Nonbinary quantum error-correcting codes from algebraic curves

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1 Dscrete Mathematcs 308 (2008) Nonbnary quantum error-correctng codes from algebrac curves Jon-Lark Km a, Judy Walker b a Department of Mathematcs, Unversty of Lousvlle, Lousvlle, KY 40292, USA b Department of Mathematcs, Unversty of Nebraska Lncoln, Lncoln, NE 68588, USA Receved 31 August 2004; accepted 9 August 2007 Avalable onlne 5 November 2007 Abstract We gve a new exposton and proof of a generalzed CSS constructon for nonbnary quantum error-correctng codes. Usng ths we construct nonbnary quantum stablzer codes wth varous lengths, dmensons, and mnmum dstances from algebrac curves. We also gve asymptotcally good nonbnary quantum codes from a Garca Stchtenoth tower of functon felds whch are constructble n polynomal tme Elsever B.V. All rghts reserved. Keywords: Algebrac geometrc codes; Nonbnary quantum codes 1. Introducton Bnary quantum error-correctng codes have been constructed n several ways. One nterestng constructon uses algebrac-geometry codes [2,6,7,15], wth the man dea beng to apply the bnary CSS constructon [4,5,19] to the asymptotcally good algebrac-geometry codes arsng from the Garca Stchtenoth [11] tower of functon felds over F q 2 (where q s a power of 2) attanng the Drnfeld Vladut bound [21]. It s natural to consder nonbnary quantum codes. Beyond the smple fact that nonbnary error-correctng codes are nterestng n the classcal case, Rans [17] ponts out that there are ndeed applcatons n whch nonbnary quantum codes would be more approprate than bnary quantum codes. Though nonbnary quantum codes have been consdered n [1,3,9,17], the majorty of attenton has been gven thus far to the bnary case. In partcular, the queston of asymptotcally good nonbnary quantum codes has not been studed untl now. In ths paper we gve a new exposton and proof of a nonbnary verson of the generalzed bnary CSS constructon based on two bnary lnear codes gven by Calderbank et al. [4]; see [13] for a dfferent approach. Usng ths constructon and algebrac curves we obtan varous parameters for nonbnary quantum codes. We further apply ths constructon to the tower of functon felds defned n [11] wth concatenaton to Reed Solomon codes to obtan asymptotcally good nonbnary quantum codes whch are constructble n polynomal tme. 2. Prelmnares In ths secton we gve some defntons and basc facts about quantum codes. Frst, we recall the generalzed bnary CSS constructon of quantum stablzer codes. We then gve a generalzaton to the nonbnary case. E-mal addresses: jl.km@lousvlle.edu (J.-L. Km), jwalker@math.unl.edu (J. Walker) X/$ - see front matter 2007 Elsever B.V. All rghts reserved. do: /j.dsc

2 3116 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) Defnton 2.1 (Calderbank et al. [4]). A bnary [[n, k, d]] 2 quantum error-correctng code s a 2 k -dmensonal subspace of C 2n (C 2 ) n whch can correct d 1 2 errors. In [4], Calderbank et al., showed how to construct bnary quantum error-correctng codes from addtve F 4 -codes. Brefly, the constructon s as follows: Let ω be a prmtve element of F 4. Any vector n F n 4 can be wrtten unquely as ωa + ωb wth a, b F 2n 2. Ths gves a bjecton map ψ : Fn 4 F2n 2, ψ(ωa + ωb) = (a b). One nterprets F2n 2 as Ē := E/{±I,±I}, where E s the quantum error group on C 2n. Now let C be an addtve F 4 -code whch s selforthogonal wth respect to the trace nner product. Then S := ψ(c) s a subgroup of Ē whose nverse mage S E s an abelan group actng on C 2n. Lettng Q be any jont egenspace of the elements of S, we have that Q s a bnary quantum error-correctng code and the parameters of Q can be computed from the parameters of C. Moreover, one may start wth a par of bnary lnear codes C 1 C 2 and form the addtve F 4 code C := ωc 1 + ωc2.then the above constructon yelds: Theorem 2.2 (Calderbank