On $[[n,n-4,3]]_{q}$ Quantum MDS Codes for odd prime power $q$

ON [[ n, n − 4 , 3]] q QUANTUM MDS CODES FOR ODD P RIME PO WER q RUIHU LI (1) (2) , ZO N GBEN XU (1) Abstract. F or eac h odd prime pow er q , let 4 ≤ n ≤ q 2 + 1. Hermitian self-orthogonal [ n, 2 , n − 1] co des ov er GF ( q 2 ) with dual distan ce three are constructed by using finite field theory . Hence, [[ n, n − 4 , 3]] q quan tum MDS codes f or 4 ≤ n ≤ q 2 + 1 are obtained. 1. Introduction The theory of qua ntu m error - correc ting codes (Q E CCs, for short) w as established a decade ago as the primary to ol for fighting decoherence in quantum computers and quantum comm unication system, see [21] and [22]. The most widely studied class o f qua ntum co de s ar e binary quantum stabilizer co des. A thor ough discuss ion on the principles of quantum co ding theory w as given in [4] and [9] for binary quantum stabilizer co des. A n app ealing a sp ect o f binary qua nt um co des is that there exist links to cla ssical co ding theory which easy the c o nstruction o f go o d quantum co des. F ollowing [4 ] and [9], many binar y quantum co des ar e constructed from binary and quaternar y classical codes , see [6-7, 13 -18, 23-25]. Almost at the same time of [4], some results of binary quant um stabilizer co des were generalized to the cas e of non-binary quantum stabilizer co des, and character- ization of non-binary quantum sta bilizer code s over GF ( q ) in term of classic al codes ov er GF ( q 2 ) was also given which genera lizes the well-known notation o f additive co des ov er GF (4) for binary cas e, see [19], [2] and [1 1] and references therein. And, many non-binary quantum stabilizer co de s are constructed fro m cla ssical co des o ver GF ( q ) or ov er GF ( q 2 ), see [2- 3 ,5,8, 10-11]. One central theme in quantum er ror-c o rrection is the co nstruction of quantum co des with go o d parameters . Except the metho d o f constructing quantum co des from classical self-o rthogona l linear co des over GF ( q ) a nd self-orthog onal additive co des ov er GF ( q 2 ) that given in [4], [9],[19], [2] and [11], Schlingemann a nd W erner [20] presented another new wa y to co nstruct qua nt um stabilizer co des by finding certain graphs (or ma trices) with sp ecific prop erties. A num ber of r esearchers use these metho ds to construct optimal quantu m co des, e.g., codes with large s t Ruihu Li i s with the College of Scienc e, Xi’an Jiaotong Unive rsity , Shaanxi 7100 49, P eople’s Republic of China, and Coll ege of Science, Air F orce Engineering Univ ersity , Xi’ an, Shaanxi 710051, People’s R epublic of Chi na (E-mail: lir uihu 2008@y ahoo.com.cn). Zongben Xu is with the Institute for Infor mation and System Sciences, Xi’an Jiaotong Univer- sity , Shaanxi 710049, People’s Republic of China (E-mail: zb xu@mail.xjtu.edu.cn) . This work is supp orted by Natural Science F oundation of China under Grant No.60573040, the National Basic R esearch Program of China (973 Program) under Grant No.2007CB3110 02, Natural Science Basic Research Plan in Shaanxi Province of China under Program No. SJ08A02. 1 2 R UIHU LI (1) (2) , ZONGBEN XU (1) po ssible k with fixed n and d . Optimal qua nt um co de s tha t saturating the qua nt um Singleton bo und received muc h attention. Lemma 1.1: [12] [1 9] (quan tum Singleto n b o und) An [[ n, k , d ]] q quantum stabilizer co des satisfies k ≤ n − 2 d + 2 . A quantum co de a tta ins the quantum Singleton b ound is called a qua ntum max- im um distance se pa rable co de o r quantum MDS c o de for sho r t. It is k nown that except trivial co des ( co des with d ≤ 2), ther e are only tw o binary q ua ntu m MDS co des, [[5 , 1 , 3]] 2 and [[6 , 0 , 4]] 2 , see [4]. Non-binary quant um MDS co des are muc h complex co mpared with the binary case. In the simplest non trivial c a se d = 3, despite many efforts to construct non-binar y quantum MDS co des, a sys tematic