Quantum Codes from Generalized Reed-Solomon Codes and Matrix-Product Codes

Transcription

1 1 Quntum Codes from Generlzed Reed-Solomon Codes nd Mtrx-Product Codes To Zhng nd Gennn Ge Abstrct rxv: v1 [cs.it] 5 Aug 015 One of the centrl tsks n quntum error-correcton s to construct quntum codes tht hve good prmeters. In ths pper, we construct three new clsses of quntum MDS codes from clsscl Hermtn self-orthogonl generlzed Reed-Solomon codes. We lso present some clsses of quntum codes from mtrx-product codes. It turns out tht mny of our quntum codes re new n the sense tht the prmeters of quntum codes cnnot be obtned from ll prevous constructons. Index Terms Quntum MDS codes, generlzed Reed-Solomon codes, quntum codes, mtrx-product codes, Hermtn constructon. I. INTRODUCTION Quntum error-correctng codes hve ttrcted much ttenton s schemes tht protect quntum sttes from decoherence durng quntum computtons nd quntum communctons. After the poneerng works n [3], [4], the theory of quntum codes hs developed rpdly. In [5], [6], Clderbnk et. l found strong connecton between lrge clss of quntum codes whch cn be seen s n nlog of clsscl group codes, nd self-orthogonl codes over F 4. Ths ws then generlzed to the nonbnry cse n [], []. Recently, mny quntum codes hve been constructed by clsscl lner codes wth Euclden or Hermtn self-orthogonlty [1], [8], [5]. Let q be prme power, q-ry ((n,k,d)) quntum code s K-dmensonl vector subspce of the Hbert spce (C q ) n C qn whch cn detect up to d 1 quntum errors. Let k log q K, we use [[n,k,d]] q to denote q-ry ((n,k,d)) quntum code. As n clsscl codng theory, one of the centrl tsks n quntum codng theory s to construct quntum codes wth good prmeters. The followng theorem gves bound on the chevble mnmum dstnce of quntum code. Theorem 1.1. ([17], [18] Quntum Sngleton Bound) Quntum codes wth prmeters [[n,k,d]] q stsfy d n k +. A quntum code chevng ths quntum Sngleton bound s clled quntum mxmum-dstnce-seprble (MDS) code. Just s n the clsscl lner codes, quntum MDS codes form n mportnt fmly of quntum codes. Constructng quntum MDS codes hs become centrl topc for quntum codes n recent yers. As we know, the length of nontrvl q-ry quntum MDS codes cnnot exceed q + 1 f the clsscl MDS conjecture holds. The quntum MDS codes of length up to q+1 hve been constructed for ll possble dmensons [11] [1], nd mny quntum MDS codes of length between q +1 nd q + 1 hve lso been obtned (see [3], [7], [14], [15], [16], [19], [0], [1] nd the references theren). However, lmost ll known q-ry quntum MDS codes hve mnmum dstnce less thn or equl to q +1. In ths pper, we construct three clsses of quntum MDS codes s follows: (1) Let q be n odd prme power wth the form m+1, then there exsts q-ry [[ q 1, q 1 d+,d]]-quntum MDS code, where d (+1)m+1. The reserch of G. Ge ws supported by the Ntonl Nturl Scence Foundton of Chn under Grnt No nd Grnt No , the Importton nd Development of Hgh-Clber Tlents Project of Bejng Muncpl Insttutons, nd Zhejng Provncl Nturl Scence Foundton of Chn under Grnt No. LZ13A T. Zhng s wth the School of Mthemtcl Scences, Cptl Norml Unversty, Bejng , Chn. He s lso wth the Deprtment of Mthemtcs, Zhejng Unversty, Hngzhou 31007, Chn (e-ml: tzh@zju.edu.cn). G. Ge s wth the School of Mthemtcl Scences, Cptl Norml Unversty, Bejng , Chn. He s lso wth Bejng Center for Mthemtcs nd Informton Interdscplnry Scences, Bejng, , Chn (e-ml: gnge@zju.edu.cn).

