Graph-Based Classification of Self-Dual Additive Codes over Finite Fields

GRAPH-BASED CLASSIFICA TION OF SELF-DUAL ADDITIVE CODES OVER FINIT E FIELDS Lars Eirik D anielsen Departmen t of Informatics Unive rsity of Be rgen PO Bo x 7803, N-5020 Bergen, Norw a y Abstract. Quan tum sta bilizer st ates o v e r F m can be represen t ed as self-dual additiv e co des ov er F m 2 . These codes can be r epresen ted as we igh ted graphs, and orbits of graphs under the generalized lo cal complement ation op eration correspond to equiv alence classes of co des. W e hav e previously used this fact to classify self-dual additiv e codes ov er F 4 . In this paper we classify self - dual additive co des ov er F 9 , F 16 , and F 25 . A ssuming that the classical MDS conjecture holds, w e are able to classify all self-dual additive MDS codes o v er F 9 b y using an exte nsion techn ique. W e pro v e that the m inimum distance of a self- dual additive co de is related to the minim um vertex degree in the associated graph orbit. Cir culan t graph codes are i n troduced, and a computer searc h reveals that this set con tains many strong co des. W e sho w that s ome of these codes ha v e highly regular graph represen tat ions. 1. Introduction It is well-known that self-ortho gonal additive c o des o v er F 4 can b e us ed to r epre- sent a class of quantum err or-c orr e cting c o des known a s binary stabilizer c o des [4]. Although the bina ry stabilizer co des hav e b een studied mos t, several a utho r s hav e considered nonbin ary s tabilizer co des ov er finite fields [1, 11, 12, 16, 22, 25], cyclic groups [14], and Abelian gro ups in gene r al [26]. W e will focus mainly on co des ov er finite fields, and exploit the fact that a stabilizer code ov er F m corres p onds to a self-orthog onal additive c o de ov er F m 2 . Quantum co des of dimension zero ar e known as s t abilize r states , which are en tangled qua ntum states with several po ssible applications. Stabilizer states corr espo nd to self-dual additive co des. It is known that such codes can be represented as graphs [12, 26]. It is also kno wn that t wo se lf- dual additive co des ov e r F 4 are equiv a len t if and only if their corr espo nding graphs are equiv alent, up to is o morphism, with resp ect to a sequence of lo c al c omplemen- tations [3, 9, 20, 21]. W e have previous ly used this fact to devise a graph-ba s ed algorithm with which we classified a ll self-dual additive co des ov er F 4 of leng th up to 12 [7]. Recent ly , the representation of equiv alence classes as g raph o r bits w as generalized to self-dual additive co des over any finite field [2]. In this pape r we use graph-ba s ed algorithms to classify a ll self-dual additive co des ov er F 9 , F 16 , and F 25 up to lengths 8, 6, and 6 , resp ectively . W e also give upp er b ounds on the num b er of co des, derived fr o m mass formulas . By using a graph extensio n tec hnique we find 2000 Mathematics Subje ct Classific ation: Primary: 94B05, 05C30; Secondary: 94B60, 05C50. Key wor ds and phr ases: Additiv e codes, self- dual codes, quantum co des, stabilizer states, graphs, local complemen tation, circulant codes, MDS codes. The author is supported by the Research Council of Norw a y . 1 2 Lars Ei rik Danielsen that ther e are only three non-trivial self-dual a dditiv e MDS co des ov er F 9 , assuming that the cla ssical MDS conjecture holds. W e prove that the minimum distance of a self-dual additive co de is related to the minim um vertex degr ee in the as so ciated graph orbit. Finally , we p erform a search of cir culant gra ph co des, a sub c la ss o f the self-dual additive co des, which is shown to contain many codes with high minimum distance. The highly regula r graph structures of some of these co des a re descr ibed. 2. St abilizer st a tes Data in a classica l computer are typically stored in bits that hav e v alues either 0 or 1. Similarly , w e can envisage a quantum co mputer where data are stored in quantum bits, also known as qubits , i.e., t wo-level quan tum systems. One qubit can then be describ ed b y a vector | x i =  α β  ∈ C 2 , wher e | α | 2 is the probabilit y of observing the v alue 0 when we measure the qubit, and | β | 2 is the probability of observing the v a lue 1. More generally , data could b e stored in m -level qu dits , describ ed by v ector s from C m . Meas uring such a qudit w o uld give a result from an alphab et with m symbols. In genera l, this alphab et could b e a n y finite Ab elian group, but w e will focus on the case where the a lphabet is a finite field. The m vectors | x i , x ∈ F m , form a n or thonormal ba sis of C m . An err or op erato r that can affect a single qudit is repre s en ted by a complex unitary m × m matrix, i.e ., a matrix U suc h that U U † = I , where † means co njugate transp ose. A state of n qudits is represented b y a v e ctor from C m n = C m ⊗ · · · ⊗ C m . Assuming that er r ors act indep enden tly on each qubit, this state is affected b y err or op erators describ ed by n -fold tensor pro ducts o f unitary m × m matric es. In the case of qubits ( m = 2), we only need to cons ider erro rs fro m the Pauli gro u p , X =  0 1 1 0  , Z =  1 0 0 − 1  , Y = iX Z =  0 − i i 0  , I =  1 0 0 1  , due to the fact that these ma trices form a bas is of all unitary 2 × 2 matrice s . The error X is called a bit-flip , since X | x i = | x + 1 i . The erro r Z is known a s a phase- flip , since Z | x i = ( − 1) x | x i . F or genera l qudits that take their v alues from F m , we consider the gener alize d Pauli gr oup , P m , a lso known as the discr ete Heisenb er g- Weyl gr oup . When our alphab et is a finite field, we must have m = p r , wher e p is a prime and r ≥ 1. The error s co n taine d in the generalized P auli group ar e shift err ors , X ( a ) | x i = | x + a i , and phase err ors , Z ( b ) | x i = ω tr m/p ( bx ) | x i , wher e a, b ∈ F m , ω is a complex p -th ro ot of unit y , and tr m/p : F m → F p is the trace function, tr m/p ( c ) = P r − 1 i =0 c p i . If m = p is a prime, i.e., r = 1, the generalized P a uli group is generated by * X (1) =      0 . . . I m − 1 0 1 0 · · · 0      , Z (1) =        1 0 ω ω 2 . . . 0 ω n − 1        + , where ω is a complex p - th ro ot of unit y , and I is the identit y ma trix of s pecified dimension. 