Quantum Goethals-Preparata Codes

Quantum Goethals-Prepara ta Codes Markus Grassl Institute for Quantum Optics and Quan tum Information Austrian Academy of Sciences T echnikerstraße 21a, 6020 Innsbruc k, Austria Email: m arkus.gr assl@oeaw .ac.at Martin R ¨ otteler NEC Laborato ries America, Inc. 4 I ndepen dence W ay , Su ite 200 Princeton, NJ 085 40, USA Email: m roetteler@nec- labs.com Abstract —W e present a family of non-additive quantum codes based on Goethals and Preparata codes with parameters ((2 m , 2 2 m − 5 m +1 , 8)) . The dimension of these codes is eight times higher th an the dimension of the b est known addi tiv e qu antum codes of equal l ength and minimum di stance. Index T erms —Non-additive quantum code, Goethals code, Preparata code I . I N T RO D U C T I O N Most of the kn own quan tum error-correcting co des (QECCs) are based on the so-called stabilizer formalism which relates quantum cod es to certain additive co des over GF (4) (see, e. g., [3], [7]). It is known that n on-add iti ve QECCs ca n have a h igher dimension compared to additi ve QE CCs with the same leng th and minimu m distance [5], [ 14], [1 7], [18]. All these examples of non-ad ditiv e QECCs are examples of so- called codew ord stabilized quan tum codes which are o btained as the complex span of som e so -called stabilizer states, wh ich correspo nd to self-dual additi ve co des. In [9] we ha ve extended the framework of stabilize r codes to the unio n of stabilizer codes (see [8]). This allows to construct no n-additive codes from any stabilizer code. In general, these non- additive QECCs correspo nd to non-ad ditiv e codes over GF ( 4) which can be decomp osed into cosets of an additive code which contains its dual. Using a con struction similar to that of so-called CSS codes (see [4], [15]), families of no n-additive quan tum cod es based on the binary Goethals and Preparata codes were derived in [9]. Here we p resent a new family of non-ad ditiv es quantum codes which have a d imension that is eig ht times high er than the dimen sion of the best known add itiv e quantu m codes. I I . U N I O N S T A B I L I Z E R C O D E S A. Stabilizer codes W e start with a brief revie w of the stabilizer formalism for quantum erro r-correcting co des and the conn ection to additive codes over GF (4) ( see, e. g., [3], [7]). A stabilizer code encodin g k qubits into n qu bits having minimu m distance d , denoted by C = [[ n, k , d ]] , is a subspace of dimension 2 k of the com plex Hilbert spac e ( C 2 ) ⊗ n of dim ension 2 n . The cod e is the jo int eigenspace of a set of n − k commuting operator s S 1 , . . . , S n − k which are tenso r prod ucts of the Pauli matrices σ x =  0 1 1 0  , σ y =  0 − i i 0  , σ z =  1 0 0 − 1  , or identity . The o perators S i generate an Abelian gr oup S with 2 n − k elements, c alled the stabilizer of th e c ode. It is a sub group o f th e n - qubit Pauli group P n which itself is generated by th e tensor pr oduct of n Pauli matrices and identity . W e fur ther require that S do es not con tain any n on- trivial multiple of iden tity . The normalizer of S in P n , deno ted by N , acts on the co de C = [[ n, k , d ]] . I t is possible to identify 2 k lo gical op erators X 1 , . . . , X k and Z 1 , . . . , Z k such th at these o perator s co mmute with any element in the stabilizer S , and such that together with S they gen erate the normalizer N of the code . The op erators X i mutually co mmute, and