2 () Let q be n odd prme power wth the form m 1, then there exsts q-ry [[ q 1 q+1, q 1 q d+3,d]]-quntum MDS code, where d (+1)m. (3) Let q be n odd prme power wth the form (+1)m 1, then there exsts q-ry [[ q 1 +1 q+1, q 1 +1 q d+3,d]]- quntum MDS code, where d (+1)m 1. In ddton, we lso show the exstence of q-ry quntum MDS codes wth length q 1 nd mnmum dstnce d for ny d q, where q s n odd prme power. Ths result extends those gven n [11], [1]. Mtrx-product codes were ntroduced n [4] s generlzton of severl well known constructons of longer codes from old ones, for exmple, the ( +b)-constructon nd the (+x b+x +b+x)-constructon. In [10], the uthors construct some new quntum codes from mtrx-product codes nd the Euclden constructon. Motvted by ths work, we wll gve new constructon of quntum codes by mtrx-product codes nd the Hermtn constructon. Some of them hve better prmeters thn the quntum codes lsted n tble onlne [9]. Ths pper s orgnzed s follows. In Secton II we recll the bscs bout lner codes, quntum codes nd mtrx-product codes. In Secton III, we gve three new clsses of quntum MDS codes from generlzed Reed-Solomon codes. In Secton IV, we present new constructon of quntum codes v mtrx-product codes nd the Hermtn constructon. II. PRELIMINARIES Throughout ths pper, let F q be the fnte feld wth q elements, where q s prme power. A lner [n,k] code C over F q s k-dmensonl subspce of F n q. The weght wt(x) of codeword x C s the number of nonzero components of x. The dstnce of two codewords x,y C s d(x,y) wt(x y). The mnmum dstnce d of C s the mnmum dstnce between ny two dstnct codewords of C. An [n,k,d] code s n [n,k] code wth the mnmum dstnce d. Gven two vectors x (x 0,x 1,,x n 1 ), y (y 0,y 1,,y n 1 ) F n q, there re two nner products we re nterested n. One s the Euclden nner product whch s defned s x,y E n 1 0 x y. When q l, where l s prme power, then we cn lso consder the Hermtn nner product whch s defned by x,y H n 1 0 x y l. The Euclden dul code of C s defned s C E {x F n q x,y E 0 for ll y C}. Smlrly the Hermtn dul code of C s defned s C H {x F n q x,y H 0 for ll y C}. A lner code C s clled Euclden (Hermtn) self-orthogonl f C C E (C C H, respectvely), nd C s clled Euclden (Hermtn) dul contnng f C E C (C H C, respectvely). For vector x (x 1,,x n ) F n q, let x q (x q 1,,xq n). For subset S of F n q, we defne S q to be the set {x q x S}. Then t s esy to see tht for q -ry lner code C, we hve C H (C q ) E. Therefore, C s Hermtn self-orthogonl f nd only f C (C q ) E,.e., C q C E. A. Quntum Codes In ths subsecton, we recll the bscs of quntum codes. Let q be power of prme number p. A qubt v s nonzero vector n C q whch cn be represented s v x F q c x x, where { x x F q } s bss of C q. For n 1, the n-th tensor product (C q ) n C q n hs bss { 1 n 1 n ( 1,, n ) F n q }, then n n-qubt s nonzero vector n C qn whch cn be represented s v F c, where c n q C. Let ζ p be prmtve p-th root of unty. The quntum errors n q-ry quntum system re lner opertors ctng on C q nd cn be represented by the set of error bses: ε n {T R b,b F q }, where T R b s defned by T R b x ζ Tr Fq/Fp (bx) p x+. The set E n {ζ l p T R b 0 l p 1, ( 1,, n ),b (b 1,,b n ) F n q } forms n error group, where ζ l pt R b s defned by ζpt l R b x ζpt l 1 R b1 x 1 T n R bn x n ζ l+tr Fq/Fp (bx) p x+,