et al. [4, Theorem 9]). Suppose C 1 C 2 F n 2 are bnary lnear codes wth dmensons k 1 and k 2, respectvely. Then there exsts a bnary [[n, k 2 k 1,d]] 2 quantum code, where d =mn{d(c 2 \C 1 ), d(c 1 \C 2 )}. In Theorem 2.2, and n the remander of ths paper, the notaton d(a\b) means the mnmum weght of any vector n A but not n B. Our next goal s to explore CSS-type constructons for nonbnary quantum codes; see also [13]. We frst gve analogs of addtve codes and the quantum error group for the nonbnary case. For the remander of the paper, we wrte q = p m, where p s an odd prme. We call C F n q an F p-lnear code f C s lnear over F p. Ths generalzes the noton of addtve F 4 -codes, snce beng an addtve subgroup of F n 4 s equvalent to beng an F 2-vector space contaned n F n 4. Addtve F 4-codes whch are self-orthogonal under the trace nner product were used to construct stablzer quantum codes n [4]. Ths dea was generalzed n [1] to the relatonshp between self-orthogonal codes over F q 2 and q-ary quantum codes for any odd prme power q. An explct error bass for p m -ary quantum codes s descrbed as follows [1]. Let T and R be the lnear operators actng on the p-dmensonal complex space C p defned by T,j = δ,j 1 (mod p) and R,j = ξ δ,j, where ξ = e 2π 1/p, the ndces range from 0 to p 1, and δ,j = 1f = j and 0 otherwse. The set of operators T R j forms an orthogonal bass under the nner product defned by A, B =Tr(A B), where A s the Hermtan transpose of A [1,18]. Fx a bass {γ 1, γ 2,...,γ m } for F p m over F p.fora,b F p m we can wrte unquely a = a 1 γ 1 + a 2 γ 2 + +a m γ m, b= b 1 γ 1 + b 2 γ 2 + +b m γ m wth a 1,...,a m,b 1,...,b m F p. Defne T a R b = (T a 1 T a 2 T a m )(R b 1 R b 2 R b m ). The set of operators of the form T a R b, where a and b ranges over all of F p m, forms an orthogonal bass of untary operators actng on the p m -dmensonal complex vector space C pm [1]. Let a = (a (1),...,a (n) ), b = (b (1),...,b (n) ) F n q. As seen above, t s enough to consder the error operators gven by E a,b = T a (1)R b (1) T a (2)R b (2) T a (n)r b (n). The set of operators E ={ξ E a,b a, b F n q and 0 p 1} form an error group of order p 2mn+1. Quantum stablzer codes are defned as jont egenspaces of the operators of a commutatve subgroup S of E [1]; see also the appendx of [3].

3 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) We are now ready to develop the q-ary CSS constructon. We begn wth a constructon gven n [1] that s analogous to the frst constructon presented n [4], and then follow the lead of [4] to derve other constructons. We note that our q-ary CSS constructon generalzes the p-ary CSS constructon [16, Theorem 5] as the latter constructon uses only self-orthogonal codes over F p 2 where p s a prme. The man result s Theorem 2.7, whch wll be used n Secton 3 to construct asymptotcally good sequences of nonbnary quantum codes. As above, we wrte q = p m, where p s an odd prme. For a = (a (1),...,a (n) ), b = (b (1),...,b (n) ) F n q, let a b = a () b () be the usual nner product on F n q.for(a b), (a b ) F 2n q, set (a b) (a b ) = Tr(a b a b), where Tr : F q F p s the trace map. We see that f q = p then (a b) (a b ) = a b a b; ths nner product on was studed n [16,17]. F 2n p Defnton 2.3 (Ashkhmn and Knll [1]). A q-ary [[n, k, d]] q quantum error-correctng code s a q k -dmensonal subspace of C qn (C q ) n whch can correct d 1 2 errors. Proposton 2.4 (Ashkhmn and Knll [1, p. 3069]). Suppose C F 2n q s an F p -lnear code of length 2n havng p r codewords. Let C be the dual of C wth respect to the nner product. If C C, then there s a q-ary [[n, n m r,d]] q quantum code wth d = d(c \C). For x, y F n q 2, defne x y = (x y q x q y ). Ths map s F q -blnear and generalzes the nner product of [16, p. 1879]. Note that for any γ 0 F q, there exsts γ F q 2\F q satsfyng γ q = γ 0 γ; ndeed snce the trace map Tr : F q 2 F q s onto and F q -lnear, we may pck γ F q 2\F q wth Tr(γ) = γ 0. Further, for any such γ, {1, γ} s a bass for F q 2 over F q snce γ / F q. Lemma 2.5. Suppose D F n s an F q 2 q -lnear code satsfyng D D, where D s the dual of D wth respect to. Fx γ 0 F q and choose γ F q 2\F q satsfyng γ q = γ 0 γ. Defne an F q -lnear map f : F n F 2n q 2 q by f(x 1,...,x n ) = (x (1) 1,...,x(1) n x (2) 1,...,x(2) n ), where x = x (1) + γx (2) for = 1,...,n. Then f(d) f(d ) = (f (D)), where (f (D)) s the dual of f(d)wth respect to. Proof. Clearly, f(d) f(d ) snce D D. It remans to show that f(d ) = (f (D)). To do ths, let x D, y D. Then 0 = x y = (x y q x q y ) = ((x (1) + γx (2) )( + γ ) q (x (1) + γx (2) ) q ( + γ )) = ((x (1) + γx (2) )( + γ q ) (x (1) + γ q x (2) )( + γ )) = (x (1) + γ q x (1) + γx (2) + γ q+1 x (2) ) (x (1) + γx (1) + γ q x (2) + γ q+1 x (2) ) = (γ q γ) (x (1) x (2) ) = (γ 0 2γ) (x (1) x (2) ). But γ 0 2γ F q 2\F q, and so (1) (x x (2) ) = 0. Therefore f(x) f(y) = Tr ( ( )) x (1) x (2) = 0. Ths shows f(d ) (f (D)). Snce these two codes have the same number of codewords, they must be equal.

4 3118 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) Proposton 2.6. Let C 1 C 2 F n q be F q-lnear codes, so that C2 C 1, where C s the dual of C under the usual nner product. Let ω be a prmtve element of F q 2 and wrte ω = ω q. Set D = ωc 1 + ωc2 Fn. Then the dual q 2 D of D s gven by D = ωc1 + ωc 2. Hence D D and d(d \D) = mn{d(c 2 \C 1 ), d(c 1 \C 2 )}. Proof. Note frst that D =q k 1+n k 2, and so D =q 2n (n+k 1 k 2 ) = q n k 1+k 2 = ωc 1 + ωc 2. Now pck x C 1, y C 2, a C 1, and b C 2. Then (ωx + ωy) ( ωa + ωb) = ((ωx + ωy )(ωa + ωb ) ( ωx + ωy )( ωa + ωb )) ( = (ω 2 ω 2 ) x a ) y b = 0, snce x a = y b = 0. The last sentence of the proposton follows snce C 1 C 2, C2 C 1, and ωc 1 ωc2 = ωc1 ωc 2 ={0}. Next, we gve a constructon whch produces a q-ary quantum code from any two F q -lnear codes C 1 C 2 F n q ; see [13] for a dfferent approach. Ths s a q-ary verson of the bnary CSS constructon [4, Theorem 9] as t s also based on two lnear codes over F q, and so t s a generalzaton of [16, Theorem 5] whch s based on self-orthogonal codes. Theorem 2.7. Let q = p m, where p s an odd prme and m 1 s an nteger. Suppose C 1 C 2 F n q are F q-lnear codes wth dmensons k 1 and k 2, respectvely. Then there exsts a q-ary [[n, k 2 k 1,d]] q quantum code, where d = mn{d(c 2 \C 1 ), d(c 1 \C 2 )}. Proof. Set D = ωc 1 + ωc2, as n Proposton 2.6. Then f(d) (f (D)) by Proposton 2.6 and Lemma 2.5. Note that f(d)s an F q -lnear code n F 2n q, hence an F p-lnear code wth p r elements, where r = m(k 1 + n k 2 ). Our clam now follows by applyng Proposton 2.4 by lettng C = f(d). Example 2.8. Let C 2 be the ternary Golay [11, 6, 5] code and let C 1 be the subcode of C 2 consstng of codewords whose weght s dvsble by 3. Then C 1 s a ternary [11, 5, 6] code and n fact s equal to C2. By Theorem 2.7, we obtan a ternary double-error correctng quantum [[11, 1, 5]] 3 code. 3. Good sequences of q-ary quantum AG codes We assume the results from Secton II of [7] and use the deas of Secton III of that paper. Note, however, that the authors of [7] used only the trval bnary MDS code n the concatenaton whle we use Reed Solomon codes over F p, whch allows us to obtan varous lengths, dmensons, and mnmum dstances of nonbnary quantum codes. We frst recall the bascs of algebrac geometry codes. For more detals, see [20] or [21]. Defnton 3.1. Let X be a smooth, projectve, absolutely rreducble curve over F q of genus g. Let P ={P 1,...,P n } be a set of dstnct F q -ratonal ponts on X, and let G be a dvsor on X wth support dsjont from P. Let L(G) ={f F q (X) (f ) + G 0} {0} be the vector space of ratonal functons assocated to G. The algebrac geometrc code C(X,P,G)assocated to X, P and G s C X (P,G):= {(f (P 1 ),...,f(p n )) f L(G)}.

