and s

andq=2:This leads to minor improvements in (4)–(6)in the coefficient of the term arising from the case k =0.For P (m;2)and the small set of Kasami sequences of length L 01we have approximately equal maximum even correlation.The Kasami set has considerably fewer sequences,however,the best known upper bound (see [7])for their maximum aperiodic correlation has 6p 2=as the coefficient of pq ln (4L= )where we have 8= .R EFERENCES[1] A.Barg,“On small families of sequences with low periodic correlation,”in Lecture Notes in Computer Science ,vol.781.Berlin,Germany:Springer-Verlag,1994,pp.154–158.[2]S.Boztas,R.Hammons,and P.V.Kumar,“4-phase sequences withnear optimum correlation properties,”IEEE rm.Theory ,vol.38,pp.1101–1113,May 1992.[3]T.Helleseth and P.V.Kumar,“Sequences with low correlation,”inHandbook of Coding Theory ,R.Brualdi,C.Huffman,and V.Pless,Eds.,preprint.[4]T.Helleseth,P.V.Kumar,O.Moreno,and A.G.Shanbhag,“Improvedestimates for the minimum distance of weighted degree Z 4trace codes,”in Proc.1995IEEE Int.Symp.on Information Theory (Whistler,B.C.,Canada,Sept.17–22,1995).[5]S.M.Krone and D.V.Sarwate,“Quadriphase sequences for spread-spectrum multiple-access communication,”IEEE rm.Theory ,vol.30,pp.520–529,May 1984.[6]P.V.Kumar,T.Helleseth,and A.R.Calderbank,“An upper bound forWeil exponential sums over Galois rings and applications,”IEEE rm.Theory ,vol.41,pp.456–468,Mar.1995.[7]htonen,“On the odd and the aperiodic correlation propertiesof the Kasami sequences,”IEEE rm.Theory ,vol.41,pp.1506–1508,Sept.1995.[8]S.Litsyn and A.Tiet¨a v¨a inen,“Character sum constructions of con-strained error-correcting codes,”Appl.Algebra in Eng.,Commun.andComp.,vol.5,pp.45–51,1994.[9] A.A.Nechaev,“Kerdock code in a cyclic form,”Discr.Math.Appl.,vol.1,pp.365–384,1991.[10] D.V.Sarwate,“An upper bound on the aperiodic autocorrelationfunction for a maximal-length sequence,”IEEE rm.Theory ,vol.IT-30,pp.685–687,July 1984.[11] A.G.Shanbhag,P.V.Kumar,and T.Helleseth,“An upper bound for theaperiodic correlation of weighted-degree CDMA sequences,”in Proc.1995IEEE Int.Symp.on Information Theory (Whistler,B.C.,Canada,Sept.17–22,1995).[12],“Improved binary codes and sequence families from Z 4-linear codes,”IEEE rm.Theory ,vol.42,pp.1582–1587,Sept.1996.[13]H.Tarnanen,“An elementary proof to the weight distribution formulaof the first order shortened Reed–Muller coset code,”preprint.[14]P.Udaya and M.U.Siddiqi,“Optimal biphase sequences with largelinear complexity derived from sequences over Z 4,”IEEE rm.Theory ,vol.42,pp.206–217,Jan.1996.[15]I.M.Vinogradov,Elements of Number Theory.New York:Dover,1954.New Construction for Families of Binary Sequenceswith Optimal Correlation PropertiesJong-Seon No,Kyeongcheol Yang,Member,IEEE ,Habong Chung,Member,IEEE ,and Hong-Yeop Song,Member,IEEEAbstract—In this correspondence,we present a construction,in a closed form,for an optimal family of 2m binary sequences of period 22m 01with respect to Welch’s bound,whenever there exists a balanced binary sequence of period 2m 01with ideal autocorrelation property using the trace function.This construction enables us to reinterpret a small set of Kasami and No sequences as a family constructed from m -sequences.New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period,Hall’s sextic residue sequences,and miscellaneous sequences of unknown type.In addition,we enumerate the number of distinct families of binary sequences,which are constructed from a given binary sequence by this method.Index Terms—Kasami sequences,Legendre sequences,No sequences,optimal correlation property,signature sequences.I.I NTRODUCTIONCode-division multiple access (CDMA)systems use pseudonoise binary sequences as signature sequences,and several spread-spectrum communication systems also use them as spreading codes for low probability of intercept [18],[20].Desirable characteristics of a family of binary sequences for such applications include long-period,low