Asymmetric and Symmetric Subsystem BCH Codes and Beyond

1 Asymmetric and Symmetric Subsys tem BCH Codes and Be yond Salah A. Aly Department of Computer Science T e xa s A&M Uni versity , College Station, TX 77843 , USA Email: salah@cs .tamu.edu Abstract —Recently , the theory of quantum err or control codes has been extended to su bsystem codes ov er symmetric and asymmetric q uantum channels – qubit-flip and phase-shi ft errors may hav e equal or different probabilities. Pr evious work in constructing quantum error control codes has focused on code constructions f or symmetric qu antum ch annels. In this paper , we deve lop a theory and establish the connection between asymmet- ric quantum codes and subsystem co d es. W e pr esent families of subsystem and asymmetric quantum codes derived, once again, from classical BCH and RS codes over fi nite fields. Particularly , we derive an in teresting asymmetric and symmetric sub system codes based on classical BCH codes with parameters [[ n, k, r, d ]] q , [[ n, k , r , d z /d x ]] q and [[ n, k ′ , 0 , d z /d x ]] q fo r arbitrary values of code len gths and dimensions. W e establi sh asymmetric Si ngleton and Hamming bound s on asymmetric quantum and subsystem code parameters ; and deriv e optimal asymmetric MDS subsystem codes. F inally , our constructions ar e well expl ained by an illustrative example. This paper is written on th e occasion of the 50th anniver sary of the discov ery of classical BCH co d es and their quantum counterparts were derived nearly 10 years ago. I . I N T RO D U C T I O N In 1996 , An drew Steane stated in his seminal work [4 3, page 2, col. 2][42], [44] ”The notation { n, K, d 1 , d 2 } is her e intr oduced to iden tify a ’quantu m code, ’ mean ing a code by which n qua ntum bits can stor e K bits of q uantum in formation and a llow co rr ection of up to ⌊ ( d 1 − 1) / 2 ⌋ amplitud e err ors, and simu ltaneously up to ⌊ ( d 2 − 1) / 2 ⌋ phase err ors. ” This paper is motiv ated by this statement, in w hich we construct efficient quan tum codes that correct amplitu de (qubit-flip) errors an d phase-shift errors separately . In [ 34], it was said that ”BCH codes are a mong the powerful codes” . W e add ress construction s of quantum and subsystem codes based on Bose- Chaudhu ri-Hocqu enghem (BCH) co des over finite fields for quantum symmetric and asymmetric chann els. Many quantum error control code s (QEC) have been co n- structed over the la st decade to protect quan tum inf ormation against noise and decoherence . In codin g theory , researchers have fo cused on bo unds a nd the co nstruction aspects of q uan- tum co des fo r large and asympto matic code len gths. On the other hand, physicists intend to study th e ph ysical r ealization and mech anical quantum oper ations of these codes for shor t code len gths. As a result, various app roaches to p rotect q uan- tum informatio n a gainst no ise a nd decoheren ce are proposed including stabilizer blo ck codes, quantu m conv o lutional codes, entangled -assisted quantum error co ntrol code s, decoh erence free sub spaces, non additive co des, and subsystem codes [1 3], [18], [21], [ 22], [38], [33 ] , [36], [27], [4 7] and r eferences therein. Asymmetric qu antum con trol codes (A Q EC), in which quantum errors h av e different probab ilities — Pr Z > Pr X , are mo re efficient than the symmetric quan tum error co ntrol codes (QEC), in which quan tum errors have equal probabilities — P r Z = P r X . It is a rgued in [26] that dephasing (loss of phase co herence, phase-shif ting) will happ en mo re freq uently than relax ation (exchang e of energy with the en v ironmen t, qubit-flipp ing). The noise le vel in a qubit is spe cified by the relaxation T 1 and d ephasing time T 2 ; fu rthermo re the relation between these two values is g i ven b y 1 /T 1 = 1 / (2 T 1 ) + Γ p ; this has been well explained by physicists in [19], [26], [46]. The ratio between the probabilities of q ubit-flip X and phase-shift Z is typically ρ ≈ 2 T 1 /T 2 . The interpretatio n is that T 1 is m uch larger than T 2 , meaning the pho tons take much more time to flip from the groun d state to the excited state. Howev er, they ch ange rapid ly f rom o ne excited state to another . Motiv ated by this, one needs to design quant um codes that are suitable for this physical phenomena. The fault tolerant op erations of a quantum com puter carry ing co n- trolled an d measur ed quantu m information over asymmetric channel h av e been investigated in [2], [14], [15], [45], [46], [1] and referen ces th erein. Fault-tolerant operations of QEC are inves tig ated for examp le in [3], [1], [ 22], [37], [41], [45], [30] and references therein. Subsystem cod es (SSC) a s we p refer to call them we re mentioned in the unpub lished work by Knill [ 31], [ 29], in which h e attemp ted to gen eralize the theory of quantum error- correcting c odes into subsystem codes. Su ch cod es with their stabilizer fo rmalism were reintrodu ced recently [11], [14], [15], [28], [ 32], [35]. Th e constru ction aspe cts of these codes are given in [9], [8], [11]. Here we expand our un derstandin g and introdu ce asy mmetric subsystem codes (ASSC). The code s deriv ed in [ 10], [12] for primitiv e and n on- primitive quantum BCH codes assume that qubit-flip errors, phase-shift errors, and their combination occur with equal probab ility , where Pr Z = P r X = Pr Y = p/ 3 , P r I = 1 − p , and { X , Z, Y , I } are the bin ary Pauli oper ators P shown in Section I I, see [18], [40]. W e aim to gen eralize these codes over asy mmetric quantum chann els. In this p aper we give families of asymmetric quan tum error co ntrol cod es (A QEC’ s) motiv ated by the work from [19], [26], [46]. Assume we have a classical good error contro l code C i with par ameters 2 [[ n, k i , d i ]] q for i ∈ { 1 , 2 } — cod es with high minimum distances d i and high r ates k i /n . W e can co nstruct a qu antum code based on these two classical codes, in whic h C 1 controls the q ubit-flip erro rs while C 2 takes ca re of the phase-shift errors, see Lemma 4. Our f ollowing theorem estab lishes the connection b etween two c lassical c odes and QEC, A QEC, SCC, ASSC. Theorem 1 (CSS A QE C and AS SC): Let C 1 and C 2 be two classical co des with parameters [ n , k 1 , d 1 ] q and [ n, k 2 , d 2 ] q respectively , and d x = min  wt( C 1 \ C ⊥ 2 ) , wt( C 2 \ C ⊥ 1 )  , and d z = max  wt( C 1 \ C ⊥ 2 ) , wt( C 2 \ C ⊥ 1 )  . i) if C ⊥ 2 ⊆ C 1 , then th ere exists an A QEC with parameters [[ n, dim C 1 − dim C ⊥ 2 , wt( C 2 \ C ⊥ 1 ) / wt( C 1 \ C ⊥ 2 )]] q that is [[ n, k 1 + k 2 − n, d z /d x ]] q . Also, the re exists a QEC with p arameters [[ n, k 1 + k 2 − n, d x ]] q . ii) From [i], th ere exists an SSC with parameter s [[ n, k 1 + k 2 − n − r , r, d x ]] q for 0 ≤ r < k 1 + k 2 − n . iii) If C ⊥ 2 = C 1 ∩ C ⊥ 1 ⊆ C 2 , then there exists an ASSC with parameters [[ n, k 2 − k 1 , k 1 + k 2 − n, d z /d x ]] q and [[ n, k 1 + k 2 − n, k 2 − k 1 , d z /d x ]] q . Furthermo re, all constructed co des a re p ure to th eir m inimum distances. A well-k nown construction on the theor y of quantum er- ror control code s is called CSS constructio ns. Th e codes [[5 , 1 , 3]] 2 , [[7 , 1 , 3]] 2 , [[9 , 1 , 3]] 2 , and [[9 , 1 , 4 , 3]] 2 have been in vestigated in several research papers that analyzed their stabilizer structure, circuits, and fault tolerant quantum com- puting operations. On this p aper, we present se veral A QEC codes, including a [[15 , 3 , 5 / 3]] 2 code, which encodes thr ee logical qub its into 15 physical qub its, detects 2 qu bit-flip and 4 ph ase-shift errors, respectively . As a r esult, many of the quantu m constru cted co des and families of QEC f or large len gths nee d further investigations. W e believe tha t their generalizatio n is a direct consequen ce. The p aper is organized as follows. Sections II, III, an d V are d ev oted to A QEC and two families o f A QEC, A QEC-BCH and A QEC-RS. W e estab lish co nditions on the existence of these families over finite fields. Sections IV and VI addr ess the subsystem co de co nstructions an d their relation to asymmetr ic quantum cod es. W e show the tra deoff between subsystem codes and A QEC. Section VI p resents the bound on A QEC and ASSC co de parameter s. Fina lly , the paper is conclu ded with a discussion in Section VII. I I . A S Y M M E T R I C Q U A N T U M C O D E S In this sectio n we shall give some primary defin itions an d introdu ce A QEC construction s. Consider a quantu m sy stem with two-dimension al state space C 2 . The basis vectors v 0 =  1 0  , v 1 =  0 1  (1) can be used to rep resent the classical bits 0 an d 1 . It is customary in qu antum information processing to use Dir ac’ s ket notation for th e basis vectors; namely , the vector v 0 is denoted by the ket | 0 i an d the vector v 1 is den oted by ket | 1 i . Any p ossible state of a two-dim ensional q uantum system is giv en by a linear comb ination of the for m a | 0 i + b | 1 i =  a b  , w here a, b ∈ C and | a | 2 + | b | 2 = 1 , (2) In quan tum informatio n pr ocessing, the o perations manip- ulating qu antum bits fo llow the rules of q uantum mech anics, that is, an oper ation that is not a measureme nt must be realize d by a unitary op erator . For examp le, a quantum bit can be flipped by a quantu m NOT gate X th at transfers the qubits | 0 i and | 1 i to | 1 i and | 0 i , respectiv ely . Thus, this o peration acts on a general quantum state as follows. X ( a | 0 i + b | 1 i ) = a | 1 i + b | 0 i . W ith respect to the computation al basis, the quantu m NOT gate X represents the q ubit-flip errors. X = | 0 ih 1 | + | 1 ih 0 | =  0 1 1 0  . (3) Also, let Z =  1 0 0 − 1  be a matrix repre sents th e q uan- tum phase-shif t errors that change s the phase of a q uantum system ( states). Z ( a | 0 i + b | 1 i ) = a | 0 i − b | 1 i . (4) Other p opular operatio ns in clude the combine d bit and p hase- flip Y = iZ X , and the Had amard gate H , which ar e represented with r espect to the computational b asis by the matrices Y =  0 − i i 0  , H = 1 √ 2  1 1 1 − 1  . (5) Connection to Classical Binary Co des. L et H i and G i be the par ity c heck an d gen erator matrices of a classical code C i with p arameters [ n, k i , d i ] 2 for i ∈ { 1 , 2 } . The commu tati v ity condition o f H 1 and H 2 is stated as H 1 .H T 2 + H 2 .H T 1 = 0 . (6) The stabilize r of a quantu m code based o n the parity check matrices H 1 and H 2 is giv en by H stab =  H 1 | H 2  . (7) One of these two classical codes co ntrols the ph ase-shift errors, while th e other codes co ntrols the bit-flip erro rs. Hen ce the CSS constru ction of a binary A QEC can b e s tated as follows. Hence the codes C 1 and C 2 are mappe d to H x and H z , respectively . Definition 2: Given two classical b inary c odes C 1 and C 2 such that C ⊥ 2 ⊆ C 1 . If we form G =  G 1 0 0 G 2  , a nd H =  H 1 0 0 H 2  , then H 1 .H T 2 − H 2 .H T 1 = 0 (8) 3 Let d 1 = wt( C 1 \ C 2 ) an d d 2 = w t ( C 2 \ C ⊥ 1 ) , su ch tha t d 2 > d 1 and k 1 + k 2 > n . If we assum e that C 1 corrects the q ubit- flip errors and C 2 corrects the phase-shift errors, then there exists A QEC with param eters [[ n, k 1 + k 2 − n, d 2 /d 1 ]] 2 . (9) W e can always ch ange the r ules of C 1 and C 2 to adjust th e parameters. A. Higher F ields and T o tal Err or Gr ou ps W e can briefly discuss the the ory in terms of higher finite fields F q . Let H be the Hilbert space H = C q n = C q ⊗C q ⊗ ... ⊗ C q . Let | x i be the vectors o f ortho normal basis of C q , where the labels x are elements in the finite field F q . Let a, b ∈ F q , the