5 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) Theorem 3.2. Let X, P and G be as n Defnton 3.1 wth g the genus of X and n the number of ponts n P. Assume 2g 2 < deg G<n. Then C X (P,G)s a lnear code over F q wth length n, dmenson k = deg G + 1 g and mnmum dstance d n deg G. Further, the mnmum dstance of the dual code C X (P,G)s at least deg G 2g + 2. Algebrac geometry codes were frst ntroduced by Goppa [12] n Only a few years later, Tsfasman, et al. [22] used modular curves to show that, for q 49 a square, there exst sequences of algebrac geometry codes over F q whch are asymptotcally better than the Glbert Varshamov bound on a certan nterval of relatve mnmum dstance. A few such sequences are explctly known (or at least the curves on whch they are based are explctly known); we wll use a sequence of curves gven by Garca and Stchtenoth [11] to construct asymptotcally good nonbnary quantum error-correctng codes. For our constructon, we need only one-pont codes, that s, algebrac geometry codes where the dvsor G s a multple of some chosen F q -ratonal pont P 0 and the set P conssts of all the other F q -ratonal ponts on X. Set N = N(X) := P =#X(F q ) 1. Pck ntegers m 1 and m 2 wth 2g 2 <m 1 <m 2 <N. We consder the codes T j := C X (P,m j P 0 ) for j = 1, 2. Then T 1 T 2 and, from Theorem 3.2, we see that T j s an [N,m j g + 1, N m j ] code over F q and the dual Tj of T j s an [N,N m j + g 1, m j 2g + 2] code over F q. As n [7], we use concatenaton (see [10]) to obtan F p -lnear codes C 1 and C 2 from T 1 and T 2. As we wll be workng wth felds of square order we wll now swtch notaton so that our ground feld s F q 2, where q = p t. We wsh to have an F p -lnear map π : F q 2 F 2t+r p, for some nonnegatve nteger r, such that the mage C of π s a [2t + r, 2t,r + 1] Reed Solomon code over F p. Snce Reed Solomon codes of over F p exst only for lengths at most p + 1, we must have 2t + r p + 1,.e., 0 r p 2t + 1. Defne π : F N q 2 F N(2t+r) p C 1 := π(t 1 ) π(t 2 ) =: C 2. by π((x 1,...,x N )) = (π (x 1 ),...,π (x N )). Then we have Thus C j,(j= 1, 2) s an F p -lnear [(2t + r)n, 2t(m j g + 1), (r + 1)(N m j )] code (see [10] or [8]). The dual of C j (j = 1, 2) s Cj = S (π (Tj )), where S s the drect sum of N copes of C and π s the F q -lnear njectve dual bass map, as