out-of-phase autocorrelation values,low crosscorrelation values,low nontrivial partial-period correlation values,large linear span,balance of symbols,large family size,and ease of implementation.A binary (0or 1)sequence f b (t );t =0;1;111;N 01g of period N =2n 01is called balanced if the number of 1’s is one more than the number of 0’s [8].It is said to have the ideal autocorrelation property if its periodic autocorrelation function R ( )is given byR ( )=if there exists an integer such that c(t)=b(t+ )for all t. Otherwise,they are said to be cyclically distinct.For an integer r, the sequence f c(t)g is called the decimation by r of the sequence f b(t)g if c(t)=b(rt)for any integer t.It is easily checked that the period of f c(t)=b(rt)g is given by N divided by gcd(r;N).It is also well known that if a sequence f b(t)g of period N has the ideal autocorrelation property,so does its decimation f b(rt)g by r,where r is an integer relatively prime to N.Two sequences f b(t)g and f c(t)g are said to be equivalent if there are some integers r and such that c(t+ )=b(rt)for all t.They are said to be inequivalent,otherwise. Consider a set of J binary sequences,each with period N,denoted byf v(j)(t);t=0;1;111;N01g;j=1;2;111;J: The periodic crosscorrelation R jk( )at shift between two se-quences f v(j)(t)g and f v(k)(t)g from this collection is defined asR jk( )=N01(t):T h e m a x i m u m o u t-o f-p h a s e p e r i o d i c a u t o c o r r e l a t i o n m a g n i t u d eR Af o r t h i s s ig n a l s e t i s d efin e d a sR A=maxjmax0< <Nj R jj( )jand the maximum crosscorrelation magnitude R C between sequences in this set is given byR C=maxj=kmax0 <Nj R jk( )j:The criterion for signal design is to minimizeR max=max(R A;R C):In signal design,the Welch bound and the Sidelnikov bound are used to test the optimality of sequence sets.Some of well-known optimal families of binary sequences include Gold sequences[6], Kasami sequences[18],[20],bent sequences[12],[20],and No sequences[15].Gold sequences form an optimal set with respect to Sidelnikov’s bound[19]which states that for any set of N or more binary sequences of period NR max>(2N02)1=2:The small set of Kasami sequences is an optimal collection of binary sequences with respect to Welch’s bound[22],which implies thatR max 1+2n=2when it is applied to a set of2n=2sequences of period N=2n01 for an even integer n.Bent and No sequences also form an optimal set with respect to Welch’s bound,respectively,but they have larger linear spans than Gold sequences and Kasami sequences.In this correspondence,we show that if a binary sequence of period 2m01in a trace expression has the ideal autocorrelation property, it can be used to construct,in a closed form,a family of2m binary sequences of period22m01with optimal correlation with respect to Welch’s bound.This construction method enables us to reinterpret the small set of Kasami sequences as well as the No sequences as a family constructed from the m-sequences.New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period,Hall’s sextic residue sequences,and miscellaneous sequences of unknown type.In addition,we enumerate the number of distinct families of binary sequences,which are constructed from a given binary sequence by this method.This correspondence is organized as follows.In Section II,we present the main results to construct an optimal family of binary sequences with respect to Welch’s bound.In Section III,the small set of Kasami sequences and the No sequences are reinterpreted asa family constructed from the m-sequences.New optimal familiesof binary sequences are constructed from the Legendre sequences of Mersenne prime period in Section IV.Hall’s sextic residue sequences and miscellaneous sequences of unknown type are also considered in Section IV.II.C ONSTRUCTION OF A F AMILY OF B INARY S EQUENCESWITH O PTIMAL C ORRELATIONLet q be a prime power and F q be thefinitefield with q elements. Let n=em>1for some