so do the opera tors Z j . T he operator X i anti-comm utes with the operator Z j if i = j and oth erwise co mmutes with it. It has been sho wn that th e n -qubit Pauli g roup correspond s to a symplectic g eometry , and th at on e can r educe the p roblem of con structing stabilizer code s to finding additive c odes over GF (4) th at ar e self-or thogon al with resp ect to a symplectic inner produ ct [2], [3]. Up to a scalar multip le, the elemen ts of P 1 can be expressed as σ a x σ b z where ( a, b ) ∈ F 2 2 is a binary vector . Choo sing the ba sis { 1 , ω } of GF (4) , wher e ω is a pr imitiv e elemen t of GF (4) with ω 2 + ω + 1 = 0 , we get th e following correspo ndence between the Pauli matr ices, elements of GF (4) , and b inary vector s of length two: operator GF (4) F 2 2 I 0 (00) σ x 1 (10) σ y ω 2 (11) σ z ω (01) This mappin g extends natura lly to tensor pr oducts of n Pauli matrices bein g mapp ed to vector s of length n over GF (4) or binary vectors of length 2 n . W e r earrang e the latter in such a way that the first n coo rdinates correspo nd to the expon ents of the oper ators σ x and write th e vector as ( a | b ) , i. e., g = σ a 1 x σ b 1 z ⊗ . . . ⊗ σ a n x σ b n z ˆ = ( a | b ) = ( g X | g Z ) . (1) T wo o perators correspon ding to th e binary vectors ( a | b ) and ( c | d ) commute if and on ly if the symplectic inner pro duct a · d − b · c = 0 . In terms of the bin ary represen tation, the stabilizer co rrespon ds to a binary code C which is self- orthog onal with r espect to this symplectic inn er pro duct, an d the norma lizer cor respond s to the symplec tic d ual code C ∗ . In terms of the correspo ndence to vecto rs over GF (4) , the stabilizer a nd normalizer co rrespon d to an add iti ve code over GF (4) a nd its dual with r espect to an symplec tic inner produ ct, respectiv ely , which we will a lso den ote by C and C ∗ . The term additive quan tum code refe rs to this cor respond ence. The minimum d istance d of the qu antum code is given as the minimum weight in th e set C ∗ \ C ⊂ GF (4) n which is lower bound ed by th e minim um distance d ∗ of the additive code C ∗ . If d = d ∗ , the code is said to b e pure , and for d ≥ d ∗ , the code is said to be pure up to d ∗ . Fixing the logical o perators X i and Z j , th ere is a can onical basis fo r the ad ditiv e q uantum code C . The stabilizer gr oup S of th e quan tum code tog ether with the lo gical operato rs Z j generate an Abelian gro up of order 2 n which corresp onds to a self-du al additive code . The joint +1 - eigenspace is one- dimensiona l, hence there is a unique quantu m state | 00 . . . 0 i ∈ C stabilized b y all elem ents of S . An or thonor mal basis of the code C is gi ven by the states | i 1 i 2 . . . i k i = X i 1 1 · · · X i k k | 00 . . . 0 i , (2) where ( i 1 i 2 . . . i k ) ∈ F k 2 . B. Union stabilizer codes The stabilizer grou p S gives rise to an orthog onal decompo- sition o f the space ( C 2 ) ⊗ n into common eigen spaces of equal dimension. The stabilizer co de C is th e jo int +1 -eige nspace of dimensio n 2 k . In ge neral, th e joint eigenspace s o f S can be labeled by the e igen values of a set of n − k generator s of S . Moreover, the n -qubit Pauli gro up P n operates transitiv ely on the eigenspaces. Hence one can id entify a set T ⊂ P n of 2 n − k operator s such that ( C 2 ) ⊗ n = M t ∈T t C . (3) Note th at each of th e spaces t C is a quantu m error-correcting code with th e same