descrbed n [7]. For any vector x C1 \C 2,wehavewt(x) m 1 2g + 2, just as n the bnary case (see [7], proof of Theorem 1.2). The followng proposton follows from Theorem 2.7. Proposton 3.3. Wth notaton as above, we get a p-ary quantum [[n, k, d]] p code B = B(X) wth n = (2t + r)n, k = 2t(m 2 m 1 ), d mn{(r + 1)(N m 2 ), m 1 2g + 2}. Example 3.4. Let X be the Hermtan curve defned by y q + y = x q+1 over F q 2 wth q = p t ; ths s the base level of the Garca Stchtenoth tower [11]. There are q F q 2-ratonal ponts on X, and the genus of X s q(q 1)/2. We choose ntegers m 1 and m 2 wth q 2 q 2 = 2g 2 <m 1 <m 2 <N = q 3 and obtan F q 2-lnear codes T j, j = 1, 2, wth parameters [q 3,m j q(q 1) 2 + 1, q 3 m j ]. For any nteger r wth 0 r p + 1 2t as above, we get a p-ary quantum [[n, k, d]] p code B wth n = (2t + r)q 3, k = 2t(m 2 m 1 ), d mn{(r + 1)(q 3 m 2 ), m 1 q(q 1) + 2}. As a fnal step before we consder the asymptotc behavor of our quantum codes, we make a few remarks. Let X be a curve of genus g wth N + 1 ratonal ponts. If we choose ntegers m 1 and m 2 wth 2g 2 <m 1 <m 2 <N, then

6 3120 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) l := m 2 m 1 satsfes 0 <l N 2g. Conversely, gven an nteger l satsfyng 0 <l N 2g, set (r + 1)N + 2g + l 2 m 2 = and m 1 = m 2 l. Then snce (r + 1)N + 2g + l 2 m 1 (2g 2) = (r + 1)N + 2g + l 2 = r + 1 (N + 1 2g l) > 0 l (2g 2) r + 1 l (2g 2) m 2 m 1 = l > 0 and (r + 1)N + 2g + l 2 N m 2 N = N 2g l > 0, we have 2g 2 <m 1 <m 2 <N. Also, snce (r + 1)(N m 2 ) = (r + 1)N ()m 2 + m 2 (r + 1)N ((r + 1)N + 2g + l 2) + m 2 = 2g k + l + m 2 = m 1 m 2 2g m 2 = m 1 (2g 2) r + 1 (N 2g l + 1), we have Proposton 3.5. Let X be a curve of genus g wth N ratonal ponts. For any ntegers l and r wth 0 <l N 2g and 0 r p + 1 2t, there s a p-ary quantum [[n, k, d]] p code B = B(X) wth parameters n = (2t + r)n, k = 2tl, d r + 1 (N 2g l + 1). Now let X ={X} be a Garca Stchtenoth tower [11] of polynomally constructble curves over F q 2 where q = p t havng ncreasng genus g = g(x) and attanng the Drnfeld Vladut bound,.e., satsfyng #X(F q 2) lm sup = q 1. X X g l Then for any sequence of ntegers {l=l(x) X X} wth 0 <l N 2g for each X,wehave0< lm sup X X N 1 2 q 1. Indeed, by choosng the values of l approprately, we can have lm sup X X l N = λ for any λ wth 0 < λ 1 2 q 1.

