.Proof.There are two cases.If V∈K,then the claim follows immediately from the correctness of the decoder in[5].Therefore,we assume that V∈B. In that case,by the definition of B,V=span {(0,eσ1),(0,eσ2),···,(0,eσℓ)} ={0ℓ}⊕V′,11whereV′=span {eσ1,eσ2,···,eσℓ} ∈C[m,ℓ,k]for someσ.In particular,V is isomorphic to V′and dim(V)=dim(V′).Let U′=U|F.Obviously,dim(U′)≤dim(U)=ℓ−κ+γ.We also have thatℓ−κ=dim(U∩V)≤dim(U′∩V′)(in particular,for any v∈U∩V,the projection of v on its last m coordinates lies in U′∩V′,and for v=u∈U∩V, the projections of u and v on the last m coordinates yield different vectors, since the only puctured coordinates are zero-coordinates).We haved(U′,V′)=dim(U′)+dim(V′)−2dim(U′∩V′)=(dim(U′)−dim(U′∩V′))+(dim(V′)−dim(U′∩V′))≤((ℓ−κ+γ)−(ℓ−κ))+(ℓ−(ℓ−κ))=κ+γ.Note that V′∈C[m,ℓ,k].The decoder D m,ℓ,k is able to correct any error pattern of size less thanℓ−k+1.Therefore,given thatκ+γ<ℓ−k+1, the decoder D m,ℓ,k will recover V′from U′.Finally,V is easily obtained as {0ℓ}⊕V′.Next,we turn to estimate the time complexity of the decoder Dℓ+m,ℓ,k.We denote the decoding time of this decoder applied to C[ℓ+m,ℓ,k]as T(ℓ+m,ℓ). Recall,that the time complexity of the algorithm D Kℓ+m,ℓ,k was shown in[5]to be O((ℓ+m)2)operations over F q m(sinceℓ≤m,this complexity is O(m2)). Then,the following recurrent relation holds:T(ℓ+m,ℓ)≤O (ℓ+m)2 +T(m,ℓ).The boundary condition is T(ℓ+m,ℓ)=O((ℓ+m)2)when m<2ℓ.We obtain thatT(ℓ+m,ℓ)=O (ℓ+m)2·mℓ .5.2General CaseConsider the case where hℓ+m=0for some m,ℓ.In this case,|B|≤ℓ.Then, the algorithm in Figure1can be modified as follows.First,the decoder D Kℓ+m,ℓ,k is applied to the input U∈W.If that decoder fails,then V/∈K,12and so one should look for V∈B such that d(U,V)<ℓ−k+1.This can be done by checking at most O(ℓ)“candidates”V∈B.If one of them lies at the distance less thanℓ−k+1from U,then this V is the codeword to be returned.If there is no such V,the failure‘?’is returned.The resulting time complexity is a sum of the two following values:•Time complexity of the decoder D Kℓ+m,ℓ,k.•Time complexity of at mostℓapplications of the algorithm for com-puting distances between two vector subspaces.Computing the distances between two vector subspaces can be done in a straight-forward manner by calculating the dimension of their intersection. It can also be done(for q=2)by an algorithm presented in[8,Sec.3.4]. AcknowledgementsThis work was supported in part by the Claude Shannon Institute for Discrete Mathematics,Coding and Cryptography(Science Foundation Ireland Grant 06/MI/006).The author would like to thank his colleagues in the Claude Shannon Institute:Carl Bracken,Eimear Byrne,Marcus Greferath,Russell Higgs,Nadya Markin,Gary McGuire and Alexey Zaytsev for stimulating discussions and advices.The author would also like to thank Emina Soljanin for helpful discussions.References[1]R.Ahlswede,N.Cai,S.Y.R.Li,and R.W.Yeung,Networkinformationflow,IEEE rm.Theory,vol.46,no.4,pp.1204–1216,July2000.[2]N.Cai,R.W.Yeung,Network coding and error correction,Proc.IEEE Inform.Theory Workshop,pp.119-122,Oct.2002.[3]E.M.Gabidulin,Theory of codes with maximal rank distance,Prob-lems of Information Transmission,vol.21,pp.1-12,July1985.13[4]M.Gadouleau,Z.Yan,Constant-rank codes and their connectionto constant-dimension codes,submitted to IEEE Trans.On Inform.Theory,available online at /abs/0803.2262.[5]R.K¨o tter,F.R.Kschischang,Coding for errors and erasures inrandom network coding,submitted to IEEE Trans.On Inform.Theory, available as arXiv report arXiv:cs/0703061v2.[6]F.Manganiello,E.Gorla,and J.Rosenthal,Spread codes andspread decoding in network coding,to appear in Proc.IEEE Intern.Symposium on Inform.Theory,Toronto,Canada,July2008.[7]R.M.Roth,Maximum-rank array codes and their application to criss-cross error correction,IEEE rm.Theory,vol.37,no.2,pp.328-336,March1991.[8]N.Silberstein,T.Etzion,Coding Theory in Projective Space,Ph.D.Research Proposal,Technion,Haifa,Israel,May2008,available online at /abs/0805.3528.[9]D.Silva, F.R.Kschischang,and R.K¨o tter,A rank-metric approach to error-control in random network coding,sub-mitted to IEEE Trans.on Inform.Theory,available online at /abs/0711.0708.[10]S.T.Xia, F.W.Fu,Johnson type bounds on constant dimensioncodes,submitted,available online at /abs/0709.1074.[11]H.W ang,C.Xing,and R.Safavi-Naini,Linear authenticationcodes:bounds and constructions,IEEE Trans.On Inform.Theory,vol.49,pp.866–873,April2003.14。