positive integers e and m.Then the trace function tr n m(1)from F2is a mapping[10],[11] given bytr n m(x)=e01a n d s e t = T w h e r e T=(2n01)=(2m01).Assume that for an index set I,the sequence f b(t1);t1=0;1;111;M01g of period M=2m01given byb(t1)=a2Itr m1([tr n m( t)]ar)also has the ideal autocorrelation property.Based on the idea of extension in Theorem1,we will provide a method to construct an optimal family of2m binary sequences of period22m01from a given binary sequence of period2m01 with ideal autocorrelation property.Throughout the correspondence, we use the following notations.Let m and n be positive integers such that m j n.Let be a primitive element in F201)=(2.From now on,we assume that the sequence f b(t1);t1=0;1;111;M01g of period M=2m01given byb(t1)=a2Itr m1([tr n m( 2t)+ j t]ar);for j2F2Then the family F is an optimal set of2m binary sequences of period N with respect to Welch’s bound.Furthermore,R ij( )takes only a value01,2m01,or02m01for any i,j,and except for the case where i=j and 0(mod N):Proof:We will show that the possible values of R ij( )are01, 2m01,or02m01for any i,j,and except for the case where i=j and 0(mod N):Let T=2m+1.Since gcd(2m01;T)=1, any integer t,0 t 22m02,can be uniquely written as t=t1T+t2(2m01);0 t1 2m02;0 t2 2m: Then each sequence f s(i)(t);t=0;1;111;2n02g becomess(i)(t)=+2t01g)+ i t(2=f22F2(2a2Itr m1f 2arta2Itr m1f 2ar(t)[f( j;t2+ 2)]ar g where ,0 22m02,is also uniquely written as= 1T+ 2(2m01);0 1 2m02;0 2 2m: Thus we get the equation at the bottom of this page.Note that the inner sum2t([f( )]+[ ;t)])) yields2m01whenf( i;t2)= 2f( j;t2+ 2)we claim that the inner sum is01.If either f( i;t2)=0or f( j;t2+ 2)=0,the exponent to(01)in the inner sum is essentially a shift of the sequence f b(2rt1)g.Since gcd(2r;M)=1, it is obvious that the sequence f b(2rt1)g is balanced and has the ideal autocorrelation property.This implies that the inner sum gives 01.On the other hand,if f( i;t2)=0and f( j;t2+ 2)=0, the inner sum is the autocorrelation of the sequence f b(2rt1)g at a nonzero shift(mod N),so it is01by the assumption.Thus the inner sum always yields01iff( i;t2)= 2f( j;t2+ 2)g: Then we haveR ij( )=(2m01)1j3j+(01)1(2m+10j3j)=2m j3j0(2m+1):(2) By defining x= 2t01)and u= 2 01),we have j3j=jf x j x+x2(ux+u2+ j)gj: Note that x2F2nf0g and x2=x,so we getx2(2= 02t01)=x01: Similarly,we have u2(ux+(ux)01+ j)gj=jf x j x2+1+ i x= 2T h e o r e m3:L e t k=2n,and let be a primitive element of F201)=(201)=(2a2Itr m1([tr n m([tr k n( 2t)+ j t]u)]ar)for j2F2a2Itr m1([tr n m( tt=0(01)s(t+ )=2t 02([f( )]+[ ;t)])):Then each sequence f s(j)(t)g becomess(j)(t)=+2t01))+ j t(2=(22F2(2a2Itr m1([tr n m( 2ut;if f( j;t3)=0c(2u(t2+l));if f( j;t3)= 2l.Since f c(2ut2);t2=0;1;111;2n02g is the decimation by2u of f c(t2)g,it also has the ideal autocorrelation property.Hence,similar arguments as in the proof of Theorem2complete theproof.a2Itr m1([tr n m( t)]ar)=a2Ia2Ia2Itr m1([tr n m([tr k n( 2t)+ j t]u)]ar)=ge r o rm o r s(j)(t)in Theorem3can be given as follows:s(j)(t)=([t r m([111tr m([tr k m)111]r)]a r01f o r e a c h i=1;2;111;d.In signal design for CDMA,it is desirable to have a lot of distinct families of binary sequences with optimal correlation for a given period.Hence it is an interesting problem tofind the number of distinct families of binary sequences constructed from a given binary sequence by Theorem2.Two families F A and F B of sequences of the same period are said to be equivalent if each sequence in F A is a cyclic shift of some sequence in F B,and vice versa.Otherwise,they are said to be distinct.Furthermore,they are said to be fully distinct if each sequence in F A is cyclically distinct from every sequence in F B. For an integer M=2m01,define the cyclotomic coset C i of an integer i,0 i M01,byC i=f j j0 j M01;j i2l(mod)M for some integer l 0g:For the sake of convenience,the cyclotomic coset representative of C i is often defined as the least integer in C i.It is easily checked