7 We put R := lm sup X X = 2t 2t + r λ, δ := lm sup X X 2tl (2t + r)n J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) r+1 r+2 (N 2g l + 1) (2t + r)n ( r + 1 = 1 2 ) ()(2t + r) q 1 λ. Note that for a sequence of p-ary [[n(b), k(b), d(b)]] p quantum codes B ={B = B(X)} comng as n Proposton 3.5 from the Garca Stchtenoth tower, we have lm sup B B k(b) d(b) = R and lm sup n(b) B B n(b) δ. To get an expresson for R n terms of δ, we solve for λ n terms of δ and substtute, yeldng R p (δ) := R = 2t ( 1 2 ) 2t() 2t + r q 1 r + 1 δ. In order to have R>0, we need δ < δ(p,r,t), where (r + 1)(p t 3) δ(p,r,t)= ()(2t + r)(p t 1). We have proved the followng. Theorem 3.6. Let p be any odd prme number. Suppose that t 1 and r 0 are ntegers satsfyng 2t + r p + 1. Let δ(p,r,t)be as above. Then for any δ wth 0 < δ < δ(p,r,t)< 4 1, there exst polynomally constructble famles of p-ary quantum codes wth n and asymptotc parameters at least (δ,r p (δ)), where R p (δ) = (2t)() (δ(p,r,t) δ). r + 1 In Fgs. 1 3, we plot some of our bounds (δ,r p (δ)) and compare them wth Ashkhmn and Knll s nonbnary quantum Glbert Varshamov (qgv) bound [1, Eq. (39)], whch s nonconstructve. We note that n the case of the bnary quantum codes, there s a large nformaton rate gap between the nonconstructve bnary quantum Glbert Varshamov bound and the constructve bounds [1, Fg. 1; 7; 15, Fg. 1] (for example, gap 0.5 atδ = 0.06), and the nonzero nformaton rate from the constructve bound s possble up to δ However our 53-ary quantum codes as seen n Fg. 3 have a small nformaton rate gap 0.1atδ = 0.06 when r = 0 and t = 1, and can have nonzero nformaton rate up to δ As p ncreases, the nformaton rate gap s gettng smaller although our bounds (δ,r p (δ)) n Theorem 3.6 are under the nonbnary qgv bound as the δ-ntercept δ(p,r,t)of the graph s < 1 4. Remark 3.7. The case when p = 2 was dscussed n [7]. In ths case we requre that t 3 s an nteger and r = 0or1. Then pluggng n p = 2 and r = 1 nto δ(p,r,t)n Theorem 3.6 gves 2(2 t 3) δ(2, 1,t)= δ t = 3(2t + 1)(2 t 1), R 2 (δ) = 3t(δ t δ), whch s Theorem 1.2 of [7]. Usng the same deas, we can construct p t -ary quantum codes.

8 3122 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) nformaton rate k/n qgv bound p=5, r=0, t=1 p=5, r=0, t=2 p=5, r=1, t= relatve dstance d/n Fg. 1. Asymptotcally good sequences of p-ary quantum codes where p = 5 wth r = 0 and t = 1, 2, or wth r = 1 and t = qgv bound p=7, r=0, t=1 p=7, r=0, t=2 p=7, r=1, t=1 0.7 nformaton rate k/n relatve dstance d/n Fg. 2. Asymptotcally good sequences of p-ary quantum codes where p = 7 wth r = 0 and t = 1, 2, or wth r = 1 and t = 1. Theorem 3.8. Let p be an odd prme, and let t 1 and r 1 be ntegers wth r p t 1. Set δ(p,r,t)= (r + 1)(pt 3) () 2 (p t 1).

9 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) qgv bound p=53, r=0, t=1 p=53, r=0, t=2 p=53, r=1, t=1 nformaton rate k/n relatve dstance d/n Fg. 3. Asymptotcally good sequences of p-ary quantum codes where p = 53 wth r = 0 and t = 1, 2, or wth r = 1 and t = 1. Then for any δ wth 0 < δ < δ(p,r,t), there exst polynomally constructble famles of p t -ary quantum codes wth n and asymptotc parameters at least (δ,r p t (δ)), where R p t (δ) = 2() (δ(p,r,t) δ). r + 1 Proof. We proceed as n the proof of Theorem 3.6. For any nteger r wth 1 r p t 1, we have a [2 + r, 2,r + 1] Reed Solomon code C over F p t. Let π : F p 2t F 2+r p t be an F p t -lnear njectve map wth π (F p 2t ) = C. The code C j wll be an F p t -lnear [(2 + r)n,2(m j g + 1), (r + 1)(N m j )] code wth Cj = S + π (Tj ), where S s the drect sum of N copes of C and π s the dual bass map correspondng to π. Applyng the CSS constructon, we get a p t -ary quantum code B = B(X) wth parameters [[ (2 + r)n,2l, d r + 1 ]] (N 2g l + 1) where l s any nteger satsfyng 0 <l N 2g as before. Now set R p t := R = lm sup X X δ = lm sup X X 2l (2 + r)n = 2 d (2 + r)n = r + 1 () 2 and wrte R p t n terms of δ to obtan the result. 