unitary operato rs X ( a ) and Z ( b ) in C q are stated as: X ( a ) | x i = | x + a i , Z ( b ) | x i = ω tr( bx ) | x i , (10) where ω = exp(2 π i /p ) is a p rimitive p th root o f unity and tr is the trace operation from F q to F p Let a = ( a 1 , . . . , a n ) ∈ F n q and b = ( b 1 , . . . , b n ) ∈ F n q . Let us deno te by X ( a ) = X ( a 1 ) ⊗ · · · ⊗ X ( a n ) and , Z ( b ) = Z ( b 1 ) ⊗ · · · ⊗ Z ( b n ) (11) the tensor produc ts of n erro r ope rators. Th e sets E x = { X ( a ) = n O i =1 X ( a i ) | a ∈ F n q , a i ∈ F q } , E z = { Z ( b ) = n O i =1 Z ( b i ) | b ∈ F n q , b i ∈ F q } (12) form an error basis on C q n . W e can define the erro r gr oup G x and G z as follows G x = { ω c E x = ω c X ( a ) | a ∈ F n q , c ∈ F p } , G z = { ω c E z = ω c Z ( b ) | b ∈ F n q , c ∈ F p } . (13) Hence the total error grou p G =  G x , G z  = n ω c n O i =1 X ( a i ) , ω c n O i =1 Z ( b i ) | a i , b i ∈ F q o (14) Let us assum e th at the sets G x and G z represent the qu bit- flip and phase-shif t errors, respectively . Many constructe d quan tum codes assume that the quan - tum errors resulted from decohere nce and n oise h av e equal probab ilities, Pr X = Pr Z . This statem ent as shown by experimental phy sics is not true [ 46], [26]. Th is means the qubit-flip and phase-shift errors happen with d ifferent p rob- abilities. Therefore, it is needed to construct quantum co des that deal with the realistic q uantum no ise. W e derive families of asymmetr ic quantum error c ontrol codes that differentiate between these two k inds of e rrors, Pr Z > Pr X . Definition 3 (AQEC): A q -ar y asymmetric quan tum co de Q , denoted by [[ n, k , d z /d x ]] q , is a q k dimensiona l subspace of the Hilbert space C q n and can contro l all bit-flip e rrors u p C 1 ^T C1 F q 2n C 2 C 1 ^dual C 2 ^dual Fig. 1. Construct ions of asymm etric quantum codes based on two classical codes C 1 and C 2 with parameters [ n, k 1 ] and [ n , d 2 ] such that C i ⊆ C 1+( i m od 2) for i = { 1 , 2 } . A QEC has paramete rs [ [ n, k 1 + k 2 − n, d z /d x ]] q where d x = wt ( C 1 \ C ⊥ 2 ) and d z = wt ( C 2 \ C ⊥ 1 ) to ⌊ d x − 1 2 ⌋ an d all phase-flip e rrors u p to ⌊ d z − 1 2 ⌋ . The co de Q detects ( d 1 − 1) q ubit-flip errors as well as d etects ( d 1 − 1) phase-shift errors. W e use different n otation from th e o ne giv en in [19]. The reason is that we would like to compar e d z and d x as a factor ρ = d z /d x not as a ratio . Th erefore, if d z > d x , then the A QEC has a factor great than on e. Hence, the phase-shift errors affect the quantu m system m ore th an qu bit-flip erro rs do. In our work, we would like to in crease b oth the factor ρ and dimension k of the q uantum code. Connection to Classical nonbinary Codes. Let C 1 and C 2 be two lin ear cod es over th e finite field F q , and let [ n, k 1 , d 1 ] q and [ n, k 2 , d 2 ] q be th eir parameters. For i ∈ { 1 , 2 } , if H i is the parity ch eck m atrix of the co de C i , then dim C ⊥ i = n − k i and rank of H ⊥ i is k i . If C ⊥ i ⊆ C 1+( i mo d 2) , then C ⊥ 1+( i mo d 2) ⊆ C i . So, the r ows of H i which form a basis for C ⊥ i can be extended to f orm a basis for C 1+( i mo d 2) by ad ding some vectors. Also, if g i ( x ) is th e gener ator po lynomial o f a cyclic code C i then k i = n − de g ( g i ( x )) , see [34], [25]. The err or grou ps G x and G z can be mapp ed, respectively , to two classical cod es C 1 and C 2 in a similar manner as in QEC. This c onnection is well-know , see fo r exam ple [18], [38], [39]. Let C i be a classical code such that C ⊥ 1+( i mo d 2) ⊆ C i for i ∈ { 1 , 2 } , then w e have a symmetric qu antum contro l code (A QEC) with par ameters [[ n, k 1 + k 2 − n, d z /d x ]] q . This can be illu strated in the following result. Lemma 4 (CSS A QEC): Let C i be a classical code with parameters [ n, k i , d i ] q such that C ⊥ i ⊆ C 1+( i mo d 2) for i ∈ { 1 , 2 } , an d d x = min  wt( C 1 \ C ⊥ 2 ) , wt( C 2 \ C ⊥ 1 )  , an d d z = max  wt( C 1 \ C ⊥ 2 ) , wt( C 2 \ C ⊥ 1 )  . Then there is asymmetric quantum cod e with param eters [[ n , k 1 + k 2 − n, d z /d x ]] q . T he quantum code is pur e to its minim um d istance meaning that if wt( C 1 ) = wt( C 1 \ C ⊥ 2 ) then th e code is p ure to d x , also if wt( C 2 ) = wt( C 2 \ C ⊥ 1 ) then the code is pure to d z . Therefo re, it is straightfo rward to derive asymmetr ic q uan- tum control codes from two classical codes as shown in Lemma 4. Of cour se, o ne wishes to increase the values o f d z vers. d x for the same code length and dimen sion. Remark 5: The notations o f p urity an d impu rity of A QEC remain the same as sho wn for QEC, the interested reader might consider any pr imary p apers on QEC. 4 I I I . A S Y M M E T R I C Q U A N T U M B C H A N D R S C O D E S In this section we derive classes of A QEC based on classical BCH and RS code s. W e will restrict ourself to the Euclidean construction f or cod es defin ed over F q . Howev er, the gen - eralization to the Hermitian construction for co des defin ed over F q 2 is straigh t fo rward. W e keep the defin itions of BCH codes to a m inimal since the y ha ve be en well-known, see example [10] or a ny textbook on classical coding th eory [3 4], [25], [24]. Let q b e a p ower of a prime and n a p ositiv e in teger such th at g cd( q , n ) = 1 . Recall that the cyclotomic coset S x modulo n is defined as S x = { xq i mo d n | i ∈ Z , i ≥ 0 } . (15) Let m be the multiplicative order of q modu lo n . L et α be a p rimitiv e e lement in F q m . A no nprimitive narrow-sense BCH code C of desig ned distance δ and length n over F q is a cyclic cod e with a gene rator mo nic polyno mial g ( x ) that has α, α 2 , . . . , α δ − 1 as zeros, g ( x ) = δ − 1 Y i =1 ( x − α i ) . (16) Thus, c is a codeword in C if and only if c ( α ) = c ( α 2 ) = . . . = c ( α δ − 1 ) = 0 . The par ity ch eck matrix of th is code can be defined as H bch =      1 α α 2 · · · α n − 1 1 α 2 α 4 · · · α 2( n − 1) . . . . . . . . . . . . . . . 