that either C i=C j or C i\C j= .Hence the set f0;1;111;M01g is partitioned into pairwise disjoint cyclotomic cosets,that is,f0;1;111;M01g=t m a n d n b e p o s i t i v e i n t e g e r s s u c h t h a tn=2m.Let be a primitive element of F2.Let f b(t1);t1=0;1;111;M01g be a binary m-sequence of period M=2m01,given byb(t1)=tr m1( twhere f s (j )(t )g is the sequence of period N given bys (j )(t )=tr m 1f [tr n m ( 2t )+ j t ]rgfor j 2F2such that[(M 01)=2m ]01i =0tr n 1( ut)(7)is exactly the Legendre sequence given in (6).Consider a decimation f s (u l t )g by u l of the sequence f s (t )g given in (7).Clearly,if l is an even integer,then f s (u l t )g is the Legendre sequence given in (6).It is also easy to show that if l is an odd integer,then f s (u l t )g is the sequence given bys (u l t )=a n d s e t= Tw h e r e T =2m+1.For an integer r ,1 r M 01,let f s (j )(t );t =0;1;111;N 01g be the sequence of period N =2n 01given bys(j )(t )=[(M 01)=2m ]01:Then the family F defined byF =ff s (j )(t )gj j =1;2;111;2m gis an optimal set of 2m binary sequences of period N =2n 01with respect to Welch’s bound.Consider the number N fam of fully distinct families F of 2m binary sequences of period N =2n 01constructed from the Legendre sequence of period M =2m 01by Theorem 7.Since we haveI=for a primitive element u in Z M ,it is easy to check that N I =2.Hence we getN fam =2'(N )n:Remark 8:By Theorem 3and Remark 4,the Legendre sequences of Mersenne prime period 2m 01can be used to construct optimal families of period 2k 01,where k is any even multiple of m..The sequence f b (t 1);t 1=0;1;111;126g given byb (t 1)=8)=8,w e d e fin es(j )(t )=8B .N e w F a m i l i e s f r o m H a l l ’s S e x t i c R e s i d u e S e q u e n c eB i n a r y s e q u e n c e s o f p e r i o d M =2m01with ideal autocorrelation property associated with Hall’s difference set appears only when m is 5,7,and 17[1],[9].They are known as the Hall’s sextic residue sequences.In the case that m =5,the Hall’s sextic residue sequences are exactly the m -sequences of period 31.Let m be one of 5,7,or 17,and set M =2m 01.Let u be a primitive element in Z M ,and let be a primitive element of F 2i =0tr m 1( utNote that its decimation by any integer r also has the ideal auto-correlation property.Hence,f b(t1)g and all of its decimations are called the Hall’s sextic residue sequences.Applying Theorem2to f b(t1)g,we have an optimal family with respect to Welch’s bound in the following.Theorem10:Let n=2m,where m is one of5,7,or17,and let u be a primitive element in Z M with M=2m01.Let be a primitive element of F2i=0tr m1f[tr n m( 2t)+ j t]ru g;for j2F2I I IISa f an a a a f2n01is a very interesting problem in both theory and practice[7],[8]. Especially,the balanced binary sequences of period2n01with ideal autocorrelation propertyfind many applications in spread-spectrum communication systems.A complete search for those sequences was conducted for period127by Baumert and Fredrickson[2],255by Cheng[4],and511by Drier[5].It is well known that there are six inequivalent binary sequences of period127with ideal autocorrelation property:an m-sequence,a Legendre sequence,a Hall’s sextic residue sequence,and three others called the miscellaneous sequences of unknown type I,II,and III. Let be a primitive element of F2)+tr71( 5t)+tr71( 11t):ii)Unknown Type IIb II(t1)=tr71( t)+tr71( 7t)+tr71( 29t)+tr71( 9t) w h e r e t1r u n s f r o m0t o126.B y T h e o r e m2,n e w o p t i m a l f a m i l i e s c a n b e c o n s t r u c t e d a b o v e s e q u e n c e s o f u n k n o w n t y p e.F o r e x a m p l e,c o n s i d e r f r o m t h e s e q u e n c ef b III(t1)g.Let n=2m=14.Let be a primitiveelement of F2and set = T where T=27+1.For any integerr,1 r 126,let f s(j)(t);t=0;1;111;N01g be the sequenceof period N=21401given bys(j)(t)=a n d I=f1;9;13g.Then the family F defined byF=ff s(j)(t)gj j=1;2;111;128gis an optimal set of128binary sequences of period N=21401withrespect to