1 The op erato rs X ( a ) and Z ( b ) are obtained by taking the a -th and b -th powers of X (1 ) a nd Z (1), resp ectively . Even if m is not prime, we can still define 1 The set of gen erators also con tains the scalar ω , exce pt for the case m = 2, where it con tains i , a 4-th r oot of unity . This ov erall phase factor can be i gnored for our purp oses. Classi fica tion of self-dual a dditive codes 3 qudits that take v alues from the cyclic group Z m , and use the same error o pera tors as defined above. Howev er, w hen m is a prime power, we get muc h better co des by using a finite field as our alphab et. When we work with qudits that take v alues from F p r , where r > 1, w e use the er r or gro up { N r i =1 E i | E i ∈ P p } [16], i.e., the op erators are r -fold tensor pro ducts of Pauli ma tr ices fro m the group P p . The error bases that we use are ex amples of nic e err or b ases [17]. Quantum c o des ar e designed to add r edundancy in order to protect quantum states ag ainst er rors due to interference from the environmen t. A co de of length n and dimension k adds redundancy by enco ding k qudits using n qudits. One type of co de that explo its the fac t that the genera lized Pauli g r oup forms a basis for a ll po ssible errors is the stabilizer c o de [10]. A st abili zer is an Abelian g roup generated by a se t of n − k comm uting error o p era to rs. An error is detected by measuring the eigenv alues of these op erators . If a s ta te is a v alid co deword that ha s not b een affected by error, w e will observe the eigenv alue +1 for a ll op erators. The qua n tum co de, i.e., the set of all v a lid co dewords, is ther efore a joint eig enspace o f the stabi- lizer. If there is a detectable error, some eigen v a lues would b e differen t from +1, due to the commutativit y prop erties of the gener a lized P a uli matrices. A stabilizer gen- erated by a set of n err or op erators defines a zero-dimensiona l qua ntum co de, also known as a stabilizer state . 2 The minimum distanc e o f a zer o-dimensional s tabilizer co de is simply the minimum nonzero weight of all error o pera tors in the stabilizer . The weigh t of a n er r or o pera tor is the num b er of m × m tenso r comp onents that are different fro m the identit y matrix. A quantum co de of length n , dimensio n k , and minimum distance d , ov e r the alphab et F m , is deno ted an [[ n, k , d ]] m co de. Stabilizer states are therefore [[ n, 0 , d ]] m co des. If the minim um distance, d , is high, the stabilizer state is r obust against er ror, which indicates that it is highly ent an- gle d . E n ta ngled quantum states hav e many p otential applicatio ns, for instance in cryptogr a phic protoc o ls, or as gr aph states [23] which can b e used as a reso urce for quantum computations. In the next s ection we will also see that z e r o-dimensional stabilizer co des corre s pond to an interesting cla ss of classical co des, known as self- dual additive c o des . Example 1. A [[4 , 0 , 3]] 3 stabilizer state is obtained fro m the s tabilizer g enerated by the following er ror op e rators. X (1) ⊗ X (1) Z (2) ⊗ I ⊗ X (1) , X (1) Z (1) ⊗ X (2) ⊗ X (1) Z (1) ⊗ X (1) , I ⊗ X (2) Z (2) ⊗ X (1 ) Z (1) ⊗ Z (2) , X (1) ⊗ X (2) Z (2) ⊗ X (2) ⊗ X (2 ) Z (2) . 3. Self-dual add itive codes W e can repr esen t a stabilizer state over F m by a n n × 2 n matrix ( A | B ) [1]. The submatrix A repres en ts shift er r ors, such that A ( i,j ) = a if X ( a ) o ccurs in the j -th tensor co mponent of the i - th err o r op erator in the set of ge ne r ators. Similarly , the submatrix B repr esent s phase er rors. 2 Stabilizer states could al s o b e called one-dimensional qua n tum codes, since they are one- dimensional Hi l bert subspaces. W e use the term dimension to mean the num ber of qudits the co de can encode. 4 Lars Ei rik Danielsen Example 2. The matrix corr espo nding to the s tabilizer state in Example 1 is ( A | B ) =     1 1 0 1 0 2 0 0 1 2 1 1 1 0 1 0 0 2 1 0 0 2 1 2 1 2 2 2 0 2 0 2     . The matr ix ( A | B ) generates a code C , and this co de is a re pr esent ation of a stabilizer state. The fact that a stabilizer is a n Abelia n group transla tes into the requirement that C must b e self-dual with resp ect to a symple ctic inner pr o duct , i.e., ( a | b ) ∗ ( a ′ | b ′ ) = tr m/p ( b · a ′ − b ′ · a ) = 0 , ∀ ( a | b ) , ( a ′ | b ′ ) ∈ C . W e define the sympl e ctic weight o f a co dew o rd ( a | b ) ∈ C as the num b er of p osi- tions i where a i , b i , or both are nonzero . (This is the same as the weigh t of the corres p onding Pauli er ror op erator .) W e can also map the linear c o de of leng th 2 n defined ab ov e to an additive co de ov er F m 2 of length n . The r e presentation of binary sta biliz e r co des as s e lf-dual a d- ditive co des ov er F 4 was first demonstra ted by Ca lde r bank et al. [4], and generalized to q udits by Ashikhmin and Knill [1], and by Ketk ar e t a l. [16]. An addi tive co de, C , over F m 2 of leng th n is defined as a n F m -linear subgr o up of F n m 2 . The code C contains m n co dew ords, and can be defined by an n × n genera tor matrix, C , with ent ries from F m 2 , such that any F m -linear combination of rows fro m C is a co de- word. 3 T o get fro m the stabilizer repr e s en ta tio n ( A | B ) to the gener ator matrix C , we s imply take C = A + ω B , wher e ω is a primitive e lemen t of F m 2 . The co de C will b e self-dua l, C = C ⊥ , where the dual is defined with res pect to the Hermitian tr ac e inner pr o duct , C ⊥ = { u ∈ F n m 2 | u ∗ c = 0 for all c ∈ C } . When m = p is prime, the Hermitian trace inner product of tw o vectors o ver F p 2 of length n , u = ( u 1 , u 2 , . . . , u n ) and v = ( v 1 , v 2 , . . . , v n ), is given by u ∗ v = tr p 2 /p ( u · v p ) = u · v p − u p · v = n X i =1 ( u i v p i − u p i v i ) , When m = p r is not a prime, w e use a modification of the Hermitian trac e inner pro duct [16], u ∗ v = tr m/p  u · v m − u m · v ω − ω m  , where ω is a primitive element of F m 2 . The Hamming weight o f a codeword u ∈ C , denoted wt( u ), is the num b er of nonzero comp onents of u . The Hamming distanc e be t ween u and v is wt( u − v ). The minimum distanc e of the co de C is the minimal Hamming distance be t ween any tw o distinct co dewords of C . Since C is a n a dditiv e co de, the minimum distance is also given b y the smalle s t nonzero weigh t of any co deword in C . A co de ov er F m 2 with minimum distance d is called an ( n, m n , d ) code. The weight distribution o f the co de C is the sequence ( A 0 , A 1 , . . . , A n ), where A i is the num b er of co dewords of weigh t i . The weight enumer ator o f C is the p olynomia l W ( x, y ) = n X i =0 A i x n − i y i F o r a n additive co de ov e r F m 2 , all A i m ust b e divisible by m − 1. 