3 3 for ny x x 1 x n, x (x 1,,x n ) F n q. For n error e ζl p T R b, ts quntum weght s defned by w Q (e) {1 n (,b ) (0,0)}. A subspce Q of C qn s clled q-ry quntum code wth length n. The q-ry quntum code hs mnmum dstnce d f nd only f t cn detect ll errors n E n of quntum weght less thn d, but cnnot detect some errors of weght d. A q-ry [[n,k,d]] q quntum code s q k -dmensonl subspce of C qn wth mnmum dstnce d. There re mny methods to construct quntum codes, nd the followng theorem s one of the most frequently used constructon methods. Theorem.1. ([] Hermtn Constructon) If C s q -ry Hermtn dul-contnng [n,k,d] code, then there exsts q-ry [[n,k n, d]]-quntum code. Then we hve the followng corollry. Corollry.. There s q-ry [[n,n k,k +1]] quntum MDS code whenever there exsts q -ry clsscl Hermtn self-orthogonl [n,k,n k +1]-MDS code. B. Mtrx-Product Codes In ths subsecton, we revew some nottons nd results of mtrx-product codes. Let C 1,C,,C s be fmly of s codes of length m over F q nd A ( j ) be n s l mtrx wth entres n F q. Then, the mtrx-product code [C 1,C,,C s ] A s defned s the code over F q of length ml wth genertor mtrx 11 G 1 1 G 1 1l G 1 1 G G l G......, s1 G s s G s sl G s where G, 1 s, s genertor mtrx for the code C. The followng theorem gves chrcterzton of mtrx-product codes. Theorem.3. [13], [10] The mtrx-product code [C 1,C,,C s ] A gven by sequence of [m,k,d ] lner codes C over F q nd full-rnk s l mtrx A s lner code whose length s ml, t hs dmenson s k nd mnmum dstnce lrger thn or equl to δ mn 1 s {d δ }, where δ s the mnmum dstnce of the code on F l q generted by the frst rows of the mtrx A. In order to construct quntum codes from mtrx-product codes, we need the followng theorem. Theorem.4. [4] Assume tht C 1,C,,C s re fmly of lner codes of length m nd A s nonsngulr s s mtrx, then the followng equlty of codes hppens where B t denotes the trnspose of the mtrx B. ([C 1,C,,C s ] A) [C 1,C,,C s ] (A 1 ) t, III. NEW QUANTUM MDS CODES FROM GENERALIZED REED-SOLOMON CODES We frst recll the bscs of generlzed Reed-Solomon codes. Choose n dstnct elements 1,, n of F q nd n nonzero elements v 1,,v n of F q. For 1 k n, we defne the code GRS k (,v) : {(v 1 f( 1 ),,v n f( n )) f(x) F q [x] nd deg(f(x)) < k}, where nd v denote the vectors ( 1,, n ) nd (v 1,,v n ), respectvely. The code GRS k (,v) s clled generlzed Reed-Solomon code over F q. It s well known tht generlzed Reed-Solomon code GRS k (,v) s n MDS code wth prmeters [n, k, n k + 1]. The followng lemm presents crteron to determne whether or not generlzed Reed-Solomon code s Hermn self-orthogonl. We ssume tht 0 0 : 1.