1 α δ − 1 α 2( δ − 1) · · · α ( δ − 1) ( n − 1)      . (17) In general th e d imensions an d minim um distan ces of BCH codes are not known. Howe ver , lower bound s on these two parameters for such co des are giv en by d ≥ δ and k ≥ n − m ( δ − 1) . Fortunately , in [10], [12] exact fo rmulas for the dimensions and minimum distances are gi ven under certain condition s. The f ollowing r esult sh ows the dime nsion o f BCH codes. Theorem 6 (Dimension BCH Codes): Let q be a prime power and g cd( n, q ) = 1 , with or d n ( q ) = m . Then a n arrow- sense BCH co de of length q ⌊ m/ 2 ⌋ < n ≤ q m − 1 over F q with designed distance δ in th e r ange 2 ≤ δ ≤ δ max = min {⌊ nq ⌈ m/ 2 ⌉ / ( q m − 1) ⌋ , n } , has dime nsion of k = n − m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ . (18) Pr oof: See [10, Theo rem 10]. Steane first derived binar y q uantum BCH codes in [43], [4 4]. In addition Gr assl el. a t. gave a family o f qu antum BCH cod es along with tables of best codes [23]. In [1 2], [ 10], while it was a ch allenging task to derive self- orthog onal or dual-con taining conditio ns for BCH codes, we can r elax a nd o mit th ese con ditions by lo oking for BCH c odes that are nested. The following re sult shows a family of QEC derived fr om no nprimitive narr ow-sense BCH cod es. W e can also switch between the code and its d ual to construct a q uantum c ode. When the BCH cod es co ntain their duals, then we can derive the f ollowing codes. T ABLE I F A M I L I E S O F A S Y M M E T R I C Q U A N T U M B C H C O D E S [ 1 6] q C 1 BCH Code C 2 BCH Code A QEC 2 [15 , 11 , 3] [15 , 7 , 5] [[15 , 3 , 5 / 3]] 2 2 [15 , 8 , 4] [15 , 7 , 5] [[15 , 0 , 5 / 4]] 2 2 [31 , 21 , 5] [31 , 16 , 7] [[31 , 6 , 7 / 5]] 2 2 [31 , 26 , 3] [31 , 16 , 7] [[31 , 11 , 7 / 3]] 2 [31 , 26 , 3] [31 , 16 , 7] [[31 , 10 , 8 / 3]] 2 [31 , 26 , 3] [31 , 11 , 11] [[31 , 6 , 11 / 3]] 2 [31 , 26 , 3] [31 , 6 , 15] [[31 , 1 , 15 / 3]] 2 [127 , 113 , 5] [127 , 78 , 15] [[127 , 64 , 15 / 5]] 2 [127 , 106 , 7] [127 , 77 , 27] [[127 , 56 , 25 / 7]] Theorem 7: Let m = ord n ( q ) and q ⌊ m/ 2 ⌋ < n ≤ q m − 1 where q is a power of a prim e and 2 ≤ δ ≤ δ max , with δ ∗ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m od d ]) , then there exists a q uantum cod e with parameters [[ n, n − 2 m ⌈ ( δ − 1 )(1 − 1 / q ) ⌉ , ≥ δ ]] q pure to δ max + 1 Pr oof: See [10, Theorem 19]. A. A QE C-BCH Fortunately , the mathematical structure of BCH co des al- ways u s easily to show th e nested r equired structur e as n eeded in Lemma 4. W e know that g ( x ) is a gener ator polynomia l of a narr ow sense BCH co de that has ro ots α 2 , α 3 , . . . , α δ − 1 over F q . W e know th at the gener ator p olynom ial has degree m ⌊ ( δ − 1 )(1 − 1 /δ ) ⌋ if δ ≤ δ max . T herefor e th e d imension is giv en by k = n − deg ( g ( x )) . Hence, the n ested structure o f BCH codes is obvious an d ca n b e d escribed as follows. Let δ i +1 > δ i > δ i − 1 ≥ . . . ≥ 2 , (19) and let C i be a BCH code that h as generato r polynomial g i ( x ) , in which it has ro ots { 2 , 3 , . . . , δ − 1 } . So, C i has param eters [ n, n − deg ( g i ( x )) , d i ≥ δ i ] q , then C i +1 ⊆ C i ⊆ C i − 1 ⊆ . . . (20) W e n eed to e nsure that δ i and δ i +1 away of each other, so th e elem ents (roo ts) { 2 , . . . , δ i − 1 } an d { 2 , . . . , δ i +1 − 1 } are different. This means th at the cyclotomic cosets generated by δ i and δ i +1 are not the same, S 1 ∪ . . . ∪ S δ i − 1 6 = S 1 ∪ . . . ∪ S δ i +1 − 1 . Let δ ⊥ i be the design ed distance of the code C ⊥ i . Then the f ollowing result g i ves a family of A QEC BCH codes over F q . Theorem 8 ( A QEC-BCH): Let q be a prim e power an d gcd( n, q ) = 1 , with ord n ( q ) = m . Let C 1 and C 2 be two narrow-sense BCH codes of length q ⌊ m/ 2 ⌋ < n ≤ q m − 1 over F q with desig ned d istances δ 1 and δ 2 in the rang e 2 ≤ δ 1 , δ 2 ≤ δ max = min {⌊ nq ⌈ m/ 2 ⌉ / ( q m − 1) ⌋ , n } and δ 1 < δ ⊥ 2 ≤ δ 2 < δ ⊥ 1 . Assume S 1 ∪ . . . ∪ S δ 1 − 1 6 = S 1 ∪ . . . ∪ S δ 2 − 1 , then there exists an asymmetr ic q uantum error c ontrol code with parameters [[ n, n − m ⌈ ( δ 1 − 1)(1 − 1 / q ) ⌉ − m ⌈ ( δ 2 − 1 )(1 − 5 1 /q ) ⌉ , ≥ d z /d x ]] q , wh ere d z = wt( C 2 \ C ⊥ 1 ) ≥ δ 2 > d x = wt( C 1 \ C ⊥ 2 ) ≥ δ 1 . Pr oof: Fro m the nested stru cture of BCH cod es, we kn ow that if δ 1 < δ ⊥ 2 , then C ⊥ 2 ⊆ C 1 , similarly if δ 2 < δ ⊥ 1 , then C ⊥ 1 ⊆ C 2 . By Lemma 6, u sing the fact th at δ ≤ δ max , the dimension o f the code C i is given by k i = n − m ⌈ ( δ i − 1 )(1 − 1 /q ) ⌉ for i = { 1 , 2 } . Since S 1 ∪ . . . ∪ S δ 1 − 1 6 = S 1 ∪ . . . ∪ S δ 2 − 1 , this means that deg ( g 1 ( x )) < de g ( g 2 ( x )) , hence k 2 < k 1 . Furthermo re k ⊥ 1 < k ⊥ 2 . By Lemma 4 and we assume d x = w t ( C 1 \ C ⊥ 2 ) ≥ δ 1 and d z = w t ( C 2 \ C ⊥ 1 ) ≥ δ 2 such that d z > d x otherwise we exchan ge the rules of d z and d x ; or the code C i with C 1+( i mo d 2) . Therefore, there exists A QEC w ith parameters [[ n, k 1 + k 2 − n, ≥ d z /d z ]] q . The p roblem with BCH codes is tha t we have lower bo unds on th eir min imum distance g iv en their arbitr ary designed distance. W e argue that their m inimum distance meets with their designed distance for small v alue s that are particularly interesting to us. On e can also use the co ndition shown in [10, Corollary 11.] to ensure that th e min imum d istance meets th e designed distance. The condition regarding the designed distances δ 1 and δ 2 allows u s to give form ulas f or the dim ensions of BCH c odes C 1 and C 2 , howev er, w e can derive A QE C-BCH witho ut this condition as shown in the fo llowing re sult. This is