3 F or additive codes o ver F 4 , each co dew ord is a sum of ro ws of the generator matrix. How ever, we also use the name “additive code” in this more general case. Classi fica tion of self-dual a dditive codes 5 Example 3. The stabilizer sta te in Ex a mple 1 corresp onds to the following gener- ator matrix of a self-dual additive (4 , 3 4 , 3) co de. C =     1 1 + 2 ω 0 1 1 + ω 2 1 + ω 1 0 2 + 2 ω 1 + ω 2 ω 1 2 + 2 ω 2 2 + 2 ω     . W e define tw o self-dual additive co des, C and C ′ ov er F m 2 , to b e e quivalent if the co dew ords of C can b e mapp ed o n to the co dewords of C ′ by a map that pr eserves the prop erties of the co de, including s elf-dualit y . A per m utation of co or dinates, or columns o f a generator matrix, is s uc h a map. Other op erations ca n also b e applied to the co ordinates of C . Let each element a + ω b ∈ F m 2 be repres e n ted as  a b  ∈ F 2 m . W e can then prem ultiply this element by a 2 × 2 matrix. (W e could equiv a len tly hav e applied tra nsformations to pairwis e columns of the 2 n × n matr ix ( A | B ).) It was shown by Rains [2 2] that by applying matrices from the symple ctic gr oup Sp 2 ( m ) to each co ordinate, w e preserve the proper ties of the co de. ( This group co n tains all 2 × 2 matrices with elements in F m and determinant o ne.) F or self-dual additiv e co des over F 4 , these symplectic op erations can be repr esent ed more simply as multiplication by nonzer o elemen ts fro m F 4 and c o njugation of co ordinates. (Conjugation of elements in F p 2 maps x to x p .) Combined, there are six p ossible tra nsformations that are equiv alent to the six p ermutations of the elements { 1 , ω , ω 2 } in the co o r dinate. The corr e sponding s ymplectic group is Sp 2 (2) =  A 1 =  0 1 1 1  , A 2 =  1 1 0 1  , where A 1 represents multip lication by ω and A 2 represents conjugation. Including co ordinate p ermutations, there are a total o f 6 n n ! ma ps for a co de of length n . F o r co des over F 9 , we observe that Sp 2 (3) is a group of o rder 24 generated by Sp 2 (3) =  A 1 =  1 1 1 2  , A 2 =  1 1 0 1  , where A 1 represents m ultiplicatio n b y ω 2 and A 2 represents the map a + ω b 7→ a + b + ω b . By ta king p ow er s of A 1 , we see that we are allow ed to multiply a co ordinate by x ∈ F 9 only if x x = 1. How ever, if we als o conjugate the co ordinate, we may multiply by x ∈ F 9 where xx = 2. Note that conjugation o n its own is not allowed. The 8 op erations just des c ribed may b e c o m bined with the o per a tions represented by A 2 and A 2 2 to give a to ta l o f 24 op era tions. In a ll there are 24 n n ! maps that take a self-dual additive co de ov er F 9 to a n equiv alent co de. In g eneral, for co des over F m 2 , the num b er of maps is | Sp 2 ( m ) | n n !. A tra nsformation that maps C to itself is ca lled an automorphism of C . All automorphisms of C make up an automorphi sm gr oup , denoted Aut( C ). The num b er of dis tinct co des equiv a len t to a self-dua l additive co de ov er F m 2 , C , is then given by | Sp 2 ( m ) | n n ! | Aut( C ) | . The e quivalenc e class o f C contains all co de s that are equiv alent to C . By adding the sizes o f all equiv a lence class es of co des of length n , we find the total nu m be r o f distinct co des of leng th n , denoted T n . The num b er T n is a lso given by a mass formula . The mas s formula for self-dual additiv e co des ov er F 4 was found by H ¨ ohn [15]. This r esult is eas ily g eneralized to F m 2 . 6 Lars Ei rik Danielsen Theorem 3.1. T n = n Y i =1 ( m i + 1) = t n X j =1 | Sp 2 ( m ) | n n ! | Aut( C j ) | , wher e t n is the nu mb er of e qu ivalenc e classes of c o des of length n , and C j is a r epr esentative fr om e ach e qu ivalenc e class. Pr o of. Let M ( n, k ) b e the total num b er of self-orthog onal ( n, m k ) c o des. One such co de, C , can be extended to a self-orthogo nal ( n, m k +1 ) c o de in m 2( n − k ) − 1 ways by adding an extra co deword from C ⊥ . Eac h ( n, m k +1 ) code can b e obtained in this wa y from m 2( k +1) − 1 differ en t ( n, m k ) co des. It follows that M ( n, k + 1) = M ( n, k ) m 2( n − k ) − 1 m 2( k +1) − 1 . Starting with M ( n, 0) = 1, the recur sion gives us the num b er of self-dual ( n, m n ) co des, M ( n, n ) = n − 1 Y i =0 m 2( n − k ) − 1 m 2( k +1) − 1 = n Y i =1 ( m i + 1) . By assuming that all codes of length n hav e a trivial automo r phism g roup, w e g et the fo llo wing lower bo und on t n , the total num b er of inequiv a len t co des. Note tha t when n is large , most co des have a trivial auto mo rphism gr oup, so the tightness of the b ound incr eases with n . Also note that this bound is m uch tight er than a bo und that was derived from res ults in g raph theory by Bahra mgiri a nd Beigi [2]. Theorem 3.2. t n ≥  c Q n i =1 ( m i + 1) | Sp 2 ( m ) | n n !  , wher e c = 1 if m is even, and c = 2 if m is o dd. Pr o of. When m is even, the trivial automorphism group includes only the identit y per m uta tio n, and the result follows from Theorem 3.1. When m = p r is o dd, where p is a prime, the trivia l a utomorphism group also contains the tra nsformation tha t applies the s ymplectic op e r ation  p − 1 0 0 p − 1  to all co ordinates. This op eration is e q uiv alent to m ultiplying e a c h co deword by p − 1, and will therefore map an additive co de to itself. It follows from the quant um singleton b oun d [18, 22] that any self-dua l additive co de must satisfy 2 d ≤ n + 2. A tight er b ound for co des o ver F 4 was given by Calderbank et a l. [4]. Co des that satisfy the singleton bound with equa lit y are known as maximu m distanc e sep ar able (MDS) c o des . Self-dual MDS co des mu st hav e even length, and MDS co des o f leng th tw o are trivial and exist for all alphab ets. The only no n- trivial MDS co de ov er F 4 is the (6 , 2 6 , 4) Hexac o de . Ketk ar et al. [16, Thm. 63] proved that a self-dua l additive ( n, m n , d ) MDS co de m us t satisfy n ≤ m 2 + d − 2 ≤ 2 m 2 − 2 . If the famous MDS conjecture holds, then n ≤ m 2 + 1 , or n ≤ m 2 + 2 when m is even a nd d = 4 or d = m 2 . Gra ssl, R ¨ otteler, and Beth [2 4] show ed that MDS co des of length n ≤ m + 1 always exist. Self-dual line ar codes over F m 2 are a subset of the self-dual additive codes. O nly additive co des that satisfy certa in constra in ts ca n be linear. Such co ns traint s for co des o ver F 4 were descr ibed by V an den Nest [20] and b y Glynn et al. [9]. An obvious co nstraint is that all coefficients of the weigh t en umera tor, ex c ept A 0 , of Classi fica tion of self-dual a dditive codes 7 a linear co de must b e divis ible b y m 2 − 1 , whereas for an additive co de they need only b e divisible by m − 1. 