4 4 Lemm 3.1. Let ( 1,, n ) F n q nd v (v 1,,v n ) (F q ) n, then GRS k (,v) GRS k (,v) H f nd only f n vq+1 qj+l 0 for ll 0 j,l k 1. Proof: Note tht GRS k (,v) GRS k (,v) H f nd only f GRS k (,v) q GRS k (,v) E. It s obvous tht GRS k (,v) q hs bss {(v q 1 q 1,,vq n q n) 0 k 1}, nd GRS k (,v) hs bss {(v 1 1,,v n n) 0 k 1}. So GRS k (,v) q GRS k (,v) E f nd only f n vq+1 qj+l 0 for ll 0 j,l k 1. Now we consder generlzed Reed-Solomon codes over F q to construct quntum codes. A. New Quntum MDS Codes of Length q 1 Theorem 3.. Let q be n odd prme wth the form m+1, ω be fxed prmtve element of F q nd n q 1. Suppose (ω,ω,,ω n ) F n q, v (ω,ω,ω,ω,,ω,ω,,ω,ω,ω,ω,,ω,ω ) F n q nd 1 k (+1)m. Then GRS k (,v) GRS k (,v) H. Note tht Proof: For 0 j,l k 1 (+1)m 1, we hve qj+l ω (q+1) (ω [s()+(+1)](qj+l) +ω [s()+(+)](qj+l) ) ω (q+1) (ω (+1)(qj+l) +ω (+)(qj+l) ) ω s()(qj+l). ω s()(qj+l) 0; f q+1 (qj +l), q+1 q+1 ; f (qj +l). Now ssume tht qj+l t q+1. We clm (t+1), otherwse t r 1 for 1 r snce 0 j,l q. But qj +l (r 1)q+1 (r Thus 1)q +r + qj+l q +1, then (+1)m l r + q 1, whch s contrdcton. ω (q+1) (ω (+1)(qj+l) +ω (+)(qj+l) ) q +1 ωt(q+1) ω (q+1)(t+1) + q +1 q+1 ωt 0. Then by Lemm 3.1, GRS k (,v) GRS k (,v) H. ω (q+1)(t+1) Theorem 3.3. Let q be n odd prme power wth the form m+1, then there exsts q-ry [[ q 1, q 1 d+,d]]-quntum MDS code, where d (+1)m+1. Proof: The proof s strghtforwrd pplcton of Corollry. nd Theorem 3.. As n mmedte consequence of Theorem 3.3, we hve the followng corollry by tkng 1. Corollry 3.4. Let q be n odd prme power, then there exsts q-ry [[q 1,q d+1,d]]-quntum MDS code, where d q. Remrk 3.5. In [11], [1], the uthors showed tht there exsts q-ry [[q 1,q d+1,d]]-quntum MDS code, where q s n odd prme power nd d q 1. Obvously, our result hs lrger mnmum dstnce.

5 5 B. New Quntum MDS Codes of Length q 1 q +1 In ths subsecton, we wll construct quntum MDS codes of length q 1 lemm. q +1. For our purpose, we need the followng Lemm 3.6. [15] Let A be n (n 1) n mtrx of rnk n 1 over F q. Then the equton Ax 0 hs nonzero soluton n F q f nd only f A (q) nd A re row equvlent, where A (q) s obtned from A by rsng every entry to ts q-th power. Theorem 3.7. Let q be n odd prme power wth the form m 1 nd n q 1 q + 1, then there exst Fn q nd v (F q ) n such tht GRS k (,v) GRS k (,v) H for 1 k (+1)m 3. Proof: Let ω be fxed prmtve element of F q. We lso let A be n (m ) (m 1) mtrx wth A j ω j(m 3+()( 1)) for 1 m, 1 j m 1. Snce (m 3 + (q 1)( 1))q (m 3 + (q 1)(m )) (mod q 1) for 1 m, then A (q) nd A re row equvlent. By Lemm 3.6, there exsts c F m 1 q such tht A c t 0. Note tht by deletng ny one column of mtrx A, the remnng mtrx s Vndermonde mtrx, hence ll coordntes of c re nonzero. So we cn represent c s c (ω 1(q+1),,ω m 1(q+1) ). Now let (ω,ω 4,,ω q+1,ω q+1+,,ω q+,,ω q q +,,ω ) F n q nd v (ω 1,, ω m 1,ω 1 (m 3),,ω m 1 (m 3),,ω 1 (m 3)(q ),,ω m 1 (m 3)(q ) ) (F q ) n. Then for 0 j,l k 1 (+1)m 4, we hve Note tht qj+l m 1 q ω (q+1)+(qj+l) ω (q+1)(qj+l m+3)s. q ω (q+1)(qj+l m+3)s 0; f (q 1) (qj +l m+3), q 1; f (q 1) (qj +l m+3). Assume qj + l m + 3 t(q 1), we clm tht t m,m 1 (mod m). Otherwse, f t m (mod m), let t rm + m, then 0 r. If r 1, then qj + l t(q 1) + m 3 (mr + m 3)q + (q mr 1) (mr + m 3)q + ( r)m nd ( r)m > ( + 1)m 4 whch s contrdcton. If r, then qj +l t(q 1)+m 3 (m+m 3)q +(m ) nd m+m 3 > (+1)m 4, whch s lso contrdcton. Smlrly, t m 1 (mod m). Hence qj +l (mod q 1 ) {t(q 1)+m 3 0 t m 3}. Thus qj+l (q 1) 0, m 1 ω (q+1)+(qj+l) where the lst equton s from the defnton of c. Then by Lemm 3.1, GRS k (,v) GRS k (,v) H. Theorem 3.8. Let q be n odd prme power wth the form m 1, then there exsts q-ry [[ q 1 q+1, q 1 q d+3,d]]- quntum MDS code, where d (+1)m. Proof: The proof s strghtforwrd pplcton of Corollry. nd Theorem 3.7. In prtculr, tkng 1, we obtn the followng corollry. Corollry 3.9. Let q 5 be n odd prme power, then there exsts q-ry [[ q 1 q +1, q 1 q d+3,d]]-quntum MDS code, where d q 1.

6 6 C. New Quntum MDS Codes of Length q 1 +1 q +1 In ths subsecton, we consder quntum MDS codes of length q 1 +1 q +1. Theorem Let q be n odd prme power wth the form (+1)m 1 nd n q 1 +1 q +1, then there exst Fn q nd v (F q ) n such tht GRS k (,v) GRS k (,v) H for 1 k (+1)m. Proof: Let ω be fxed prmtve element of F q. We lso let A be n (m ) (m 1) mtrx wth A j ω (() 1)j for 1 m, 1 j m 1. Snce ((q 1) 1)q ((q )(q 1) 1) (mod q 1) for 1 q 1, then A (q) nd A re row equvlent. By Lemm 3.6, there exsts c F m 1 q such tht A c t 0. Snce by deletng ny one column of mtrx A, the remnng mtrx s Vndermonde mtrx, then ll coordntes of c re nonzero. Hence we cn represent c s c (ω 1(q+1),,ω m 1(q+1) ). Now let (ω +1,ω (+1),,ω q,ω q++,,ω q+1,,ω q +,,ω q ) F n q nd v (ω 1,,ω m 1,ω 1+1,,ω m 1+1,,ω 1+q,,ω m 1+q ) (F q ) n. Then for 0 j,l k 1 (+1)m 3, we hve Note tht qj+l m 1 q ω (q+1)+(qj+l) ω (q+1)(qj+l+1)s. q ω (q+1)(qj+l+1)s 0; f (q 1) (qj +l+1), q 1; f (q 1) (qj +l+1). Assume qj + l +1 t(q 1), then t 0,m 1 (mod m). Otherwse, f t 0 (mod m), let t rm. Then qj + l t() 1 (rm 1)q+(+1 r)m nd mn{rm 1,(+1 r)m }> (+1)m 3, whch s contrdcton. Smlrly, t m 1 (mod m). Thus qj+l (q 1) 0, m 1 ω (q+1)+(qj+l) where the lst equton s from the defnton of c. Then by Lemm 3.1, GRS k (,v) GRS k (,v) H. Theorem Let q be n odd prme power wth the form (+1)m 1, then there exsts q-ry [[ q 1 +1 q +1, q 1 +1 q d+3,d]]-quntum MDS code, where d (+1)m 1. Proof: The proof s strghtforwrd pplcton of Corollry. nd Theorem In prtculr, tkng 0, we obtn the followng corollry. Corollry 3.1. Let q be n odd prme power, then there exsts q-ry [[q q,q q d +,d]]-quntum MDS code, where d q. IV. NEW QUANTUM CODES FROM MATRIX-PRODUCT CODES Let A ( j ) be n s s mtrx wth entres n F q, we defne A (q) ( q j ). Then we hve the followng result. Lemm 4.1. Let A ( j ) be nonsngulr s s mtrx such tht A (q) s lso nonsngulr. Suppose there exst lner codes C such tht C H C for 1,,,s. Then ([C 1,C,,C s ] A) H [C 1,C,,C s ] [(A (q) ) 1 ] t.