explained by an example in the n ext section. Lemma 9: Let q b e a prim e p ower , gcd( m, q ) = 1 , and q ⌊ m/ 2 ⌋ < n ≤ q m − 1 for some integers m = o rd n ( q ) . Let C 1 and C 2 be two BCH codes with p arameters [ n, k 1 , d x ≥ δ 1 ] q and [ n, k 2 , d z ≥ δ 2 ] q , respectively , such th at δ 1 < δ ⊥ 2 ≤ δ 2 < δ ⊥ 1 , an d k 1 + k 2 > n . Assume S 1 ∪ . . . ∪ S δ 1 − 1 6 = S 1 ∪ . . . ∪ S δ 2 − 1 , then ther e exists an asymmetric qu antum error control code with p arameters [[ n, k 1 + k 2 − n, ≥ d z /d x ]] q , where d z = wt( C 1 \ C ⊥ 2 ) = δ 2 > d x = wt( C 2 \ C ⊥ 1 ) = δ 1 . In fact the previous th eorem can be used to d eriv e any asym- metric cyclic qu antum contr ol code s. Also, one can constru ct A QEC based on co des th at ar e defined over F q 2 . B. RS Codes W e can also der iv e a family of asymme tric qu antum co ntrol codes based on Redd-Solomo n codes. Recall that a RS cod e with length n = q − 1 an d designed distanc e δ over a finite field F q is a code with p arameters [[ n, n − d + 1 , d = δ ]] q and generato r poly nomial g ( x ) = d − 1 Y i =1 ( x − α i ) . ( 21) It is much easier to d erive con ditions for A QEC derived from RS as shown in the following th eorem. Theorem 10: Let q be a p rime power and n = q − 1 . Let C 1 and C 2 be two RS cod es with param eters [ n, n − d 1 + 1 , d 1 ]] q and [ n, n − d 2 + 1 , d 2 ] q for d 1 < d 2 < d ⊥ 1 = n − d 1 . Then there exists A QEC code with parameters [[ n, n − d 1 − d 1 + 2 , d z /d x ]] q , where d x = d 1 < d z = d 2 . Pr oof: since d 1 < d 2 < d ⊥ 1 , then n − d ⊥ 1 +1 < n − d 2 +1 < n − d 1 + 1 and k ⊥ 1 < k 2 < k 1 . Hen ce C ⊥ 2 ⊂ C 1 and C ⊥ 1 ⊂ C 2 . Let d z = wt( C 2 \ C ⊥ 1 ) = d 2 and d x = wt( C 1 \ C ⊥ 2 ) = d 1 . C 1 C 2 dual C 1 dual C 2 F q 2n k=dim C 2  - dim C 1 r= dim C 1 - dim C 2 ^dual Yellow Detectable errors Green Undetectable errors Fig. 2. A quantum code Q is decomposed into two subsystem A (info) and B (gauge) Therefo re there must exist A QEC with param eters [[ n, n − d 1 − d 1 + 2 , d z /d x ]] q . It is obvious fro m this theor em that the construc ted cod e is a pure code to its minimum d istances. One can also deriv e asymmetric quan tum RS co des based on RS c odes over F q 2 . Also, generalized RS cod es ca n be used to deriv e similar results. In fact, one can deri ve A QEC f rom any two classical cyclic cod es ob eying the pair-nested struc ture over F q . I V . AQ E C A N D C O N N E C T I O N W I T H S U B S Y S T E M C O D E S In this section we establish the conn ection b etween A QEC and sub system codes. Further more we derive a larger class o f quantum codes called asymmetric subsystem codes (ASSs). W e derive families o f subsystem BCH code s and cyclic subsystem cod es over F q . In [8], [9] we co nstruct sev era l families o f subsystem cyclic, BCH, RS and MD S codes over F q 2 with much more details W e expand o ur und erstanding o f the th eory of quantum error co ntrol co des by c orrecting th e quantum err ors X and Z separately using two dif fe rent classical codes, in a ddition to correcting only er rors in a sm all subspace . Su bsystem code s are a generalization o f the th eory of quantum error control codes, in wh ich erro rs can be corrected as well as a voided (isolated). Let Q be a qu antum code such that H = Q ⊕ Q ⊥ , where Q ⊥ is the orth ogon al comp lement o f Q . W e can define the subsystem code Q = A ⊗ B , see Fig.2, as follows Definition 11 (Subsystem Codes): An [[ n, k , r, d ]] q subsys- tem code is a de composition of the sub space Q into a tensor produ ct o f two vector spaces A and B such that Q = A ⊗ B , where dim A = q k and dim B = q r . The c ode Q is able to detect all errors of weight less than d on subsystem A . Subsystem codes can be constru cted fro m th e classical codes over F q and F q 2 . Such codes do not n eed the classical c odes to be self-ortho gonal (or dual-containing ) as shown in the Euclidean constru ction. W e ha ve given g eneral constructio ns of subsy stem co des in [ 11] known as th e subsystem CSS and Hermitian Con structions. W e p rovide a pr oof for th e following special c ase of the CSS constru ction. Lemma 12 (SSC Euclidean Construction): If C 1 is a k ′ - dimensiona l F q -linear co de of len gth n that has a k ′′ - 6 dimensiona l subco de C 2 = C 1 ∩ C ⊥ 1 and k ′ + k ′′ < n , then there exist [[ n, n − ( k ′ + k ′′ ) , k ′ − k ′′ , wt( C ⊥ 2 \ C 1 )]] q [[ n, k ′ − k ′′ , n − ( k ′ + k ′′ ) , wt( C ⊥ 2 \ C 1 )]] q subsystem codes. Pr oof: Let us define the code X = C 1 × C 1 ⊆ F 2 n q , therefor e X ⊥ s = ( C 1 × C 1 ) ⊥ s = C ⊥ s 1 × C ⊥ s 1 . Hence Y = X ∩ X ⊥ s = ( C 1 × C 1 ) ∩ ( C ⊥ s 1 × C ⊥ s 1 ) = C 2 × C 2 . Thus, dim F q Y = 2 k ′′ . Hence | X || Y | = q 2( k ′ + k ′′ ) and | X | / | Y | = q 2( k ′ − k ′′ ) . By Theo rem [11, Theorem 1 ], there exists a subsystem code Q = A ⊗ B with param eters [[ n, log q dim A, lo g q dim B , d ]] q such that i) dim A = q n / ( | X || Y | ) 1 / 2 = q n − k ′ − k ′′ . ii) dim B = ( | X | / | Y | ) 1 / 2 = q k ′ − k ′′ . iii) d = swt( Y ⊥ s \ X ) = wt ( C ⊥ 2 \ C 1 ) . Exchang ing the rules of the codes C 1 and C ⊥ 1 giv es us the other subsystem code with the given p arameters. Subsystem codes (SCC) require the c ode C 2 to b e self- orthog onal, C 2 ⊆ C ⊥ 2 . A QEC and SSC are bo th can be constructed from th e p air-nested classical codes, as we call them. Fro m this r esult, we can see that any two classical codes C 1 and C 2 such that C 2 = C 1 ∩ C ⊥ 1 ⊆ C ⊥ 2 , in which they can be used to construct a subsystem code (SCC), can be also u sed to con struct asymmetric qu antum code (A QEC). Asy mmetric subsystem co des (ASSCs) are muc h larger class than th e class of symmetr ic subsystem codes, in which the quantu m errors occur with different probab ilities in the former one and have equal p