Welch’s bound.It is easily checked that N I='(127)=7=18.Hence we have N fam=18'(21401)=14=13608optimalfamilies from a binary sequence of each miscellaneous type.At period255,it is found that there are four inequivalent binarysequences with ideal autocorrelation property:an m-sequence,aGMW sequence,and two others of unknown type.New optimalfamilies can be constructed from a binary sequence of each unknowntype.Note that N I='(255)=8=16in this case.At period511,there arefive inequivalent binary sequences withideal autocorrelation property:an m-sequence,a GMW sequence,and three others of unknown type.New optimal families can beconstructed from a binary sequence of each unknown type.Note thatN I='(511)=9=48in this case.In the case of period1023,a computer search found that thereis at least one binary sequence f b(t1);t1=0;1;111;1022g withideal autocorrelation property,which is inequivalent to any of knownbinary sequences such as the m-sequences,the GMW sequences,andthe extensions of the Legendre sequences.It is given byb(t1)=tr101( t)+tr101( 57t)+tr101( 121t[11] F.J.MacWilliams and N.J.A.Sloane,The Theory of Error-CorrectingCodes.Amsterdam,The Netherlands:North-Holland,1977.[12]J.D.Olsen,R.A.Scholtz,and L.R.Welch,“Bent-function sequences,”IEEE rm.Theory,vol.IT-28,pp.858–864,Nov.1982. [13]J.-S.No,“A new family of binary pseudorandom sequences havingoptimal periodic correlation properties and large linear span,”Ph.D.dissertation,Univ.of Southern California,Los Angeles,CA,May1988.[14],“Generalization of GMW sequences and No sequences,”IEEErm.Theory,vol.42,pp.260–262,Jan.1996.[15]J.-S.No and P.V.Kumar,“A new family of binary pseudorandomsequences having optimal periodic correlation properties and large linear span,”IEEE rm.Theory,vol.35,pp.371–379,Mar.1989.[16]J.-S.No,H.-K.Lee,H.Chung,H.-Y.Song,and K.Yang,“Tracerepresentation of Legendre sequences of Mersenne prime period,”IEEE rm.Theory,vol.42,pp.2254–2255,Nov.1996.[17]J.-S.No,K.Yang,H.Chung,and H.-Y.Song,“Extension of binarysequences with ideal autocorrelation property,”in Proc1996IEEE Int.Symp.on Information Theory and Its Applications(ISITA’96)(Victoria, BC,Canada,Sept.17–20,1996),pp.837–840.[18] D.V.Sarwate and M.B.Pursley,“Crosscorrelation properties of pseu-dorandom and related sequences,”Proc.IEEE,vol.68,pp.593–619, May1980.[19]V.M.Sidelnikov,“On mutual correlation of sequences,”Sov.Math.–Dokl.,vol.12,pp.197–201,1971.[20]M.K.Simon,J.K.Omura,R.A.Scholtz,and B.K.Levitt,SpreadSpectrum Communications,vol. 1.Rockville,MD:Computer Sci.Press,1985.[21]H.Y.Song and S.W.Golomb,“On the existence of cyclic Hadamarddifference sets,”IEEE rm.Theory,vol.40,pp.1266–1268, July1994.[22]L.R.Welch,“Lower bounds on the maximum cross-correlation ofsignals,”IEEE rm.Theory,vol.IT-20,pp.397–399,May 1974.Is Code Equivalence Easy to Decide?Erez Petrank and Ron M.Roth,Member,IEEE Abstract—We study the computational difficulty of deciding whether two matrices generate equivalent linear codes,i.e.,codes that consist of the same codewords up to afixed permutation on the codeword coordinates. We call this problem Code ing techniques from the area of interactive proofs,we show on the one hand,that under the assumption that the polynomial-time hierarchy does not collapse,Code Equivanence is not NP-complete.On the other hand,we present a polynomial-time reduction from the Graph Isomorphism problem to Code Equivalence. Thus if one couldfind an efficient(i.e.,polynomial-time)algorithm for Code Equivalence,then one could settle the long-standing problem of determining whether there is an efficient algorithm for solving Graph Isomorphism.Index Terms—Code Equivalence,Graph Isomorphism,interactive proofs,polynomial hierarchy.I.I NTRODUCTIONLet F be afinitefield and let G1and G2be generator matrices of two linear codes C1and C2over F:We say that G1and G2are code-equivalent,denoted G1 G2,if the sets C1and C2are the Manuscript received November30,1995;revised February12,1997.E.Petrank