7 7 Proof: By Theorem.4, we hve ([C 1,C,,C s ] A) H ([C q 1,Cq,,Cq s ] A(q) ) E [(C q 1 ) E,(C q ) E,,(C q s) E ] [(A (q) ) 1 ] t [C1 H,C H,,C H ] [(A (q) ) 1 ] t [C 1,C,,C s ] [(A (q) ) 1 ] t. s Now we would lke to use mtrx-product codes to construct quntum codes. Corollry 4.. Let q p t be n odd prme power, where p s prme number. Suppose C 1,C re lner codes over F q wth prmeters [n,k,d ] nd C H C, 1,. Then there exsts Hermtn dul contnng [n,k 1 +k, mn{d 1,d }] code over F q. Proof: Tke A ( p 1 ), then [(A (q) ) 1 ] t ( p+1 p+1 p+1 p 1 ). By Lemm 4.1, we hve ([C 1,C ] A) H [C 1,C ] [(A (q) ) 1 ] t [C 1,C ] A. Applyng Theorem.3, [C 1,C ] A s Hermtn dul contnng [n,k 1 +k, mn{d 1,d }] code. The followng result cn be found n [11], [14], [15]. Theorem 4.3. Let q be n odd prme power, then 1) there exsts q -ry Hermtn dul contnng [q +1,q + d,d] code for 1 d q +1; ) there exsts q -ry Hermtn dul contnng [q,q +1 d,d] code for d q. By combng Theorems 4.3, 3. nd Corollry 4., we cn mmedtely get the followng lemm. Lemm 4.4. Let q be n odd prme power, then 1) there exsts q -ry Hermtn dul contnng [q +,q +4 d d,d] code, where d q +1 s even; ) there exsts q -ry Hermtn dul contnng [q +,q +3 d d 1,d] code, where d q +1 s odd; 3) there exsts q -ry Hermtn dul contnng [q,q + d d,d] code, where d q s even; 4) there exsts q -ry Hermtn dul contnng [q,q +1 d d 1,d] code, where d q s odd; 5) there exsts q -ry Hermtn dul contnng [q,q d d,d] code, where d q s even; 6) there exsts q -ry Hermtn dul contnng [q,q 1 d d 1,d] code, where d q s odd. Then by the Hermtn constructon nd Lemm 4.4, we hve the followng theorem. Theorem 4.5. Let q be n odd prme power, then 1) there exsts q-ry [[q +,q +6 3d, d]] quntum code, where d q +1 s even; ) there exsts q-ry [[q +,q +5 3d, d]] quntum code, where d q +1 s odd; 3) there exsts q-ry [[q,q +4 3d, d]] quntum code, where d q s even; 4) there exsts q-ry [[q,q +3 3d, d]] quntum code, where d q s odd; 5) there exsts q-ry [[q,q + 3d, d]] quntum code, where d q s even; 6) there exsts q-ry [[q,q +1 3d, d]] quntum code, where d q s odd. In Tble I, we lst some quntum codes obtned from Theorem 4.5. The tble shows tht our quntum codes hve better prmeters thn the prevous quntum codes vlble.

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