para meters as the code C and stabilizer group t S t − 1 . The decompo sition (3) co rrespond s to th e de- composition of th e n -qu bit Pauli gro up P n into cosets with respect to the normalizer N of the code C and like wise to the decomp osition of the full vector space GF (4) n into cosets of the additive code C ∗ . The main idea of u nion stabilizer cod es is to find a subset T 0 of the translations T such that th e space L t ∈T 0 t C is a good quantu m code (see [8], [9]). Definition 1 (unio n stabilizer cod e): Let C 0 = [[ n, k ]] be a stabilizer code an d let T 0 = { t 1 , . . . , t K } be a subset of the coset representatives o f the n ormalizer N 0 of the code C 0 in P n . Then the union stabilizer code is defined as C = M t ∈T 0 t C 0 . W ithout loss of g enerality we assume tha t T 0 contains iden tity . The dimension of C is K 2 k , and we will use the notatio n C = (( n, K 2 k , d )) . Similar to (2) a cano nical b asis of the un ion stabilizer code C is g iv en b y | j ; i 1 i 2 . . . i k i = t j X i 1 1 · · · X i k k | 00 . . . 0 i , (4) where j = 1 , . . . , K , ( i 1 i 2 . . . i k ) ∈ F k 2 , and X i are log ical operator s of the stabilizer code C 0 . In or der to compu te th e m inimum distance of this co de, we first con sider the d istance b etween two spaces t 1 C 0 and t 2 C 0 . As fo r a fixed stabilizer cod e C 0 two spaces t 1 C 0 and t 2 C 0 are either id entical or orthog onal, we can define the distance of them as follows: dist( t 1 C 0 , t 2 C 0 ) := min { wgt( p ) : p ∈ P n | pt 1 C 0 = t 2 C 0 } . (5) Here wgt( p ) is the number o f tensor factors in the n -q ubit Pauli op erator p that are different from identity . Clearly , dist( t 1 C 0 , t 2 C 0 ) = dist ( t − 1 2 t 1 C 0 , C 0 ) . The two spaces are identical if and o nly if t − 1 2 t 1 is an element o f th e normalize r group N 0 , or equiv alently , if the cosets C ∗ 0 + t 1 and C ∗ 0 + t 2 of the additiv e no rmalizer code C ∗ 0 are identical. (Note that we denote both an n - qubit Pauli o perator a nd the correspond ing vector over GF (4) by t i .) Hence the distance (5) can also be expressed in term s of the assoc iated vector s over GF (4) . Lemma 2 : The distance o f th e spaces t 1 C 0 and t 2 C 0 equals the m inimum weight in the coset C ∗ 0 + t 1 − t 2 . Pr o of: D irect co mputation shows dist( t 1 C 0 , t 2 C 0 ) = dist( C ∗ 0 + t 1 , C ∗ 0 + t 2 ) = dist ( C ∗ 0 + ( t 1 − t 2 ) , C ∗ 0 ) = min { wgt( c + t 1 − t 2 ) : c ∈ C ∗ 0 } = min { wgt( v ) : v ∈ C ∗ 0 + t 1 − t 2 } . While th e distance b etween the co sets C ∗ 0 + t j is an u pper bound on the minimum d istance of the un ion code C , the true minimum d istance can be derived from the following cod e over GF (4) . Definition 3 (unio n normalizer code) : W ith th e union sta- bilizer co de C we associate the ( in ge neral n on-add iti ve) union normalizer c ode given by C ∗ = [ t ∈T 0 C ∗ 0 + t = { c + t j : c ∈ C ∗ 0 , j = 1 , . . . , K } , where C ∗ 0 denotes th e ad ditiv e code associated with the normalizer N 0 of the stabilizer code C 0 . W e will r efer to both, the vectors t i and the correspo nding unitary operator s, as translations . Theor em 4: Th e m inimum distance o f a union stabilizer code with union normalizer code C ∗ is given by d = min { wg t( v ) : v ∈ ( C ∗ − C ∗ ) \ e C 0 } ≥ d min ( C ∗ ) = min { dist( c + t i , c ′ + t i ′ ) : t i , t i ′ ∈ T 0 , c, c ′ ∈ C ∗ 0 c + t i 6 = c ′ + t i ′ } , where C ∗ − C ∗ := { a − b : a, b ∈ C ∗ } d enotes th e set of a ll differences of vectors in C ∗ , a nd e C 0 ≤ C 0 is th e add itiv e code that corr esponds to all e lements of the stabilizer group S that commute with all t j ∈ T 0 . Pr o of: Let E ∈ P n be an n -qub it Pauli error of weight 0 < wgt( E ) < d . For two cano nical basis states | ψ a i an d | ψ b i as giv en in (4) w e consider the inner produ ct h ψ a | E | ψ b i = h j ; i 1 i 2 . . . i k | E | j ′ ; i ′ 1 i ′ 2 . . . i ′ k i = h 00 . . . 