4. Correspondence to weighted graphs A gr aph is a pair G = ( V , E ) where V is a set of vertic es and E ⊆ V × V is a set of e dges . Let an m -weighte d gr aph b e a triple G = ( V , E , W ) where W is a set of w eights fro m F m . Each edge has an asso ciated non-zero w eight. (An edge with weight zero is the same as a no n-edge.) An m -weigh ted graph with n vertices ca n be repre s en ted by an n × n adja c en cy matrix Γ, where the elemen t Γ ( i,j ) = W ( { i, j } ) if { i, j } ∈ E , and Γ ( i,j ) = 0 otherwise. W e will only consider simple undir e cte d gr aphs whose adjacency matrices are symmetric with all diag onal elements being 0. The neighb ourho o d of v ∈ V , denoted N v ⊂ V , is the set of vertices connected to v b y a n edge. The n umber of v ertices adjacen t to v , | N v | , is ca lled the de gr e e of v . The induc e d sub gr aph of G on U ⊆ V contains vertices U and a ll edges from E whose endpoints a re b oth in U . The c omplement of a 2-weigh ted gra ph G is found by replacing E with V × V − E , i.e., the edge s in E are changed to non-edges , and the non- edges to edg es. Two graphs G = ( V , E ) and G ′ = ( V , E ′ ) are isomorphic if and only if there exists a p ermutation π of V such that { u, v } ∈ E ⇐ ⇒ { π ( u ) , π ( v ) } ∈ E ′ . W e also r equire that weigh ts a re prese r v ed, i.e., W { u,v } = W { π ( u ) ,π ( v ) } . A p ath is a sequence of vertices, ( v 1 , v 2 , . . . , v i ), such that { v 1 , v 2 } , { v 2 , v 3 } , . . . , { v i − 1 , v i } ∈ E . A gr aph is c onne cte d if there is a path from any vertex to any other vertex in the g raph. A c omplete gr aph is a g raph where a ll pairs of vertices a re connected by an edge. A clique is a co mplete subgra ph. Definition 4.1 . A gr aph c o de is a n additive co de over F m 2 that has a generator matrix of the form C = Γ + ω I , where I is the identit y matrix, ω is a primitive element of F m 2 , and Γ is the adjacenc y matrix of a simple undire c ted m -weigh ted graph. Theorem 4. 2. Every self-dual additive c o de over F m 2 is e quivalent t o a gr aph c o de. Pr o of. The genera tor matrix, C , of a self-dual additive co de over F m 2 corres p onds to an n × 2 n matrix ( A | B ) w ith elements from F m , such that C = A + ω B . W e m ust pr o ve that a n equiv alent co de is generated by (Γ | I ), where I is the identit y matrix a nd Γ is the a djacency matrix of a simple undirec ted m -weighted graph. A basis change c a n b e acco mplished b y ( A ′ | B ′ ) = M ( A | B ), where M is an n × n inv ertible matrix with elements from F m . If B has full ra nk, the solution is simple, since B − 1 ( A | B ) = (Γ ′ | I ). W e o btain (Γ | I ) after c hang ing the diag onal elemen ts of Γ ′ to 0, b y appro priate symplectic trans fo rmations. An y tw o rows of (Γ | I ) will b e o r thogonal with re spect to the symplectic inner pro duct, which means that Γ I T − I Γ T = 0, a nd it follows that Γ will alwa ys be a symmetric matrix. In the case where B has r ank k < n , w e can p erform a ba sis change to get ( A ′ | B ′ ) =  A 1 B 1 A 2 0  , where B 1 is a k × n matrix with full rank, and A 1 also has size k × n . Since the row- space of ( A ′ | B ′ ) defines a self-dual co de, and B ′ contains an all- zero row, it must b e true that A 2 B T 1 = 0 . A 2 m ust hav e full rank, and the row spa ce o f B 1 m ust b e the orthogo nal co mplemen t o f the r o w s pa ce of A 2 . W e assume that B 1 = ( B 11 | B 12 ) where B 11 is a k × k invertible ma trix. W e also write A 2 = ( A 21 | A 22 ) where A 22 has size ( n − k ) × ( n − k ). Assume that there exists an x ∈ F n − k m such that 8 Lars Ei rik Danielsen 2 1 1 1 Figure 1 : Graph Repre s en ta tio n of the (4 , 3 4 , 3) Co de A 22 x T = 0. Then the vector v = (0 , . . . , 0 , x ) of length n satisfies A 2 v T = 0. Since the row space of B 1 is the orthogona l complement of the row space of A 2 , we can write v = y B 1 for some y ∈ F k m . W e see that y B 11 = 0, and since B 11 has full ra nk, it must therefore b e true that y = 0. This means that x = 0, whic h pro ves that A 22 is an inv ertible matrix. Two of the symplectic op erations that we can apply to columns of a g enerator matrix are  0 m − 1 1 0  and  0 1 m − 1 0  . This means tha t we c a n in ter c ha nge column i of A ′ and column i of B ′ if we also multiply one of the columns b y m − 1. In this wa y we sw ap the i -th columns of A ′ and B ′ for k < i ≤ n to ge t ( A ′′ | B ′′ ). Since B 11 and A 22 are inv er tible, B ′′ m ust also be an in vertible matrix. W e then find B ′′− 1 ( A ′′ | B ′′ ) = (Γ | I ), and set all diagonal e lemen ts o f Γ to 0 b y symplectic tra nsformations. Example 4. The matrix from Example 2 can b e transformed in to the follo wing matrix, using the metho d g iv en in the pro of of Theorem 4.2. (Γ | I ) =     0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 2 0 0 1 0 0 1 2 0 0 0 0 1     . This means that the stabilizer state from Example 1 is equiv alent to the graph co de generated by C = Γ + ω I . The gr aph defined by Γ is depicted in Fig. 1. Note that Theor em 4.2 is a g eneralization o f the same theorem fo r co des ov er F 4 [7], which was prov e d by V a n den Nest e t al. [2 1]. The fact that s tabilizer co des can be r epresented by graphs was also shown by Sc hlingemann and W erner [26] and by Gra ssl, Klapp enec ker, and R ¨ otteler [12]. W e hav e seen that every m - w eig h ted g raph r e presents a self-dual a dditiv e co de ov er F m 2 , and that every s elf-dual additive co de ov er F m 2 can b e r epresented by an m -weigh ted gra ph. It fo llows that w e can, without loss of genera lit y , restrict our study to co des with genera tor ma trices of the for m Γ + ω I , where Γ is an adjacency matrix of an unlab eled simple undirected m - w eig hed g raph. 5. Graph equiv al ence and code equiv al ence Swapping vertex i and vertex j of a gr aph with adjacency matrix Γ can be accomplished b y exchanging column i a nd c o lumn j of Γ and then exchanging r o w i and row j of Γ. W e ca ll the res ulting matr ix Γ ′ . Exac tly the sa me co lumn and row op erations map Γ + ωI to Γ ′ + ωI , which are gener a tor matrices for equiv alent co des. It follows that tw o co des ar e eq uiv alent if their corresp onding gra phs are isomor phic. How ever, the s ymplectic transfor mations that map a co de to an eq uiv alent co de do Classi fica tion of self-dual a dditive codes 9 2 1 3 4 (a) The Graph G 2 1 3 4 (b) The