Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes

1 Quantum generalized Reed-S olomon codes: Unified framework for quantum MDS codes Zhuo Li , Li-Juan Xing , and Xin-Mei Wang State Key Laboratory of Integrated Serv ice Networks, Xidian University, Xi’an, Shannxi 710071, China We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exis ts, then a quantum code of this type with the same parameters also exists. Thus as far as is known at present, they are the most important family of quantum MDS codes. PACS numbers: 03.67.Pp, 03.67.Hk, 03.67.Lx I. INTRODUCTION The quest to build a scalable quantu m computer that is resilient against decoherence errors and operational noise has sparked a lot of interest in quantum error-correcting codes [1-11]. The early rese arch has been confined to the study of binary quantum error-correcting codes, but more recently the theory was extended to nonbinary codes that are useful in the rea lization of fault-tolerant computations. 2 Among these codes quantum MDS codes are op timal in the sense that the minimum distance is maximal, since they meet the quantum Singleton bound [10]. Recently many families of quantum MDS codes ha ve been found by various different approaches [12-15]. In this paper we derive a new family of quantum MDS codes that are based on classical generalized Reed-Solomon codes [16]. We call them quantum generalized Reed-Solomon (QGRS) codes. Firstly we give the definition of QG RS code. Then we give the necessary and sufficient condition for the existence of QGRS code, which shows that the problem of finding QGRS codes can be transformed into the problem of finding the weight distributions of certain classical codes. So it is po ssible to search for codes of this type. But this is no t practic al for large codes. Thus we also show how to construct QGRS codes analytically in the end. Another achievement of this paper is the unification of various quantum MDS codes under QGRS codes. Recall that vari ous existing quantum MDS codes have been found by different approaches, whic h brings inconvenience for application. Occurrence of QGRS codes changes this situation. Each prec eding quantum MDS code has a counterpart in QGRS codes. In the other word, it will tu rn out that in all the known cases, when a quantum MDS code exis ts, then a QGRS code with the sam e parameters also exists. Thus as far as is known at present, QG RS codes are the most important family of quantum MDS codes. II. PRELIMINARIES Firstly, let us recall the d efinition and pr operty of classical ge neralized RS codes briefly [16]. Let 1 (, , ) n α α = α … where the i α are distinct elements of , and let () m GF q 1 (, , ) n ν ν = ν … where the i ν are nonzero elements of . Then the generalized RS code, denoted by , consists of all vectors () m GF q (, , ) GRS C αν k ) 11 2 2 (( ) , ( ) , , ( ) nn FF F ν αν α ν α … where is any polynomial of degree with coefficients from . () Fz k < () m GF q Let be an [, (, , ) GRS C αν k , 1 ] q nk n k − + generalized RS code. The extended generalized RS code, , has generator matrix [1 , , 2 ] q nk n k +− + (, , ) GRS C ∗ αν k 1 11 22 11 11 11 0 0 0 1 n nn nn kk nn νν αν α ν αν αν αν αν ∗ −− ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ G … … … … ……… … . When and even, one more parity check can always be added, producing an extended generalized RS code, , by using the generator matrix 3 k = q [2 , 3 , ] q nn + (, , 3 ) GRS C ∗ αν 1 11 22 11 00 10 01 n nn nn νν αν α ν αν αν ∗ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ G … … … . Additionally, a typical method which we w ill use for construction is as follows. Lemma 1 [11]. Let be an classical code contained in its Herm itian dual, , such that . Then there exists an quantum code. C 2 [, ] q nk h C ⊥ min { ( \ )} h dw t C ⊥ = C ,2 , q nn k d −  3 III. QUANTUM GENERALIZED RS CODES Now we start to show our contributions. Definition 2. Quantum generalized Reed-Solom on (QGRS) code is the quantum code that is derived from classical He rmitian self-orthogon al generalized RS or extended generalized RS code by lemma 1. Let be a -linear code. Define C 2 () GF q 1 () ( ) : , ( ) qn n ii i PC c d C G F q ⊥ = =∈ ∩ cd , which is equivalent to the punctu re code introduced by Rains [10]. Theorem 3. A QGRS code ,2 ,1 q rr k k − +  ) ) exists if and only if there ex ists a codeword of weight in where vector r * (( , , GRS PC k α 1 2 1 (, , ) q α α = α … contains all elements of and 1 denotes vector of 1’s. 2 () GF q Proof. (Only if.) Suppose a QGRS code ,2 ,1 q rr k k − +  r r exists, and is the corresponding Hermitian self-orthogonal generalized RS code with generator matrix C 2 [, , 1 ] q rk r k −+ 1 1 1 1 11 1 r r r ii kk ii νν α να ν α να −− ν ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ G … … …… … … . Then for 1 11 () () jj j rr st q s t q q ij ij i j jj αν αν α ν ++ == == ∑∑ 0 1 0, st k ≤ ≤− . Setting the coordinate j i equal to 1 q j ν + for 1 j r ≤≤ and the others to gives a codeword of weight in . The case of extended generalized RS code is handled in the same way. The proof of the converse is similar. Q.E.D. 0 r * (( , , GRS PC k α 1 ) ) 4 5 ) ) This theorem tells us th at the problem of finding QGRS codes can be transformed into the problem of finding th e weight distributions of cer tain classical codes. Once we find out the weight distribution of , we can derive all possible QGRS codes by theorem 3. * (( , , GRS PC k α 1 Corollary 4. (Bounds.) Let be an L ,2 , 1 q nn k k − +  QGRS code. Then: (i) If , 3 q ≥ kq ≤ ; (ii) For all and except for k q 3 k = and q even, 2 1 nq ≤ + ; (iii) For an d even, 3 k = q 2 2 nq ≤ + . Proof. (i) Suppose , then has parity check matrix 1 kq ≥+ * (( , , GRS PC k α 1 ) ) 0 2 22 2 2 1 11 1 ( 1)( 1) ( 1)( 1) 1 11 0 0 1 q qq q kq kq q αα αα αα −− −+ −+ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ … … ………… … … whose determinant is nonzero. Thus there ex ists no QGRS code by theorem 3. (ii) and (iii) are obvious by theorem 3. Q.E.D. In the next section, we will show that QGRS code exists. For and 22 1, 2 1, 1 q qq q q +− + +  3 k = 2, 4 q = , we also find QGRS codes and . Hence the bounds given by corollary 4 are tight. 2 6, 0, 4  4 18 ,12, 4  IV. ANALYTICAL CONSTRUCTI ON FOR QGRS CODES Theorem 3 makes it possible to search for QGR S codes. But this is not p ractical for large codes. In this sectio n we show how to construct QGRS codes analytically. We begin with a lemma. Lemma 5. Let 1 ,, n α α … be distinct elements of any field. Then we have n 1 1 0 () h n i n i ij j ji α αα = = ≠ = − ∑ ∏ for . 