robabilities in the later one. In sho rt, A QEC does d oes not require the intersection code to be self-orth ogonal. The constru ction in Lemm a 1 2 can b e gen eralized to ASSC CSS con struction in a similar way . This m eans that we can look at an A Q EC with parameter s [[ n, k , d z /d x ]] q . as subsystem code with parameters [[ n , k , 0 , d z /d x ]] q . Therefo re all resu lts shown in [9], [8 ], [11] are a dir ect conseq uence b y just fixing the minimum distance condition . W e h av e shown in [9], [8] that All stabilizer co des (pure and impu re) can b e red uced to subsystem codes as shown in the following result. Theorem 13 (T rading Dimension s of SSC an d Co-SCC): Let q be a power of a prime p . I f there exists an F q -linear [[ n, k , r , d ]] q subsystem code (stabilizer code if r = 0 ) with k > 1 that is p ure to d ′ , then there exists an F q -linear [[ n, k − 1 , r + 1 , ≥ d ]] q subsystem code that is pur e to min { d, d ′ } . If a pu re ( F q -linear) [[ n, k , r, d ]] q subsystem code exists, then a p ure ( F q -linear) [[ n, k + r , d ]] q stabilizer co de exists. W e ha ve shown in [1 0], [ 12] that narrow sense BCH codes, primitive and non -primitive, with leng th n and designed distance δ are Euclidea n dual- containing cod es if an d o nly if 2 ≤ δ ≤ δ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m odd ]) . (2 2) W e use th is result and [9, Theorem 2] to der i ve nonp rimitive subsystem BCH codes fro m classical BCH co des over F q and F q 2 [11], [12]. The subsystem co des deriv ed in [8] are o nly for the primitive c ase. Lemma 14: If q is power of a prime, m is a positive integer , and q ⌊ m/ 2 ⌋ < n ≤ q m − 1 . Let 2 ≤ δ ≤ δ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m odd ]) , then there exists a subsystem BCH code with para meters [[ n, n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ − r , r, ≥ δ ]] q where 0 ≤ r < n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ . Pr oof: W e know that if 2 ≤ δ ≤ δ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m od d ]) , the the classical BCH co des con tain the ir Euclidean du al code by [ 10, Theor em 3.]. But existence o f this code gives a stabilizer code with par ameters [[ n, n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ , ≥ δ ]] q using [1 0, Theorem 19.]. W e know that e very stabilizer code can be r educed to a subsystem co de by Theor em 13. Le t r be an integer in the range 0 ≤ r < n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ . Fro m [9, Th eorem 2] or Theor em 13, then there must e xist a subsystem BCH code with p arameters [[ n , n − 2 m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ − r, r , ≥ δ ]] q . W e can also constru ct subsystem BCH code s from stabilizer codes usin g the Hermitian construction s wh ere the c lassical BCH codes are defined over F q 2 . Lemma 15: If q is a power o f a prime, m = ord n ( q 2 ) is a positive integer , and δ is an integer in the range 2 ≤ δ ≤ δ max = ⌊ n ( q m − 1) / ( q 2 m − 1) ⌋ , then th ere exists a subsystem code Q with parameter s [[ n, n − 2 m ⌈ ( δ − 1 )(1 − 1 / q 2 ) ⌉ − r , r, d Q ≥ δ ]] q that is pu re up to δ , wher e 0 ≤ r < n − 2 m ⌈ ( δ − 1)(1 − 1 / q 2 ) ⌉ . Pr oof: W e knot that if 2 ≤ δ ≤ δ max = ⌊ n ( q m − 1) / ( q 2 m − 1 ) ⌋ , then exists a classical BCH code w ith param - eters [ n, n − m ⌈ ( δ − 1)(1 − 1 / q 2 ) ⌉ , ≥ δ ] q which con tains its Hermitian du al code using [10, Theorem 14.]. But e x istence of the classical c ode that conta ins its Herm tian code gives us qu antum codes b y [1 0, Theo rem 21.]. Fro m [9, Theorem 2], then the re m ust exist a subsy stem cod e with the given parameters [[ n, n − 2 m ⌈ ( δ − 1)(1 − 1 /q 2 ) ⌉ − r, r, d Q ≥ δ ]] q that is pure up to δ , fo r all range of r in 0 ≤ r < n − 2 m ⌈ ( δ − 1 )(1 − 1 /q 2 ) ⌉ .. If fact there is a tradeoff between the con struction of sub- system codes and asymmetric q uantum c odes. T he condition C 2 = C 1 ∩ C ⊥ 1 used f or the co nstruction of SSC, is n ot n eeded in th e constructio n of A QEC. Instead of constructing subsystem codes from stabilizer BCH codes as shown in Lem mas 14, 1 5, we can also con struct subsystem codes from classical BCH co des over F q and F q 2 under som e restrictions on the designed distance δ . Let S i be a cycloto mic c oset d efined as { iq j mo d n | j ∈ Z } . W e will derive o nly SSC fr om n onprim iti ve BCH codes over F q ; for codes over F q 2 and furth er deta ils see [8]. Also, th e g enerator polyno mial can be u sed in stead o f the defining set ( cylotomic cosets) to deriv e BCH codes. Lemma 16: If q is a power of a prime, m = o rd n ( q ) is a positive integer and 2 ≤ δ ≤ δ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2)[ m odd ]) . Let C 2 be a BCH cod e with length q ⌊ m/ 2 ⌋ < n ≤ q m − 1 an d defining set T C 2 = { S 0 , S 1 , . . . , S n − δ } , such th at g cd( n, q ) = 1 . L et T ⊆ { 0 } ∪ { S δ , . . . , S n − δ } be a nonemp ty set. Assume C 1 ⊆ F n q be a BCH co de with the defining set T C 1 = { S 0 , S 1 , . . . , S n − δ } \ ( T ∪ T − 1 ) where T − 1 = {− t mo d n | t ∈ T } . Then there exists a subsystem 7 T ABLE II S U B S Y S T E M B C H C O D E S U S I N G T H E E U C L I D E A N C O N S T RU C T I O N Subsystem Code Parent Designed BCH Code distanc e [[15 , 4 , 3 , 3]] 2 [15 , 7 , 5] 2 4 [[15 , 6 , 1 , 3]] 2 [15 , 5 , 7] 2 6 [[31 , 10 , 1 , 5]] 2 [31 , 11 , 11] 2 8 [[31 , 20 , 1 , 3]] 2 [31 , 6 , 15] 2 12 [[63 , 6 , 21 , 7]] 2 [63 , 39 , 9] 2 8 [[63 , 6 , 15 , 7]] 2 [63 , 36 , 11] 2 10 [[63 , 6 , 3 , 7]] 2 [63 , 30 , 13] 2 12 [[63 , 18 , 3 , 7]] 2 [63 , 24 , 15] 2 14 [[63 , 30 , 3 , 5]] 2 [63 , 18 , 21] 2 16 [[63 , 32 , 1 , 5]] 2 [63 , 16 , 23] 2 22 [[63 , 44 , 1 , 3]] 2 [63 , 10 , 27] 2 24 [[63 , 50 , 1 , 3]] 2 [63 , 7 , 31] 2 28 [[15 , 2 , 5 , 3]] 4 [15 , 9 , 5] 4 4 [[15 , 2 , 3 , 3]] 4 [15 , 8 , 6] 4 6 [[15 , 4 , 1 , 3]] 4 [15 , 6 , 7] 4 7 [[15 , 8 , 1 , 3]] 4 [15 , 4 , 10] 4 8 [[31 , 10 , 1 , 5]] 4 [31 , 11 , 11] 4 8 [[31 , 20 , 1 , 3]] 4 [31 , 6 , 15] 4 12 [[63 , 12 , 9 , 7]] 4 [63 , 30 , 15] 4 15 [[63 , 18 , 9 , 7]] 4 [63 , 27 , 21] 4 16 [[63 , 18 , 7 , 7]] 4 [63 , 26 , 22] 4 22 ∗ punctured code + Extended code BCH cod e with the par ameters [[ n, n − 2 k − r , r, ≥ δ ]] q , wher e k = m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ and 0 ≤ r = | T ∪ T − 1 | < n − 2 k . Pr oof: The proo f can be divided into the f ollowing p arts: i) W e know that T C 2 = { S 0 , S 1 , . . . , S n − δ } and T ⊆ { 0 } ∪ { S δ , . . . , S n − δ } be a nonempty set. Hence T ⊥ C 2 = { S 1 , . . . , S δ − 1 } . Furthermo re, if 2 ≤ δ ≤ δ max = n q m − 1 ( q ⌈ m/ 2 ⌉ − 1 − ( q − 2 )[ m odd ]) , then C 2 ⊆ C ⊥ 2 . Furthermo re, let k = m ⌈ ( δ − 1 )(1 − 1 /q ) ⌉ , then dim C ⊥ 2 = n − k and dim C 2 = k . ii) W e kn ow that C 1 ∈ F n q is a BCH code with d efining set T C 1 = T C 2 \ ( T ∪ T − 1 ) = { S 0 , S 1 , . . . , S n − δ } \ ( T ∪ T − 1 ) where T − 1 = { − t mo d n | t ∈ T } . T hen the dual cod e C ⊥ 1 has d efining set T ⊥ C 1 = { S 1 , . . . , S δ − 1 } ∪ T ∪ T − 1 = T C ⊥ 2 ∪ T ∪ T − 1 . W e can comp ute the union set T C 2 as T C 1 ∪ T ⊥ C 1 = { S 0 , S 1 , . . . , S n − δ } = T C 2 . Therefo re, C 1 ∩ C ⊥ 1 = C 2 . Fu rthermor e, if 0 ≤ r = | T ∪ T − 1 | < n − 2 k , then dim C 1 = k + r . iii) From step (i) an d (ii), and for 0 ≤ r < n − 2 k , and by Lemma 12, th ere exits a subsystem cod e with p arameters [[ n, dim C ⊥ 2 − dim C 1 , dim C 1 − dim C 2 , d ]] q = [[ n, n − 2 k − r , r, d ]] q , d = min w t ( C ⊥ 2 − C 1 ) ≥ δ . One can also construct asymmetric subsystem BCH codes in a n atural way mean ing the distances d x and d z can be defined using th e A QE C definitio n. In o ther words one can obtain ASSCs with param eters [[ n , n − 2 k − r , r, d z /d x ]] q and [[ n, r , n − 2 k − r, d z /d x ]] q . The extension to ASSCs based on RS codes is straight forward an d similar to ou r c onstruction s in [9], [8]. A. Cyclic Subsystem Codes Now , we shall give a general constructio n for subsystem cyclic cod es. This would app ly f or all cyclic codes includin g BCH, RS, RM and du adic codes. W e sho w that if a classical cyclic co de is self-orth ogon al, then o ne can e asily construct cyclic subsystem codes. W e say that a cod e C 2 is self- orthog onal if an d only if C 2 ⊆ C ⊥ 2 . W e will derive subsystem cyclic codes over F q , an d the case of F q 2 is illustrated in [8]. Theorem 17: Let C 2 be a k -dimensional self-or thogon al cyclic code of length n over F q . Let T C 2 and T C ⊥ 2 respectively denote the defining sets of C 2 and C ⊥ 2 . If T is a subset of T C 2 \ T C ⊥ 2 that is the u nion of cyclotomic co sets, then one can define a cyclic code C 1 of length n over F q by the defining set T C 1 = T C 2 \ ( T ∪ T − 1 ) . If r = | T ∪ T − 1 | is in the range 0 ≤ r < n − 2 k , and d = min wt( C ⊥ 2 \ C ) , then th ere exists a subsystem code with parameters [[ n, n − 2 k − r, r, d ]] q . Pr oof: see [8] and more details are shown in in [4]. Now it is straight forward to deriv e asymm etric cyclic subsystem codes with parameter s [[ n, n − 2 k − r, r, d z /d x ]] q for all 0 ≤ r < n − 2 k using Theorem 17 where d x = min { wt( C ⊥ 2 \ C 1 ) , wt( C ⊥ 2 \ C ⊥ 1 ) } and d z = max { wt( C ⊥ 1 \ C 2 ) , wt( C ⊥ 1 \ C 2 ) } . V . I L L U S T R A T I V E E X A M P L E W e ha ve demonstrated a f am ily of as y mmetric qu antum codes with arbitr ary leng th, dimen sion, and minim um distance parameters. W e will present a simple examp le to explain our construction . Consider a BCH code C 1 with parameters [15 , 11 , 3] 2 that has designed distance 3 and generator matrix given b y                   1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1                   (23) and th e code C ⊥ 1 has parameters [15 , 4 , 8] 2 and g enerator matrix     1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1     (24) Consider a BCH cod e C 2 with param eters [15 , 7 , 5] 2 that has designed distance 5 and generator matrix given b y 8           1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1           (25) and the code C ⊥ 2 has parameters [15 , 8 , 4] 2 and generator matrix             1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1             (26) A QEC. W e can consider the code C 1 corrects the bit-flip errors such that C ⊥ 2 ⊂ C 1 . Furthermo re, C ⊥ 1 ⊂ C 2 . Furthermo re and d x = wt( C 1 \ C ⊥ 2 ) = 3 and d z = wt( C 2 \ C ⊥ 1 ) = 5 . Hence, the quantu m code can de tect f our phase-shift errors and two bit-flip error s, in o ther word s, th e code can corr ect two pha se- shift errors and one bit-flip errors. Ther e must e xist asymmetric quantum error contro l codes (A QEC) with parame ters [[ n, k 1 + k 2 − n, d z /d x ]] 2 = [[15 , 3 , 5 / 3 ]] 2 . W e ensu re that this q uantum code en codes th ree qubits in to 1 5 qu bits, and it migh t also be easy to desig n a fault tolerant circ uit for this code similar to [[9 , 1 , 3]] 2 or [[7 , 1 , 3]] 2 , but o ne can use the cyclotom ic structure o f this co de. W e ensure that ma ny other quantu m BCH can be construc ted using the approach gi ven in this paper that may or may no t h av e better fault tolerant oper ations and better threshold values. SSC. W e can a lso co nstruct a subsystem co de (SSC) based on the co des C 1 and C 2 . First we no tice that C ⊥ 1 = C 2 ∩ C ⊥ 2 6 = ∅ , C 2 ⊂ C 1 and C ⊥ 2 ⊂ C 1 . Let k = dim C 1 − dim C 2 = 4 and r = dim C 2 − dim C ⊥ 1 = 3 . Furtherm ore d = wt( C 1 \ C 2 ) = 3 . Therefore, there exists a subsystem code with parameters [[15 , 4 , 3 , 3]] 2 also an ASSC code with p arameters [[15 , 4 , 3 , 5 / 3 ]] 2 . Remark 18: An [7 , 3 , 4] 2 BCH cod e is used to deri ve Steane’ s code [[7 , 1 , 4 / 3]] 2 . A QEC might not b e interesting for Stean e’ s cod e becau se it can only detect 3 shift-erro rs a nd 2 bit-flip erro rs, f urthermo re, the code c orrects one bit-flip an d one pha se-shift at mo st. The refore, on e need s to desig n A QEC with d z much larger than d x . One mig ht argue o n h ow to choo se the distances d z and d x , we th ink the answer co mes f rom the physical system point of view . The time n eeded to phase-shift er rors is m uch less that the time need ed fo r qub it-flip erro rs, hen ce d ependin g on the factor between them, o ne c an design A QEC with factor a d z /d x . V I . B O U N D S O N A S Y M M E T R I C Q E C A N D S U B S Y S T E M C O D E S One might wonder wh ether th e known b ounds on QE C and SSC p arameters would also app ly f or A QEC a nd ASSC code parameters. W e can show that A QECs and ASSCs o bey the asymmetric Singleton bound as fo llows. In fact we can trad e the d imensions o f SCC a nd ASSC in a similar manner as shown in [9], [8]. A. Singleto n Bound [Asymmetric Singleton Bound] Theorem 19: An [[ n, k , d z /d x ]] q asymmetric pure quan tum code with k ≥ 1 satisfies d x ≤ ( n − k + 2) / 2 , an d th e bou nd d x + d z ≤ ( n − k + 2) . ( 27) Pr oof: From the construction o f A QE C, existence of the A QEC with pa rameters [[ n, k, d z /d x ]] q implies existence of two codes C 1 and C 2 such that C ⊥ 2 ⊆ C 1 and C ⊥ 1 ⊆ C 2 . further more d x = wt( C 1 \ C ⊥ 2 ) and d z = wt ( C 2 \ C ⊥ 1 ) . Hence we hav e d x ≤ ( n − k 1 + 1) a nd d z ≤ ( n − k 2 + 1) , and by addin g these two terms we obtain d x + d z ≤ n − ( k 1 + k 2 − n ) + 2 = n − k + 2 . It is much easy to show th at th e b ound for d x than the bou nd for d z since QEC’ s with parameter s [[ n, k , d x ]] q obey this bound . Also, impure A QECs obey this bound d x + d z ≤ ( n − k + 2) . The pr oof is straight fo rward to th e case QECs and we omit it here. One can also show that Asymm etric subsystem codes o bey the Singleton boun d Lemma 20: Asymmetric subsystem cod es with parameters [[ n, k , r , d z /d x ]] q for 0 ≤ r < k satisfy k + r ≤ n − d x − d z + 2 . (28) Remark 21: In fact, the A QEC RS codes deriv ed in Sec- tion III ar e o ptimal and a symmetric MDS co des in a sense that they meet asymm etric Singleton bound with eq uality . The co nclusion is that M DS QE Cs are also MDS A QEC. Furthermo re, MDS SCC are also MDS ASSC. B. Hamming Bound Based on the d iscussion pre sented in the previous sections, we can treat subsystem code constructio ns as a special class of asym metric q uantum cod es wh ere C ⊥ i ⊂ C 1+( i mo d 2) , for i ∈ { 1 , 2 } and C 2 = C 1 ∩ C ⊥ 1 . Fu rthermor e, the more general theory of qu antum error contro l co des would be asymm etric subsystem codes. Lemma 22: A pure (( n, K , K ′ , d z /d x )) q asymmetric sub- system co de satisfies ⌊ d x − 1 2 ⌋ X j =0  n j  ( q 2 − 1) j ≤ q n /K K ′ . (29) Pr oof: W e know that a pu re (( n, K , K ′ , d z /d x ))) q code implies the existence of a p ure (( n, K K ′ , d x )) q stabilizer code this is direct b y lo oking at an A QEC as a QEC. But th is o beys 9 the qua ntum Hammin g bou nd [20], [11]. Therefo re it follows that ⌊ d x − 1 2 ⌋ X j =0  n j  ( q 2 − 1) j ≤ q n /K K ′ . In terms of packing codes, it is easy to show that the imp ure asymmetric subsystem codes does not obey the quan tum Hamming bo und. Since the special case does not o bey this bound , so why th e ge neral case does. Lemma 23: An impure (( n, K, K ′ , d z /d x )) q asymmetric subsystem code does not satisfy ⌊ d x − 1 2 ⌋ X j =0  n j  ( q 2 − 1) j ≤ q n /K K ′ . It is o bvious th at the distan ce of p hase-shift w ou ld not obey this boun d as well, d z > d x . Finally o ne can always look at asymmetric quan tum cod es (A QECs) as a special class o f asym metric subsy stem cod es (ASSCs). In o ther words ev er y an [[ n, k , d z /d x ]] q is also a n [[ n , k , 0 , d z /d x ]] q , and this is the main c ontribution of this paper . Also, a SSC with parameters [[ n , k , r , d x ]] q can p roduc e AS S C with param eters [[ n, k , r , d z /d x ]] q . One ca n also go fro m ASSCs to A QECs using th e results derived in [9], [8]. an d Fin ally an ASSC with parameters [[ n , k , r , d z /d x ]] q is also an ASSC w ith para meters [[ n, r , k , d z /d x ]] q . The proof fo r all these facts is a direct consequen ce by writing the F q bases fo r the co des A QEC and ASSC. V I I . C O N C L U S I O N A N D D I S C U S S I O N This pap er in troduced a new th eory of asymmetric quantu m codes. It establishes a link be tween asymme tric and symmetric quantum contro l codes, as well as subsystem c odes. Families of A QEC are derived ba sed on RS and BCH code s over finite fields. Furthermo re we in troduced families of sub system BCH codes. T ables of A QEC-BCH and CSS-BCH are shown over F q . W e p ose it as open quantum to study the fault to lerance operation s o f the constructed quantum BC H co des in this paper . Some BCH codes are turn ed out to be also LDPC codes. Therefo re, one can use th e same metho d shown in to constru ct asymmetric quantum LDPC codes [5]. A C K N OW L E D G M E N T S . I thank A. Klap penecker fo r his support and I think my family , teachers, and colleague s. Part of this research o n SSC an d QEC has been d one at CS/T AMU in Sp ring ’07 and during a research visit to Bell- Labs & alcatel- Lucent in Summ er ’0 7, the g eneralization to ASSC is a consequ ence. Sharing knowledge , in which we all born kno wing no thing , is better than proving or canceling it . S . A . A . V I I I . A P P E N D I X A. Quan tum BCH Codes This paper is written on the occasion of the 50th an niv er sary of the discovery of classical BCH codes and their quantum counterp arts were derived nearly 10 years ag o. 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