0 | X i 1 1 · · · X i k k t j E t j ′ X i ′ 1 1 · · · X i ′ k k | 00 . . . 0 i = ± h 00 . . . 0 | X i 1 + i ′ 1 1 · · · X i k + i ′ k k t j t j ′ E | 00 . . . 0 i . If E ∈ S commutes with all t j ∈ T 0 , then h ψ a | E | ψ b i = δ ab . Otherwise, E / ∈ C ∗ − C ∗ since 0 < wgt( E ) < d , and h ence the inner produ ct vanishes. I I I . T H E B I N A RY G O E T H A L S A N D P R E P A R A T A C O D E S In this section we recall some p roperties of the binary Goethals c odes [ 6] an d th e Prepa rata co des [1 3]. It has b een shown that v ariations o f these cod es have a simp le d escription as Z 4 -linear co des [1 0], but in our c ontext the description in terms of cosets of linear binar y cod es is u sed. In th e following m is an even integer ( m ≥ 4 ) and n = 2 m − 1 − 1 . Let α b e a p rimitive elemen t of the finite field GF (2 m − 1 ) . By µ i ( z ) we deno te th e minimal po lynomial of α i over GF (2) , i. e., the poly nomial with roots α j for j = i 2 k . The idempoten t θ i ( z ) is the un ique po lynomial satisfying θ i ( α i ) = 1 an d θ i ( α j ) = 0 for j 6 = i 2 k . Codew ords of a cyclic code can be represented by polynomials f ( z ) , and we use ( f ( z ); f ( 1)) to deno te the cod ew ord of the extended cyclic cod e obtained b y add ing an overall parity check. Similar, we use ( f ( z ); f (1 ); g ( z ); g (1)) to d enote the juxtaposition of codewords of two extended cyclic co des. Definition 5 (Goeth als code [6]): The Goethals code G ( m ) of le ngth 2 m is the union o f 2 m − 1 cosets of the linear binary code C G = [2 m , 2 m − 4 m + 2 , 8] . The cod e C G is obtained via th e | u | u + v | con struction applied to the extended cyclic codes C 1 and C 2 . Th e cyclic co de C 1 is a single- error correcting code with gen erator p olynom ial µ 1 ( z ) , an d C 2 is generated by µ 1 ( z ) µ r ( z ) µ s ( z ) where r = 1 + 2 m/ 2 − 2 and s = 1 + 2 m/ 2 − 1 . The non-zer o coset repr esentativ es are given by ( z i ; 1; z i θ 1 ( z ); 0) f or i = 1 , . . . , n − 1 . An alternative description of Goeth als cod es h as been given in [1]. Th e codew ords are described by pairs ( X , Y ) of subsets of GF (2 m − 1 ) . The corre sponding codeword is given by the juxtaposition of the character istic f unctions χ X and χ Y of the two set X and Y , i. e. ( X, Y ) ˆ = (1 X ( α i ); 1 X (0); 1 Y ( α i ); 1 Y (0)) , where 1 X ( α i ) is a short- hand for the vector 1 X ( α i ) = (1 X ( α 0 ) , 1 X ( α 1 ) , . . . , 1 X ( α n − 1 )) and 1 S ( x ) = ( 1 if x ∈ S , 0 if x / ∈ S . The non -zero elem ents of X and Y give rise to the po lyno- mials f X ( z ) an d f Y ( z ) g iv en by f S ( z ) = n − 1 X i =0 1 S ( α i ) z i . (6) Definition 6 (Goeth als code [1]): The Goethals code G ( m ) of length 2 m consists of th e codewords descr ibed by all pairs ( X, Y ) satisfying : a) | X | is even, | Y | is even, b) X x ∈ X x = X y ∈ Y y , c) X x ∈ X x r + X x ∈ X x ! r = X y ∈ Y y r , d) X x ∈ X x s + X x ∈ X x ! s = X y ∈ Y y s . In order to relate the two definitions, we distinguish three cases. 