2 hn ≤− Proof. It is easy to verify that 12 12 1 22 2 22 12 12 1 11 12 12 11 1 11 1 () nn h n i n i nn n nn n ij nn j hh h nn n ji nn 2 1 α αα α α α αα αα α αα α αα α αα α = −− − −− = α − − −− ≠ = − ∑ ∏ …… …… … ……… … ……… …… …… . Q . E . D . Now we can give the m ain theorem of this paper : Theorem 6. There exist QGRS codes ,2 , 1 q nn k k − +  where code parameters satisfy , , 2 1 nq kq ⎧ =+ ⎨ = ⎩ 2 1 02 nq l kq l lq ⎧ =− ⎪ ≤− − ⎨ ⎪ ≤≤ − ⎩ 0 1 nm q l km l lm mq = − ⎧ ⎪ ≤ − ⎪ ⎨ ≤< ⎪ ⎪ < < ⎩ , or 2 nq kn ≤ ⎧ ⎨ ≤ ⎢ ⎥ ⎣ ⎦ ⎩ . Proof. (Constructive proof .) Let 11 () { , , , 0 } qq GF q β ββ − = = … , and let {1 , } γ be the basis of 2 () ( ) GF q GF q . Then , 2 1 () { | , 0 1 , 1 } iq j iq j i j GF q i q j q αα β γ β ++ + =+ ≤ ≤ − ≤ ≤  11 1 1 11 1 ( ) [( ) ( )] ( ) ( ) qq q s qj t q i s j t i s t i ii i α αβ γ β β γ β β β γ β ++ + + + + == = −= + − + = − + ∏∏ ∏ st ≠ 1 q , , (1) 1 11 11 () [ ( ) ( ) ] qq s qj s q i s j s i i ii ij ij i α α βγ β βγ β β − ++ + + == ≠≠ −= + − + = ∏∏ = ∏ . ( 2 ) When , set 2 1 nq =+ 2 12 (, , , ) q α αα = α … . It is easy to verify that is Hermitian self-orthogonal. Thus QGRS code exists. (, , ) GRS Cq ∗ α 1 22 1, 2 1, 1 q qq q q +− + +  6 When with 02 , set 2 nq l =− lq ≤≤ − 2 12 (, , , ) ql α αα − = α … and set 2 12 (, , , ) ql ν νν − = ν … where 2 2 1 ( q qq ii jq l ) j ν αα =− + =− ∏ . Then for and , (, , ) q GRS Ck αν (, , ) GRS Cn − α 1 k 2 22 2 2 2 2 22 2 2 1 1 1 11 1 1 1 1 11 1 11 11 1 1 () () ( ) ( ) () () () () () q st ij i qq nn n jq l qs q t s t ii i i j i j q ii i j jq l j j q st ij i st qq nn jq l i jj n q ii jj ij ij j j ji ji αα α να α α α α α α αα α α αα αα αα + − =− + + − == = = =− + = + + −− =− + == == = = ≠ ≠ − =− = − == − − ∏ ∑∑ ∑ ∏∏ ∏ ∏ ∑∑ ∏∏ ∏ ∏ 0 = for 1 s k ≤− , by lemma 5. So is Hermitian dual to . Now if , . 1 tn k ≤− − (, , ) q GRS C αν k k k (, , ) GRS Cn − α 1 1 kq l ≤− − (, , ) ( , , ) q GRS GRS Ck C n ⊆− αν α 1 When with 1 , nm q l =− mq << 0 lm ≤ < , from (1) and (2), there is a nonzero constant () i GF q λ ∈ , with , such that 1 im q ≤≤ 1 1 () mq m ij i j ji α αλ ζ − = ≠ −= ∏ where 1 ( q i i ) ζ γβ = =+ ∏ . Set 12 (, , , ) mq l α αα − = α … , 1 (, , ) mq l μ μ − = μ … , and 1 (, , ) mq l ν ν − = ν … where 1 () mq qq ii i j jm q l μ να =− + =− ∏ α 1 i , 1 q i ν λ + − = . Then for and , (, , ) q GRS Ck αμ (, , ) GRS Cn − αν k 1 11 1 11 1 1 1 11 11 1 1 () () ( ) ( ) () 0 () () mq st ii mq nn n jm q l qs q t q s t m ii ii i i i j m ii i jm q l i mq st ii j st nn jm q l mm i mq n ii ij ij j j ji ji j α αα μα ν α ν α α α ζ λζ αα α α ζζ αα αα + =− + ++ − − == = =− + + + =− + −− == = = ≠ ≠ − =− = − == − − ∏ ∑∑ ∑ ∏ ∏ ∑∑ ∏ ∏ = 7 for 1 s k ≤− , by lemma 5. So is Hermitian dual to . Now if , . 1 tn k ≤− − (, , ) q GRS C αμ k k k (, , ) GRS Cn − αν km l ≤− (, , ) ( , , ) q GRS GRS Ck C n ⊆− αμ α ν When , set nq ≤ 12 (, , , ) n β ββ = α … and 1 (, , ) n ν ν = ν … where 1 1 1( n q ii j ji ) j ν ββ + = ≠ =− ∏ . Then for and (, , ) GRS Ck αν (, , ) GRS Cn k − αν , 1 11 1 1 () ( ) () st nn n sq t q s t i ii ii i i n ii i ij j ji β νβ νβ ν β ββ + ++ == = = ≠ 0 = == − ∑∑ ∑ ∏ for 1 s k ≤− , by lemma 5. So is Hermitian dual to . Now if 1 tn k ≤− − (, , ) GRS C αν k k (, , ) GRS Cn − αν 2 kn ≤ ⎢⎥ ⎣⎦ , . Q . E . D . (, , ) (, , ) GRS GR S Ck Cn ⊆ αν αν k −   ) nq In theorem 6 we just give the smallest range of code parameters among which QGRS codes exist. We have proved that the ran ge can be enlarged slightly. This is to say, more QGRS codes can be constructed analytica lly. V. DISCUSSION Now let us study existing quantum MD S codes. For quantum MDS codes in [12], there exist QGRS codes with param eters by theorem 6. For quantum MDS codes and in [13], by direct search and theorem 6 we can find QGRS codes with corr esponding param eters easily. For quantum MDS codes for 5, 1 , 3 q  5, 1 , 3 q  6, 2, 3 p  7, 3 , 3 p  (3 p ≥ ,22 , q nn d d −+  ≤ , and for some s in [14], corresponding QGRS codes can be obtained from theorem 6 and theorem 3 respectively. 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