Graph G ∗ 1 Figure 2 : Example of Lo cal Complementation not in genera l pr oduce iso morphic g raphs, but we will see tha t they can b e describ ed as graph op erations. It is known that tw o self-dual a dditiv e co des over F 4 are equiv a len t if and only if their corr e sponding graphs are equiv alent, up to isomorphism, with resp ect to a sequence of lo c al c omplementations [3, 9, 20, 2 1]. W e ha ve previously used this fact to devise a graph-bas ed algor ithm with whic h we class ified all self-dual additiv e co des ov er F 4 of length up to 12 [7]. Definition 5.1 ([3 ]) . Given a graph G = ( V , E ) and a vertex v ∈ V , let N v ⊂ V b e the neig h b ourho o d o f v . L o c al c omplementation (LC) o n v tra nsforms G in to G ∗ v by repla cing the induced s ubgraph of G on N v by its complement. Theorem 5.2 ([3, 9, 20, 21]) . Two self-dual additive c o des over F 4 , C and C ′ , with gr aph r epr esentations G and G ′ , ar e e quivalent if and only if t her e is a finite se quenc e of not ne c essarily distinct vert ic es ( v 1 , v 2 , . . . , v i ) , su ch that G ∗ v 1 ∗ v 2 ∗ · · · ∗ v i is isomorphi c to G ′ . The LC op eration can b e ge ner alized to weighted graphs, and it w a s first shown by Bahramgiri and Beig i [2] that the equiv alence of nonbinary stabilizer states ov er F m , i.e., se lf-dua l additive codes over F m 2 , can b e descr ibed in terms o f gr a ph op erations. 4 Definition 5.3 ([2]) . Given an m -weigh ted graph G = ( V , E , W ) a nd a vertex v ∈ V , weight shifting on v by a ∈ F m \ { 0 } transforms G in to G ◦ a v by m ultiplying the weigh t o f each edg e incident on v by a . Definition 5.4 ([2]) . Given an m -weigh ted graph G = ( V , E , W ) a nd a vertex v ∈ V , gener alize d lo c al c omplementation on v by a ∈ F m \ { 0 } tr ansforms G into G ∗ a v . Let Γ and Γ ′ be the adjacency matr ices of G a nd G ∗ a v , r espe c tiv ely . Then Γ ′ ( i,j ) = Γ ( i,j ) + a Γ ( v, i ) Γ ( v, j ) , for all i 6 = j , and Γ ′ ( i,i ) = 0 for all i . Theorem 5.5 ([2]) . Two self-dual additiv e c o des over F m 2 , C and C ′ , with gr aph r epr esentations G and G ′ , a r e e qu ivalent if and only if we get a gr aph isomo rphic to G ′ by applying s ome fi nite se quenc e of weight shifts and gener alize d lo c al c omple- mentations to G . 4 Bahramgiri and Beigi [2] only state their theorem f or F m where m is prime, but the result holds for an y finite field, as their proof do es not depend on m bei ng pr i me. 10 Lars Ei rik Danielsen 2 1 3 4 v x y z u (a) The Graph G 2 1 3 4 v ax ay az u (b) The Graph G ◦ a 1 Figure 3 : Example o f W eight Shifting 2 1 3 4 v x y z u (a) The Graph G 2 1 3 4 v+axy x y z u+axz ayz (b) The Graph G ∗ a 1 Figure 4: Example of Generalized Lo cal Complement ation A pro of of Theo rem 5 .5 w a s given by by Bahra mg iri and Beigi [2], as a gener - alization o f the pr o o f given by V an den Nest et al. [21] for self-dual a dditiv e co des ov er F 4 . Definition 5.6. The LC orbit of a weigh ted gr aph G is the set of all no n- isomorphic graphs that can be o btained by per forming any seq uence of weight shifts and gen- eralized LC o per a tions on G . Theorem 5. 7 . The minimum distanc e of a self-dual ad ditive ( n, m n , d ) c o de is e qual to δ + 1 , wher e δ is the minimum vertex de gr e e over al l gra phs in t he asso ciate d LC orbit. Pr o of. A v ertex with degre e d − 1 in the LC or bit co rresp onds to a c odeword of weigh t d , a nd w e will now show that suc h a vertex alw ays exists. Cho ose any graph representation of the co de and let G = (Γ | I ) be the corr espo nding genera tor matrix. Find a co deword c of weight d genera ted by G . Let the i -th row of G b e one of the rows that c is linear ly dep endent on. Apply s y mplectic transfo rmations to the co or dinates of the co de suc h that c is mapped to c ′ with 1 in coo rdinate n + i , and with 0 in all other of the la st n co ordina tes. Since we do not care ab out changes in the cor resp onding fir st n co ordinates, a s long as the symplec tic weigh t of c is preserved, there will alwa ys b e transforma tio ns that achieve this. Apply the same tr ansformations to the columns of G , and then replace the i -th row with c ′ , to get G ′ . Note that the right ha lf of G ′ still ha s full r ank, s o we c a n transform G ′ int o a matrix of the form (Γ ′ | I ) by Gaussia n elimination, where the symplectic weigh t of the i -th row is d . Finally , we set all diago nal elements of Γ ′ to zero by appr opriate s y mplectic transfor mations. V er tex i o f the gr aph with adjac e ncy matrix Γ ′ has degr ee d − 1. Classi fica tion of self-dual a dditive codes 11 6. Classifica tion It follo ws from Theorem 5.5 that tw o self-dual a dditiv e co des o ver F m 2 are equiv- alent if and o nly if their gr aph representations are in the same LC orbit. The LC orbit of a graph can easily b e generated b y a recursive alg o rithm. W e hav e used the progra m nauty [19] to chec k for gra ph iso morphism. Let G n,m be the s e t consisting of all non-isomor phic simple undirected connected m -weigh ted g raphs on n vertices. Note that connec ted graphs corre s pond to inde- c omp osable codes . A co de is decomp osable if it can b e written as the dir e ct s um of t wo sma ller co des. F o r ex a mple, let C be a n ( n, m n , d ) co de and C ′ an ( n ′ , m n ′ , d ′ ) co de. The direct s um, C ⊕ C ′ = { u || v | u ∈ C , v ∈ C ′ } , where || means concatenation, is an ( n + n ′ , m n + n ′ , min { d, d ′ } ) c o de. It follows that all decomp osa ble co des of length n can b e classified e a sily once all indecomp osable co des of length les s than n are known. The set of all distinct LC or bits of co nnected m -weigh ted graphs o n n vertices is a partitioning of G n,m int o i n,m disjoint sets. i n,m is a lso the n umber of indeco m- po sable self-dual additive co des ov er F m 2 of leng th n , up to equiv alence. Let L n,m be a set containing one repr e sen ta tiv e from each LC or bit of co nnected m -weigh ted graphs on n vertices. The simplest a lgorithm for finding such sets of representativ es is to start with the set G n,m and genera te L C orbits of its members until w e have a partitioning o f G n,m . The following more efficient technique is based o n a metho d describ ed by Glynn et al. [9]. Let the m n − 1 extensions of a n m - w eig h ted graph on n vertices b e formed by adding a new vertex and joining it to all p oss ible com- binations of at least one o f the old vertices, using a ll p o ssible combinations of edge weigh ts. The set E n,m , containing i