1) X = Y : Conditions c) and d) imp ly that P x ∈ X x = 0 . This is tru e fo r all cod ew ords of th e cyclic code gene r- ated by µ 1 ( z ) . Adding an overall parity check im plies Condition a). 2) X = ∅ : The left hand side of Con ditions b), c ), and d) vanishes, so the solutions for Y correspond to an extend ed cyclic code with genera tor polyn omial µ 1 ( z ) µ r ( z ) µ s ( z ) . 3) X = { 0 , x = α i } : From (6) it f ollows that f Y ( α ) = P y ∈ Y y . So Condition b) holds for th e set Y cor re- sponding to f Y ( z ) = z i θ 1 ( z ) . The lef t hand side of Conditions c) an d d) vanishes, so the solution s for Y are elements of th e extended cyclic code with gen erator polyno mial µ r ( z ) µ s ( z ) . As n either r nor s is a p ower of two, th e polynomial θ 1 ( z ) and hence f Y ( z ) = z i θ 1 ( z ) vanishes for α r and α s , i. e., Conditions c) and d) hold. Finally , all codewords of the Goeth als code as giv en in Definition 5 ar e the ju xtaposition of two b inary vectors of ev en weight, i. e., Condition a) ho lds. Hence any codeword giv en b y Definition 5 fulfills the co nditions of Definitio n 6. The equiv alence o f the definitions follows from the fact th at the c odes h av e eq ual size. Next we consider th e definition of Prepara ta codes similar to Defin ition 6 given in [1]. Definition 7 (Prepar ata code [1]): The extended Preparata code P ( m ) of leng th 2 m and param eter σ consists of th e codewords described b y all pairs ( X , Y ) satisfying : a) | X | is even, | Y | is even, b) X x ∈ X x = X y ∈ Y y , c) X x ∈ X x σ +1 + X x ∈ X x ! σ +1 = X y ∈ Y y σ +1 , Here σ is a power of two and gcd( σ ± 1 , n ) = 1 . For σ = 2 m/ 2 − 1 and n = 2 m − 1 − 1 we com pute  2 m − 1 − 1  −  2 m/ 2 − 1 ± 1   2 m/ 2 ∓ 2  = 1 , showing that g cd( σ ± 1 , n ) = 1 . Hence for this p articular choice of σ , the Preparata code of De finition 7 co ntains the Goethals code. What is e ven mor e, we can describe the Preparata code similar to Definition 5 as the union of cosets of the linear binary code C P which contains the linear binary code C G . Definition 8: The extended Preparata code P ( m ) o f length 2 m is the un ion of 2 m − 1 cosets of the linear bin ary code C P = [2 m , 2 m − 3 m + 1 , 6] . The code C P is obtained via the | u | u + v | constru ction ap plied to the extend ed cyclic cod es C 1 and C 3 . T he cyclic co de C 1 is a single- error co rrecting code with generato r po lynomial µ 1 ( z ) , and C 3 is gen erated by µ 1 ( z ) µ s ( z ) where s = 1 + 2 m/ 2 − 1 . T he no n-zero co set representatives are given by ( z i ; 1; z i θ 1 ( z ); 0) . Comparing Definitions 5 and 8 we see that we can use the very same coset rep resentatives to co nstruct the Goethals and th e Preparata code as union of cosets of the linear binary codes C G and C P , respectively . Mor eover , all cod es lie between codes that ar e equiv alent to the Reed-Muller codes RM ( m − 3 , m ) and RM ( m − 2 , m ) = [2 m , 2 m − m − 1 , 4] (see [11]). This is illustrated by the following diag ram: RM( m − 3 , m ) [2 m , 2 m − 4 m + 2 , 8] = C G [2 m , 2 m − 3 m + 1 , 6] = C P ✘ ✘ ✘ ✘ ✘ ✘ G ( m ) = S i C G + t i P ( m ) = S i C P + t i [2 m , 2 m − m − 1 , 4] = RM( m − 2 , m ) The compon ents of the codes are summ arized as f ollows: C 1 : c yclic co de generated by µ 1 ( z ) C 3 : c yclic co de generated by µ 1 ( z ) µ s ( z ) C 2 : c yclic co de generated by µ 1 ( z ) µ r ( z ) µ s ( z ) r = 1 + 2 m/ 2 − 2 , s = 1 + 2 m/ 2 − 1 C G : | u | u + v | construc tion app lied to th e extended cyclic codes C 1 and C 2 C P : | u | u + v | construction applied to the extended cyclic codes C 1 and C 3 t i : n + 1 coset re presentatives with t i = ( ( z i ; 1; z i θ 1 ( z ); 0) for i = 0 , . . . , n − 1 , (0 , . . . , 0) for i = n. I V . T H E Q U A N T U M G O E T H A L S - P R E PA R A T A C O D E S Before pr esenting the ne w family of no n-additive quantum codes, we recall Steane’ s c onstruction to enlarge the dimension of CSS codes. Theor em 9 (see [16]): Let C = [ n, k , d ] and C ′ = [ n, k ′ > k + 1 , d ′ ] be linear bin ary code s with C ⊥ ≤ C < C ′ . Then there exists an a dditive quan tum cod e C = [[ n, k + k ′ − n, ≥ min( d, 3 d ′ / 2)]] . Given a generator m atrix G of the code C and a g enerator matr ix D of the c omplemen t of C in C ′ , the normalizer o f the code C is generate d by   G 0 0 G D AD   , where A is a fixed-point free linear transform ation. As the co de C G contains a co de that is isomorp hic to the Reed-Muller co de RM ( m − 3 , m ) it follows that C ⊥ G ≤ C G . Hence we can app ly Steane’ s construction [16] to the chain C ⊥ G ≤ C G < C P of linea r binar y codes an d obtain an additive quantum code with parameters C 0 = [[2 m , 2 m − 7 m + 3 , 8]] . In a second step we u se the K = 2 m − 1 coset represen- tati ves t i of the decompo sition of both the Goethals and the Pre parata cod e. This yields a non-ad ditive code with dimension K 2 2 2 m − 7 m +3 = 2 ℓ where ℓ = 2 m − 5 m + 1 .     G 0 0 G D AD                              t 1 t 1 . . . . . . t 1 t K . . . . . . t K t 1 . . . . . . t K t K                          Fig. 1. S tructu re of the non-additi ve union normalizer code of the quantum Goethal s-Preparat a codes. Theor em 10 : Let C 0 = [[2 m , 2 m − 7 m + 3 , 8]] b e the additi ve quantum co de ob tained fro m the chain of linear bin ary cod es C ⊥ G ≤ C G ≤ C P using Stea ne’ s enlargemen t co nstruction. Furthermo re, let T 0 = { ( t i | t j ) : i, j = 0 , . . . , 2 m − 1 − 1 } wh ere t i are th e coset represen tati ves used to o btain the Goethals a nd Preparata cod e. Then the quantu m G oethals-Preparata code is a u nion stabilizer co de given by C 0 and T 0 . Th e minimum distance of the quan tum Goethals-Prep arata cod e is eigh t. Pr o of: Le t G d enote a gen erator m atrix o f the c ode C G and let D be such that ( G D ) genera tes C P . T he stru cture of the non-ad ditiv e u nion-n ormalizer co de of the quantum Goethals- Preparata co des is illustrated in Fig. 1. A gen erator matrix of the normalize r of the add iti ve quan tum cod e C 0 is given above the horizon tal line, while the set of tra nslations is listed below the horizon tal line. Every codew ord of the non- additive unio n normalizer co de is of the form g = ( g X | g Z ) = ( c 1 + v + t i | c 2 + w + t j ) , where c 1 , c 2 ∈ C G = [2 m , 2 m − 4 m + 2 , 8] and v , w ∈ C P /C G . For g , g ′ ∈ C ∗ , g 6 = g ′ we compu te dist( g , g ′ ) = dist(( c 1 + v + t i | c 2 + w + t j ) , ( c ′ 1 + v ′ + t ′ i | c ′ 2 + w ′ + t ′ j )) = wg t(( c ′′ 1 + v ′′ + t i − t ′ i | c ′′ 2 + w ′′ + t j − t ′ j )) , where c ′′ 1 = c 1 − c ′ 1 and c ′′ 1 = c 1 − c ′ 1 are co dew ords of C G , and v ′′ = v − v ′ , w ′′ = w − w ′ are c odewords of C P /C G . In general, the weight of g = ( g X | g Z ) is given by wgt(( g X | g Z )) = 1 2 (wgt( g X ) + wgt( g Z ) + wgt( g X + g Z )) . Hence we get