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

The quest to build a scalable quantum computer that is resilient against decoherence errors and operational noise has sparked a lot of interest in quantum error-correcting codes [1][2][3][4][5][6][7][8][9][10][11]. The early research 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 realization of fault-tolerant computations. Among these codes quantum MDS codes are optimal in the sense that the minimum distance is maximal, since they meet the quantum Singleton bound [10]. Recently many families of quantum MDS codes have been found by various different approaches [12][13][14][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 QGRS 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 possible to search for codes of this type. But this is not practical 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 various existing quantum MDS codes have been found by different approaches, which brings inconvenience for application.

Occurrence of QGRS codes changes this situation. Each preceding quantum MDS code has a counterpart in QGRS codes. In the other word, it will turn out that in all the known cases, when a quantum MDS code exists, then a QGRS code with the same parameters also exists. Thus as far as is known at present, QGRS codes are the most important family of quantum MDS codes.

Firstly, let us recall the definition and property of classical generalized RS codes briefly [16].

( ( ), ( ), , ( )

where is any polynomial of degree with coefficients from .

( )

When and even, one more parity check can always be added, producing an extended generalized RS code, , by using the generator

Additionally, a typical method which we will use for construction is as follows.

Lemma 1 [11]. Let be an classical code contained in its Hermitian dual, , such that . Then there exists an quantum code.

Now we start to show our contributions.

Definition 2. Quantum generalized Reed-Solomon (QGRS) code is the quantum code that is derived from classical Hermitian self-orthogonal generalized RS or extended generalized RS code by lemma 1.

) : , ( )

which is equivalent to the puncture code introduced by Rains [10].

exists if and only if there exists a codeword of weight in where vector r * ( ( , ,

… contains all elements of and 1 denotes vector of 1’s.

Proof. (Only if.) Suppose a QGRS code , 2 , 1 q r r k k -+ r r exists, and is the corresponding Hermitian self-orthogonal generalized RS code with generator matrix

(ii) For all and except for k q 3 k = and q even, 2 1 n q ≤ + ;

(iii) For and even,

Proof. (i) Suppose , then has parity check matrix

whose determinant is nonzero. Thus there exists 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

1, 2 1, 1 q q q q q + -+ +

, we also find QGRS codes and .

Hence the bounds given by corollary 4 are tight.

Theorem 6. There exist QGRS codes , 2 , 1 q n n k k -+ where code parameters satisfy , ,

, and let {1, } γ be the basis of 2 ( ) ( ) GF q GF q . Then , 2

( , , , )

Hermitian self-orthogonal. Thus QGRS code exists.

( , , )

( , , , )

( , , , )

. Then for and , ( , , )

)

and ( 2), there is a nonzero constant ( )

, with , such that 1 i mq ≤ ≤

Then for and , ( , , )

. Then for and ( , , ) ( ,,) ( ,, )

In theorem 6 we just give the smallest range of code parameters among which QGRS codes exist. We have proved that the range can be enlarged slightly. This is to say, more QGRS codes can be constructed analytically.

Now let us study existing quantum MDS codes. For quantum MDS codes in [12], there exist QGRS codes with parameters by theorem 6. For quantum MDS codes and in [13], by direct search and theorem 6 we can find QGRS codes with corresponding parameters easily. For quantum MDS codes for 5,1, 3 q 5,1,3 q 6, 2,3 p 7,3,3 p ( 3 p ≥ , 2 2, q n n d d -+ ≤ , and for some s in [14], corresponding QGRS codes can be obtained from theorem 6 and theorem 3 respectively. For quantum MDS codes for and 0 2 2 , 2 2 , q q s q d s d

, 2 2 , q q q q q v v α α —-+ 2 1 0 2 v q ≤ ≤q v α ≤ ≤ -in [15], QGRS codes with corresponding parameters can be derived from theorem 6 immediately.

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