n − 1 ,m ( m n − 1 − 1) gr aphs, is formed by making all p ossible e xtensions of all gra phs in L n − 1 ,m . Theorem 6.1. L n,m ⊂ E n,m , i.e., the set E n,m wil l c ontain at le ast one r epr esen- tative fr om e ach LC orbit of c onne cte d m -weighte d gr aphs on n vertic es. Pr o of. Let G = ( V , E , W ) ∈ G n,m , a nd choose any subset U ⊂ V of n − 1 vertices. By doing weight shifts and generalized LC op era tio ns on vertices in U , we can transform the induced subgr aph o f G on U into one of the g raphs in L n − 1 ,m that were extended whe n E n,m was constr ucted. It follows that for all G ∈ G n,m , some graph in the LC orbit o f G must b e pa rt o f E n,m . The set E n,m will b e muc h smaller than G n,m , so it will b e more efficien t to se a rch for a set of LC o rbit repr esent atives within E n,m . Another fa c t that simplifies our classification alg o rithm is that weigh t shifting and gener alized lo cal complementa- tion commute. This means that to genera te the L C orbit of a weigh ted g raph, we may first generate the orbit with respect to gener alized lo cal complementation only , and then apply weight shifting to the r esulting set o f gra phs. Using the describ ed techniques, we w er e able to classify all self-dual a dditiv e co des o ver F 9 , F 16 , and F 25 up to lengths 8, 6, and 6, r e spectively . T able 1 gives the v a lues of i n,m , the num b er o f distinct LC orbits of connected m - w eig h ted g raphs on n vertices, w hich is a ls o the num be r of inequiv alent indecomposa ble self-dual additive co des ov er F m 2 of leng th n . The to ta l num b er of inequiv a len t co des of length n , t n , is shown in T able 2 together with lower bo unds derived from Theor em 3.2. The 12 Lars Ei rik Danielsen T a ble 1: Num b er ( i n,m ) of Indecomp osable Co des of Length n ov e r F m 2 n i n, 2 i n, 3 i n, 4 i n, 5 1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 4 2 3 3 3 5 4 5 6 7 6 11 21 25 38 7 26 73 8 101 659 9 440 10 3,132 11 40,457 12 1,274,06 8 nu m be r s t n are easily derived fro m the num b ers i n by using the Euler tr ansform [27], c n = X d | n di d t 1 = c 1 t n = 1 n c n + n − 1 X k =1 c k t n − k ! . T a bles 3, 4, and 5 list by minimum dis ta nce the num b ers of indeco mp osa ble co des ov er F 9 , F 16 , and F 25 . A database cont aining one representative from ea c h equiv- alence class is av ailable a t http ://www. ii.uib.no/~larsed/nonbinary/ . F o r the classification o f se lf-dua l additive co des ov er F 4 , we refer to previous work [7], a nd the web page htt p://www .ii.uib.no/~larsed/vncorbits/ . Note that a pply ing the gr aph ex tension technique describ ed previo us ly is equiv- alent to lengthening [8] a self-dua l a dditiv e co de. Given an ( n, m n , d ) co de, we add a row and column to its gener ator matr ix to o btain an ( n + 1 , m n +1 , d ′ ) co de, where d ′ ≤ d + 1 . If follows that giv en a clas sification of all co des of length n and minim um distance d , w e can clas s ify all co des of length n + 1 and minimum distance d + 1. All length 8 c odes o ver F 9 hav e b een clas sified a s describ ed ab o ve. B y extending the 77 (8 , 3 8 , 4) co des, w e found 4 (9 , 3 9 , 5) co des, and from those we obtained a single (10 , 3 10 , 6) co de. Assuming tha t the MDS conjecture holds, there are no self- dual additive MDS co des ov er F 9 with length ab ov e 10. This would mean that the three MDS co des with parameter s (4 , 3 4 , 3), (6 , 3 6 , 4), and (10 , 3 10 , 6) are the only non-trivial self-dua l additive MDS co des over F 9 . The (6 , 3 6 , 4) and (10 , 3 10 , 6) are constructed as c irculant co des in Section 7. A ge nerator matr ix for the (4 , 3 4 , 3) co de is given in E xample 4 . In fact, a (4 , m 4 , 3) co de, for any m ≥ 3, is genera ted by     ω 1 1 0 1 ω 0 1 1 0 ω α 0 1 α ω     , Classi fica tion of self-dual a dditive codes 13 T a ble 2: T otal Num ber ( t n,m ) of Co des of Length n ov er F m 2 n t n, 2 t n, 3 t n, 4 t n, 5 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 6 7 7 7 5 11 13 14 15 6 26 39 44 58 7 59 121 ? ? 8 182 817 ≥ 946 ≥ 21,161 9 675 ≥ 9,646 ≥ 458,993 ≥ 38,267, 406 10 3,990 ≥ 2,373,100 11 45,144 12 1,323,3 63 13 ≥ 72,573 ,549 T a ble 3: Number of Indeco mposa ble Co des of Leng th n and Distance d ov er F 9 d \ n 2 3 4 5 6 7 8 9 10 2 1 1 2 4 15 51 3 88 ? ? 3 1 1 5 2 0 19 4 ? ? 4 1 2 77 ? ? 5 4 ? 6 1 All 1 1 3 5 21 73 659 ? ? T a ble 4: Num b er o f Indecomp osable Co des of Length n and Distance d ov er F 16 d \ n 2 3 4 5 6 2 1 1 2 4 16 3 1 2 6 4 3 5 6 All 1 1 3 6 25 where α ∈ F m \ { 0 , 1 } . This co de has w eight enumerator W (1 , y ) = 1 + 4( m 2 − 1) y 3 + ( m 2 − 3)( m 2 − 1) y 4 . There are four (9 , 3 9 , 5) co des, all with weigh t enum erator W (1 , y ) = 1 + 252 y 5 + 1176 y 6 + 3672 y 7 + 7794 y 8 + 6788 y 9 . None of these are equiv alent to circulant graph 14 Lars Ei rik Danielsen T a ble 5: Num b er o f Indecomp osable Co des of Length n and Distance d ov er F 25 d \ n 2 3 4 5 6 2 1 1 2 4 21 3 1 3 1 1 4 6 5 6 All 1 1 3 7 38 co des. The g e ne r ator matrices a r e:               ω 2 2 1 0 0 0 2 2 2 ω 1 0 0 0 1 2 1 2 1 ω 0 0 2 2 1 1 1 0 0 ω 2 1 0 0 1 0 0 0 2 ω 2 1 1 1 0 0 2 1 2 ω 1 1 0 0 1 2 0 1 1 ω 0 2 2 2 1 0 1 1 0 ω 2 2 1 1 1 1 0 2 2 ω               ,               ω 2 2 2 2 2 2 2 1 2 ω 0 0 0 0 2 1 2 2 0 ω 0 2 0 1 0 2 2 0 0 ω 1 0 1 2 0 2 0 2 1 ω 1 2 2 1 2 0 0 0 1 ω 0 1 1 2 2 1 1 2 0 ω 2 1 2 1 0 2 2 1 2 ω 1 1 2 2 0 1 1 1 1 ω               ,               ω 2 0 0 0 2 1 2 1 2 ω 0 2 1 2 0 1 0 0 0 ω 0 1 2 0 1 1 0 2 0 ω 0 1 0 1 1 0 1 1 0 ω 1 2 2 1 2 2 2 1 1 ω 1 2 1 1 0 0 0 2 1 ω 1 1 2 1 1 1 2 2 1 ω 1 1 0 1 1 1 1 1 1 ω               ,               ω 0 2 0 2 2 0 2 2 0 ω 2 2 0 0 2 0 1 2 2 ω 0 0 1 0 2 0 0 2 0 ω 0 0 1 2 1 2 0 0 0 ω 0 2 1 1 2 0 1 0 0 ω 1 1 2 0 2 0 1 2 1 ω 1 0 2 0 2 2 1 1 1 ω 1 2 1 0 1 1 2 0 1 ω               . The t wo (7 , 3 7 , 4) co des, the three (6 , 4 6 , 4), a nd five o f the six (6 , 5 6 , 4) co des are equiv a len t to circulant gr a ph co des genera ted in Section 7 . The la s t (6 , 5 6 , 4) co de has weigh t enum erator W (1 , y ) = 1 + 3 60 y 4 + 30 24 y 5 + 12 240 y 6 and generato r matrix         ω 2 2 1 0 0 2 ω 0 1 4 1 2 0 ω 0 1 1 1 1 0 ω 3 4 0 4 1 3 ω 3 0 1 1 4 3 ω         . 