dist( g , g ′ ) = 1 2 wgt( c ′′ 1 + v ′′ + t i − t ′ i ) (7a) + 1 2 wgt( c ′′ 2 + w ′′ + t j − t ′ j ) (7b) + 1 2 wgt( c ′′ 1 + c ′′ 2 + v ′′ + w ′′ + t i − t ′ i + t j − t ′ j ) . (7 c) By Stean e’ s con struction the vectors v ′′ and w ′′ are either both zero , or both are non- zero and th ey are different. For v ′′ = w ′′ = 0 , we can assume without loss of gener ality that the vectors in (7a) and (7 b) are both non -zero. Th e weight of th ese vecto rs equals th e distance between two cod ew ords of the Go ethals code, so it is at lea st 8. For v ′′ 6 = 0 6 = w ′′ the terms (7a) and (7b) eq ual the distance of two codewords of the Pre parata code, so th ey are lower bou nded b y 6. W e will show th at for v ′′ 6 = w ′′ , the vecto r in (7c) is a n on-zer o codeword of the line ar cod e iso morphic to the Reed-Muller code RM ( m − 2 , m ) , hen ce its weigh t is at least 4. For this, consider the vectors a = ( a 1 ; a 2 ) = c ′′ 1 + c ′′ 2 + v ′′ + w ′′ 6 = 0 and b = ( b 1 ; b 2 ) = t i − t ′ i + t j − t ′ j . Th e co set rep resentatives are of th e form t i = ( z i ; 1; z i θ 1 ( z ); 0) , so the second half b 2 of b is a codeword o f the extended cyclic cod e gen erated by θ 1 ( z ) , while a 2 is a codeword o f the extended cyclic code generated by µ 1 ( z ) . The intersection of the two codes is tri vial, so a 2 = b 2 only if a 2 = b 2 = 0 . Then wgt( b ) ≤ 4 while wgt( a ) ≥ 6 since 0 6 = a ∈ C P . Hence a 6 = b . T o o ur best k nowledge, the best additive qu antum co de with the same le ngth and min imum distance has dimen sion 2 m − 5 m − 2 . Codes with these par ameters can, e. g., be ob tained by app lying Steane’ s constructio n to extended primitive BCH codes [2 m , 2 m − 2 m − 1 , 8] and [2 m , 2 m − 3 m − 1 , 6] (see [1 6]). In the following table we g iv e the p arameters of the first cod es in these families. Additionally , we g i ve the param eters of the non-ad ditiv e quantum co des der iv ed from Goethals codes in [9]. Goethals enlarged BCH Goethals-Prepa rata ((64 , 2 30 , 8)) [[64 , 32 , 8 ]] ((64 , 2 35 , 8)) ((256 , 2 210 , 8)) [[256 , 214 , 8]] ((256 , 2 217 , 8)) ((1024 , 2 966 , 8)) [[1024 , 972 , 8]] ((1024 , 2 975 , 8)) V . C O N C L U S I O N S W e have constructed some new non-additive quantu m codes from nested non-lin ear binary codes which can be decomposed into cosets of lin ear codes which co ntain their dual. It is interesting to find mor e g ood n on-linear b inary or quatern ary codes with this pro perty . Recently , Ling and Sol ´ e hav e constructed some non-add itiv e quantum codes fro m Z 4 -linear c odes using a CSS-like con- struction [1 2]. So far it is not clear whether the non-additive codes presented here can also be put into the fra mew ork of Z 4 -linear codes. A C K N OW L E D G M E N T S W e ackn owledge fru itful discussions with V aneet Ag garwal and Robert Calderbank . Mark us Gr assl would like to thank NEC Labs., Prin ceton f or the h ospitality d uring his visit as well as T ero Laih onen, Kalle Ran to, and Sanna Ranto fo r discussions on v ariations o f Goethals co des. This work was partially suppo rted by the FWF (proje ct P17 838). R E F E R E N C E S [1] R. D. Baker , J. H. v an Lint, and R. M. W ilson, “On the Prepa rata and Goethal s Codes, ” IE EE T ran sactions on Informat ion Theory , vol . 29, no. 3, pp. 342–3 45, May 1983. [2] A. R. Calderba nk, E. M. Rains, P . W . Shor , and N. J. A. 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