4. Concluson p t, 2 + r λ, ( 1 2 p t 1 λ In ths paper we gve an ndependent proof of [14, Theorem 3] for a generalzed CSS constructon for nonbnary quantum error-correctng codes. Usng ths constructon and algebrac curves, we obtan varous parameters for nonbnary quantum codes. In partcular, usng a Garca Stchtenoth tower of functon felds, we have constructed famles )

10 3124 J.-L. Km, J. Walker / Dscrete Mathematcs 308 (2008) of asymptotcally good nonbnary quantum codes. It s noted that the decodng of our quantum codes s related to the decodng algorthm [14] of the correspondng algebrac geometry codes. Acknowledgments The authors would lke to thank one of the referees for detaled comments. The frst author would lke to thank the Math Department of the Unversty of Nebraska at Lncoln where most of ths work was done. The second author was supported n part by NSF Grant No. DMS References [1] A. Ashkhmn, E. Knll, Nonbnary quantum stablzer codes, IEEE Trans. Inform. Theory 47 (2001) [2] A. Ashkhmn, S. Ltsyn, M.A. Tsfasman, Asymptotcally good quantum codes, quant-ph/ , [3] H. Barnum, C. Crépeau, D. Gottesman, A. Smth, A. Tapp, Authentcaton of quantum messages, quant-ph/ , [4] A.R. Calderbank, E.M. Rans, P.W. Shor, N.J.A. Sloane, Quantum error correcton va codes over GF (4), IEEE Trans. Inform. Theory 44 (1998) [5] A.R. Calderbank, P.W. Shor, Good quantum error-correctng codes exst, Phys. Rev. A 54 (1996) [6] H. Chen, Some good quantum error-correctng codes from algebrac geometry codes, IEEE Trans. Inform. Theory 47 (2001) [7] H. Chen, S. Lng, C. Xng, Asymptotcally good quantum codes exceedng the Ashkhmn Ltsyn Tsfasman bound, IEEE Trans. Inform. Theory 47 (2001) [8] I. Dumer, Concatenated codes and ther multlevel generalzatons, Handbook of codng theory, Vol. II, North-Holland, Amsterdam, 1998, pp [9] K. Feng, Quantum codes exst [[6, 2, 3] p and [[7, 3, 3] p (p 3) exst, IEEE Trans. Inform. Theory 48 (2002) [10] G.D. Forney Jr., Concatenated codes, M.I.T. Research Monograph, No. 37, The MIT Press, Cambrdge, MA, [11] A. Garca, H. Stchtenoth, A tower of Artn Schreer extensons of functon felds attanng the Drnfeld Vladut bound, Invent. Math. 121 (1995) [12] V.D. Goppa, Codes assocated wth dvsors, Problemes Peredach Informats 13 (1977) (Englsh translaton n Problems Inform Transmsson 13 (1977) 22 27). [13] M. Grassl, T. Beth, M. Rötteler, On optmal quantum codes, Internat. J. Quantum Inform. 2 (2004) [14] V. Guruswam, M. Sudan, Improved decodng of Reed Solomon and algebrac-geometry codes, IEEE Trans. Inform. Theory 45 (1999) [15] R. Matsumoto, Improvement of Ashkhmn Ltsyn Tsfasman bound for quantum codes, IEEE Trans. Inform. Theory 48 (2002) [16] R. Matsumoto, T. Uyematsu, Constructng quantum error-correctng codes for p m -state systems from classcal error-correctng codes, IEICE Trans. Fundamentals E83-A (2000) [17] E.M. Rans, Nonbnary quantum codes, IEEE Trans. Inform. Theory 45 (1999) [18] J. Schwnger, Untary operator bases, Proc. Nat. Acad. Sc. 46 (1960) [19] A.M. Steane, Multple partcle nterference and quantum error correcton, Proc. Roy. Soc. London Ser. A 452 (1996) [20] H. Stchtenoth, Algebrac Functon Felds and Codes, Sprnger, Berln, [21] M.A. Tsfasman, S.G. Vladut, Algebrac Geometrc Codes, Kluwer, Dordrecht, [22] M.A. Tsfasman, S.G. Vlǎduţ, Th. Znk, Modular curves, Shmura curves, and Goppa codes, better than the Varshamov Glbert bound, Math. Nachr. 109 (1982)

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