7. Cir culant graph codes It is clea rly infea s ible to study all self-dual additive codes o f lengths m uch hig he r than those clas s ified in the previous sec tio n. W e therefore restrict our sea rch space to the m ⌈ n − 1 2 ⌉ co des over F m 2 of length n corr espo nding to gr aphs with cir culant adjacency matric e s. A matrix is cir culant if the i -th r o w is equal to the first row, cyclically shifted i − 1 times to the right. W e hav e p erformed a n exhaustive search Classi fica tion of self-dual a dditive codes 15 T a ble 6: Highest F o und Minimum Distance of Co des ov er F m 2 of Le ng th n n \ m 2 3 4 5 2 2 2 2 2 3 2 2 2 2 4 2 3 ∗ 3 ∗ 3 ∗ 5 3 3 3 3 6 4 4 4 4 7 3 4 4 4 8 4 4 4 4 9 4 5 s 5 5 10 4 6 6 6 11 5 s 5 6 6 12 6 6 6 6 13 5 6 6 7 14 6 6 7 8 15 6 6 7 7 16 6 6 8 8 17 7 7 8 9 18 8 ∗ 8 8 10 19 7 8 20 8 8 21 8 ∗ 8 22 8 9 23 8 9 24 8 9 25 8 26 8 27 9 s 28 10 29 11 30 12 of such graphs, the result o f which is summarized in T able 6. This table shows the highest found minim um distance of self-dual a dditiv e co des over v ario us alphab ets. A co de with the given minimum distance has b een found in our s earch, except for the cases marked ∗ , where a b etter co de is obtained in some other wa y and do es not hav e a cir culan t gra ph repres en tatio n, 5 and ca s es marked s , which ar e not circulant, but obtained b y a triv ial shortening [8 ] of a longer cir culant c o de. Minimum distances printed in b old font are optimal a ccording to the qua ntum singleton b ound. If n is even and the quantum s ingleton b ound is satisfied with equality , w e hav e an MDS co de. W e here g iv e the fir st r ow o f a circulant generato r matrix for those co des cla ssified in Sectio n 6 that are eq uiv ale n t to cir culan t gr aph co des. There is a unique (6 , 3 6 , 4) co de with w eight enum erator W (1 , y ) = 1 + 120 y 4 + 24 0 y 5 + 368 y 6 generated b y 5 See the we b page http://www .codetabl es.de/ for details on how co des ov er F 4 of length 18 and 21 can be obtained. 16 Lars Ei rik Danielsen ( ω 0111 0). There ar e t wo inequiv a len t (7 , 3 7 , 4) co des g enerated by ( ω 11 0011) and ( ω 0222 20), b oth with weight enumerator W (1 , y ) = 1 + 70 y 4 + 336 y 5 + 812 y 6 + 968 y 7 . There is a unique (10 , 3 10 , 6) c o de with weigh t enumerator W (1 , y ) = 1 + 16 80 y 6 + 2880 y 7 + 1 4 040 y 8 + 2216 0 y 9 + 1828 8 y 10 generated by ( ω 0121 11210). There are three inequiv alent (6 , 4 6 , 4) codes with weight enumerator W (1 , y ) = 1 + 225 y 4 + 1080 y 5 + 27 90 y 6 generated b y ( ω 0111 0), ( ω 01 α 10), and ( ω 0 1 α 2 10), where α = ω 5 is a primitive element of F 4 . There are fiv e inequiv alent (6 , 5 6 , 4) co des genera ted by ( ω 0111 0), ( ω 01 210), ( ω 0222 0), ( ω 10 201), and ( ω 122 21), a ll with w e ig h t enumerator W (1 , y ) = 1 + 360 y 4 + 302 4 y 5 + 122 40 y 6 . F o r circulant gr a ph codes o f higher length that are o ptimal a ccording to the quantum single to n bo und, we find that all co des o f the same length hav e the same weigh t enumerator. In the list b elow, we give the first r o w of o ne generato r matrix for each w eight enumerator. • (7 , 4 7 , 4), ( ω 11 αα 11), W (1 , y ) = 1 + 105 y 4 + 100 8 y 5 + 483 0 y 6 + 104 40 y 7 . • (9 , 4 9 , 5), ( ω 001 αα 10 0), W (1 , y ) = 1 + 378 y 5 + 3 780 y 6 + 232 20 y 7 + 881 55 y 8 + 146 610 y 9 . • (10 , 4 10 , 6), ( ω 010 α 1 α 010), W (1 , y ) = 1 + 3150 y 6 + 180 00 y 7 + 111 375 y 8 + 366 000 y 9 + 550 050 y 10 . • (11 , 4 11 , 6), ( ω 00 α 11 11 α 00), W (1 , y ) = 1 + 1386 y 6 + 138 60 y 7 + 994 95 y 8 + 505 560 y 9 + 151 1598 y 10 + 206 2404 y 11 . • (7 , 5 7 , 4), ( ω 01111 0), W (1 , y ) = 1 + 140 y 4 + 218 4 y 5 + 170 80 y 6 + 587 20 y 7 . • (9 , 5 9 , 5), ( ω 00211 200), W (1 , y ) = 1 + 504 y 5 + 840 0 y 6 + 842 40 y 7 + 507 420 y 8 + 135 2560 y 9 . • (10 , 5 10 , 6), ( ω 001 2 22100), W (1 , y ) = 1 + 5 040 y 6 + 547 20 y 7 + 508 680 y 8 + 270 4560 y 9 + 649 2624 y 10 . • (11 , 5 11 , 6), ( ω 001 2 222100 ), W (1 , y ) = 1 + 1848 y 6 + 3 1680 y 7 + 370 260 y 8 + 297 7480 y 9 + 142 82664 y 10 + 311 64192 y 11 . • (13 , 5 13 , 7), ( ω 010 0 111100 10), W (1 , y ) = 1 + 6864 y 7 + 118 404 y 8 + 153 8680 y 9 + 148 67424 y 10 + 972 22320 y 11 + 388 930776 y 12 + 718 018656 y 13 . • (14 , 5 14 , 8), ( ω 101 1 331331 101), W (1 , y ) = 1 + 7207 2 y 8 + 816 816 y 9 + 104 74464 y 10 + 906 79680 y 11 + 544 536720 y 12 + 201 044188 8 y 13 + 344 649398 4 y 14 . Classi fica tion of self-dual a dditive codes 17 • (17 , 5 17 , 9), ( ω 001 0 111001 110100), W (1 , y ) = 1 + 9724 0 y 9 + 163 3632 y 10 + 245 04480 y 11 + 296 652720 y 12 + 273 362040 0 y 13 + 187 494033 60 y 14 + 899 945689 92 y 15 + 269 984494 620 y 16 + 381 154477 680 y 17 . • (18 , 5 18 , 10), ( ω 1213 4 242124 243121), W (1 , y ) = 1 + 1050 192 y 10 + 114 56640 y 11 + 180 442080 y 12 + 196 481376 0 y 13 + 168 776136 00 y 14 + 107 991522 432 y 15 + 485 972877 960 y 16 + 137 215593 4320 y 17 + 182 954155 4640 y 18 . As mentioned in the introductio n, stabilizer co des can be defined ov e r a n y Abelia n group, not only finite fields. F or compar ison, w e also generated circulant co des over Z 2 4 . As expected, the minimum distance of these co des are muc h w o rse tha n for co des ov er F 16 . W e found a (7 , 4 7 , 4)-co de over Z 2 4 , but for all other lengths up to 16, the b est minim um dis ta nce was equal to the best minim um distance of co des ov er F 4 of the same length. Gulliver and Kim [13] per formed a computer search of cir culan t s elf-dual additive co des over F 4 of length up to 30 . Their se a rch was no t re s tricted to graph co des, so our sear c h space is a subset of theirs. It is interesting to note that for every length, the highest minimum distance found was the sa me in bo th searches. This suggests that the circulant gra ph co de construction can pro duce co des as s tr ong as the more gener a l circulant co de construction. Besides a smaller se a rch s pace, the sp ecial form of the genera to r matrix of a graph co de makes it easier to find the minim um distance, since a n y code word obtained as a linear combination of i rows of the generator matrix m ust have weight a t least i . If, for example, w e w ant to determine whether a co de has minimum distance at least d , we only need to consider combinations of d or fewer rows of its ge nerator matrix. Circulant graphs m us t be r e gular , i.e., all vertices must hav e the same num b er of neighbours. W e have previo usly discovered [5, 6] that many strong cir culan t s elf- dual additive co des ov er F 4 can b e represented as hig hly structured nest e d clique gr aphs . Some of thes e graphs ar e shown in Fig. 5 . F o r instance, Fig. 5b shows a graph representation of the (12 , 2 12 , 6) “ Dodeca co de ” cons is ting of three 4- cliques. The remaining edges fo rm a Hamiltonian cycle , i.e., a cycle that visits every vertex of the gra ph exa ctly o nce. Notice that a ll graphs shown in Fig. 5 have minimum r e gular vertex de gr e e , i.e., each vertex has d − 1 neigh b ours, where d is the minim um distance of the cor resp onding co de. W e have disc o vered some new highly structured weigh ted g raph repr esen tations of self-dual additive co de s ov er F 9 and F 16 . Fig . 