is with the DIMACS Center,P.O.Box1179,Piscataway,NJ 08855USA.R.M.Roth is with Hewlett-Packard Laboratories,Palo Alto,CA94304 USA,on leave from the Computer Science Department,Technion–Israel Institute of Technology,Haifa32000,Israel.Publisher Item Identifier S0018-9448(97)05216-4.same,up to afixed permutation on the coordinates of the codewordsof C1:In other words,G1 G2if and only if both matrices have the same order k2n,and there exist an n2n permutation matrix P and a nonsingular k2k matrix S over F such that G1=SG2P: The problem of deciding whether two generator matrices are code-equivalent will be referred to as the Code Equivalence problem. The purpose of this correspondence is to study the computational difficulty of the Code Equivalence problem.As one application of a related problem,we mention the public-key cryptosystems due to McEliece[9]and Niederreiter[11].Recall that an alternant code overGF(q)is defined by a parity-check matrix of the form[y j i j]r01;n01i=0;j=0, where the j’s are distinct elements in GF(q m)and the y j’s are nonzero elements in GF(q m)[8,ch.12].Goppa codes are special cases of alternant codes where certain restrictions are imposed on the values y j’s,and generalized Reed–Solomon codes are special cases of alternant codes where m=1:The mentioned cryptosystems are based on the assumption that it is difficult to identify the values j and y j out of an arbitrary generator matrix(or parity-check matrix)of an alternant ly,it is difficult to obtain a code-equivalent matrix of the form[y j i j]r01;n01i=0;j=0:On the other hand,as shown in[12],it is easy to extract the values j and y j from any systematic generator matrix of a generalized Reed–Solomon code; hence,cryptosystems based on such a code are breakable.This was pointed out explicitly by Sidelnikov and Shestakov in[13].For related work,see also the references cited in[10,p.317].The significance of the Code Equivalence problem can also be exhibited through the results of Kasami,Lin,and Peterson[6], and Kolesnik and Mironchikov[7],who showed that Reed–Muller codes are equivalent to subcodes of extended Bose–Chaudhuri–Hocquenghem(BCH)codes.Thus it should be interesting to design an efficient algorithm that decides whether two codes are indeed equivalent,and thus infer from the properties that arise from one code representation to the other.On the positive side,wefirst show that the Code Equivalence problem is unlikely to be NP-complete.The proof of this assertion relies on techniques developed in thefield of interactive proofs, which we summarize in Section II.In Section III,we invoke results of Goldwasser,Micali,and Rackoff[4],Goldreich,Micali,and Wigderson[3],Goldwasser and Sipser[5],and Boppana,H˚a stad, and Zachos[2],to show that if Code Equivalence is NP-complete, then the polynomial hierarchy collapses.Yet,we do state also a negative result,namely,that Code Equiv-alence is also unlikely to be too easy.We do this by relating Code Equivalence to the Graph Isomorphism problem.Let G1=(V;E1) and G2=(V;E2)be two undirected graphs with the same set of vertices V,and with sets of edges E1and E2,respectively. We say that G1is isomorphic to G2if there exists a permutation (isomorphism) :V!V such that f u;v g2E1if and only if f (u); (v)g2E2(we assume here that the graphs have no parallel edges;if they do,then E1and E2are multisets,in which case isomorphism requires equality of the multiplicities of f u;v g and f (u); (v)g in E1and E2,respectively).The problem of deciding efficiently(i.e.,in polynomial time)whether two graphs are isomorphic is a notoriously open question in Computer Science.The problem has been studied extensively in recent decades,but the state of the art is that there is no known efficient algorithm for determining whether two given graphs are isomorphic.In Section IV,we show a polynomial-time reduction from Graph Isomorphism to Code Equivalence.This implies that presenting an0018–9448/97$10.00©1997IEEE。