6 shows t wo interconnected 5- cliques wher e all edges hav e weigh t one, and a 10-cy cle where a ll edges hav e weight t wo. The sum of these tw o gr aphs, s uc h that no edg es overlap, corres p onds to the (10 , 3 10 , 6) co de. Up to isomor phism, there is only one wa y to add a Hamilto nian cycle of weigh t tw o edges to the double 5-cliq ue, since there cannot be b oth weight one and weigh t tw o edges betw een the same pair of vertices. The firs t row of a circulant gener ator ma trix corr espo nding to this gra ph is ( ω 01 211121 0). As a seco nd example, Fig. 7 shows tw o pair s of 4-cliques, each of which is con- nected by a length 8 cycle, and tw o 16-cycles wher e all edges have weight α and α 2 , re spectively , wher e α = ω 5 is a primitive element of F 4 . The (16 , 4 16 , 8) co de generated by ( ω 0 α 2 1 α 1000 10001 α 1 α 2 ) corr espo nds to a sum of these thre e g r aphs. 18 Lars Ei rik Danielsen (a) (6 , 2 6 , 4) (b) (12 , 2 12 , 6) (c) (20 , 2 20 , 8) (d) (25 , 2 25 , 8) Figure 5 : Examples of Nested Clique Graphs Co r resp onding to Co des over F 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 Figure 6: Two Graphs Whose Sum Corres p onds to the (10 , 3 10 , 6) Co de Classi fica tion of self-dual a dditive codes 19 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 P Sf r a g r epl a cem en t s α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α P Sf r a g r epl a cem en t s α α α α α α α α α α α α α α α α α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 α 2 Figure 7 : Three Gra phs Whose Sum Corr e sponds to the (16 , 4 16 , 8) Co de Note that the vertices of the graphs corr espo nding to circulant (10 , 3 10 , 6) and (16 , 4 16 , 8) g raph codes hav e degree highe r than d − 1. W e hav e tried to obtain similar graph representations for other co des in T a ble 6, but without success. Many of the circulant g raph co des hav e vertex degr ee m uch higher than d − 1, for in- stance the (14 , 5 14 , 8) co de generated b y ( ω 12212 020212 21), and the (18 , 5 18 , 10) co de genera ted by ( ω 1213 424212 4243121). Ackno wledgements The author would like to thank Matthew G. Parker for helpful discussio ns and comments. Also thanks to Markus Gra ssl for helpful comments. References [1] A. Ashikhmin and E. K ni ll, Nonbinary quantum stabilizer c o des , IEEE T rans. Inform. Theory , 47 (20 01), 3065–3072. [2] M. Bahramgir i and S. Beigi, Gr aph states under the action of lo c al Cliffor d g r oup in non- binary c ase (2006), preprint , arXiv:quant-ph /0610267 [3] A. Bouche t, Gr aphic pr esent ations of isotr opic systems , J. Com bi n. Theory Ser. B, 45 (1988), 58–76. [4] A. R . Calderbank, E. M . Rains, P . M. Shor and N. J. A . Sloane, Quantum err or c orr e ct ion via c o des over GF(4), IEEE T rans. Inform. Theory , 44 (1998), 1369–1387. [5] L. E. D aniels en, “On Self-Dual Quant um Codes, Graphs, and Bo olean F unctions” , Master’s thesis, Univ. Bergen, 2005. [6] L. E. Danielsen and M. G. P arker, Sp e ct ra l orbit s and p e ak- to-aver age p ower r atio of Bo ole an functions with r esp e c t to the { I , H, N } n tr ansform , i n “Sequences and Their Applications – SET A 2004” (eds. T. Helleseth, D. Sarwate, H.- Y. Song and K. Y ang), Spri nger-V erlag, Berlin, (2005), 373–388 . [7] L. E. Danielsen and M. G. Park er, On the classific ation of al l self-dual additive c o des over GF(4) of length up to 12 , J. Combin. Theory Ser. A, 113 (2006), 1351–13 67. [8] P . Gaborit, W. C. Huffman, J.-L. Kim and V. Pless, O n add it ive GF(4) c o des , in “Co des and Asso ciation Sche mes” (eds. A. Barg and S. Litsyn), Amer. Math. So c., Pr o vidence, RI, (2001), 135–149 . [9] D. G. Glynn, T. A. Gulliver, J. G. Maks and M. K. Gupta, “The Geo metry of A dditive Quan tum Co des” (2004), submitted to Springer-V erlag. [10] D. Gottesman, “St abilizer Co des and Quantu m Error Correction” , Ph.D. thesis, Caltec h, 1997. [11] M. Grassl, T. Beth and M. R ¨ otteler, On optimal quantum c o des , In ternat. J. Quan tum Inf., 2 (2004) , 55–64. [12] M. Grassl, A. Klapp eneck er and M. R ¨ otteler, Gr aphs, quadr atic forms, and quantum co des , in “Proc. 2002 IEEE In t. Symp. Inform. Theory” , (2002) , 45. 20 Lars Ei rik Danielsen [13] T. A. Gulliver and J.-L. Kim, Cir culant ba se d e xtr emal additive se lf- dual c o des over GF(4), IEEE T r ans. Inform. Theo ry , 50 (2004), 359–366. [14] E. Hostens, J. Dehaen e and B. De Moor, Stabilizer st ates and Cliffor d op er ations for systems of arbitr ary dimensions and mo dular arithmetic , Phys. Rev. A, 71 (20 05), 042315. [15] G. H ¨ ohn, Self-dual c o des over the Kleinian four gr oup , M ath. Ann., 32 7 (200 3), 227–255. [16] A. Ketk ar, A. Klappeneck er, S. Kumar and P . K. Sarvepalli, Nonbinary stabilizer co des over finite fields , IEEE T rans. Inform. Theory , 52 (2006), 4892–4914 . [17] A. Kl appenec ker and M. R ¨ otteler, Beyond stabilizer c o des I: Nic e err or b ases , IEEE T r ans. Inform. Theory , 4 8 (20 02), 2392–2395. [18] E. Knill and R. Laflamme, The ory of q uantum err or-c orr e c t ing c o des , Phys. Rev. A, 55 (1997), 900–911. [19] B. D. McKa y , nauty User’s Guide, v ersion 2.2 (2004), http://c s.anu.ed u.au/~bdm/nauty/ . [20] M. V an den Nest, “Local Equiv alence of Stabilizer States and Co des” , Ph.D. thesis, K. U. Leu- v en, 2005. [21] M. V an den Nest, J. Dehaene and B. De M oor, Gr aphic al description of the acti on of lo c al Cliffor d t r ansformations on gr aph states , Phys. Rev. A, 69 (2004), 022316. [22] E. M. Rains, Nonbinary quantum co des , IEEE T rans. Inform. Theory , 45 (1999) , 1827–1832. [23] R. Raussendorf, D. E. Browne and H. J. Briegel, Mea sur e ment-b ase d quantum c omputation on cluster states , Ph ys. Rev. A, 68 (20 03), 022312. [24] M. R ¨ otteler, M . Grassl and T. Beth, O n quantum MDS c o des , i n “Pro c. 2004 IEEE Int. Symp. Inform. Theory” , (2004) , 355. [25] D. Schlingemann, Stabilizer co des c an b e r e alize d as gr aph c o des , Quan tum Inf. Comput., 2 (2002), 307–323 . [26] D. Sc hli ngemann and R. F. W erner, Q uantum e rro r- c orr e cting c o des asso ciated with gr aphs , Ph ys. Rev. A, 65 (2002), 012308. [27] N. J. A. Sloane and S. Plouffe, “The Encyclop edia of Integer Sequence s ” , Acad emic Press, San Diego, CA, 199 5.