construction for all leng ths has not b een achieved yet, see [2 -3,5,8, 10 -11]. Kno wn results on non-binary quantum MDS co des ar e a s follows: [1 9] proved the existence of [[5 , 1 , 3]] p for o dd prime p , [3] prov ed the exis tence of [[ q 2 + 1 , q 2 − 3 , 3]] q quantum MDS co des for prime p ow er q , [6] pro ved the existence of [[ n, n − 2 d + 2 , d ]] q for a ll 3 ≤ n ≤ q and 1 ≤ d ≤ n 2 + 1, a nd [[ q 2 , q 2 − 2 d + 2 , d ]] q for 1 ≤ d ≤ q , all these three pap ers used self-orthog onal co des to construc t quantum MDS co des. Based on graph states metho d, [20] proved the e x istence of [[5 , 1 , 3 ]] p for prime p ≥ 3, [5] prov ed the existence o f [[6 , 2 , 3]] p and [[7 , 3 , 3]] p for prime p ≥ 3. With a co mputer search a nd gra ph states metho d, [10] constr uc ted four families of qua ntum MDS co des [[6 , 2 , 3]] p , [[7 , 3 , 3]] p , [[8 , 4 , 3]] p , and [[8 , 2 , 4]] p , for o dd p ≥ 3. If d ≥ 3, [11] prov ed that the maximal length n of [[ n, k , d ]] q quantum MDS co des satisfies n ≤ q 2 + d − 2. In this pap er , w e will use Hermitian self-o rthogona l co des ov er F q 2 to construct q -ar y qua nt um MDS co des of distance three, where q = p r and p ≥ 3 is an o dd prime. Our ma in result of this pap er is as follow: Theorem 1.1: If q = p r and p ≥ 3 is an o dd prime, then there are [[ n, n − 4 , 3]] q quantum MDS co des for 4 ≤ n ≤ q 2 + 1. 2. Preliminaries In order to pr esent our main r esult, we make so me pr eparation on qua ntu m co des and finite fields. Let GF ( q 2 ) n be the n - dimensional row spa ce over the finite field GF ( q 2 ). F or X = ( x 1 , x 2 , ..., x n ), Y = ( y 1 , y 2 , ..., y n ) ∈ GF ( q 2 ) n , the Her mitian inner pr o duct of X and Y is defined as follow: ( X, Y ) = x 1 y q 1 + x 2 y q 2 + ... + x n y q n . If C is an [ n, k ] linear co de ov er GF ( q 2 ), its dual co de C ⊥ = { X | X ∈ GF ( q 2 ) n , ( X , Y ) = 0 for any Y ∈ GF ( q 2 ) n } . C is self-or thogonal if C ⊆ C ⊥ , and self-dual if C = C ⊥ . [6] gav e the following theor e m of constructing quantum codes fro m self-orthogo nal co des ov er GF ( q 2 ). Theorem 2 .1: If C is an [ n, k ] linear co de ov er GF ( q 2 ) s uch tha t C ⊥ ⊆ C , a nd d = min { wt ( v ) : v ∈ C \ C ⊥ } , then there exists [[ n, 2 k − n, d ]] q quantum code. In the construc tio n of self-orthog onal co des, we also need the following r esults on finite fields. Lemma 2.1 : If α is a primitive element of GF ( q 2 ), for ea ch non zero element β of GF ( q ), there are q + 1 elements α i of GF ( q 2 ) such that ( α i ) q +1 = β . Pro of: Supp ose ( α ) q +1 = γ , then γ is a pr imitive e lement o f GF ( q ). Let β = γ i , 0 ≤ i ≤ q − 2. Then ( α i +( q − 1) j ) q +1 = β for 0 ≤ j ≤ q , thus the Lemma holds. 3 Lemma 2.2: If α is a pr imitive element of GF ( q 2 ), then 1 + ( α ) q +1 + ( α 2 ) q +1 + ... + ( α q 2 − 2 ) q +1 = 0. Pro of: Supp ose ( α ) q +1 = γ , then 1 + ( α ) q +1 + ( α 2 ) q +1 + ... + ( α q 2 − 2 ) q +1 = ( q + 1)(1 + γ + γ 2 + ... + γ q − 2 ) = 0 T o simplify s ta temen t of the following tw o section, we divide the non-zer o ele- men ts of GF ( q 2 ) into t w o subsets, s ay A and B . Let A = { 1 , − 1 , α q − 1 2 , − α q − 1 2 , α q − 1 2 + 1 , − α q − 1 2 − 1 } , B = { x 1 , − x 1 , ..., x k , − x k } = GF ( q 2 ) \ ( A ∪ { 0 } ), w her e 2 k = q 2 − 7. According to Lemma 2.2 , one can deduce that Lemma 2.3: Suppo se A and B b e defined as above, then 2Σ k i =1 ( x i ) q +1 + 2( α q − 1 2 + 1) q +1 = 0. 3. [[ n, n − 4 , 3]] q f or q = 3 r In this section, we will prov e Theore m 1.1 holds for q = 3 r . First we discuss the construction of [[ n, n − 4 , 3]] 3 quantum code. Let GF (3) = { 0 , 1 , 2 } = { 0 , 1 , − 1 } be the Galois field with three elements. then f ( x ) = x 2 + x + 2 is irreducible ov er GF (3). Using f ( x ), one can construct the Galois field GF (9) with nine element s as GF (9) = { 0 , 1 , 2 , α, α + 1 , α + 2 , 2 α, 2 α + 1 , 2 α + 2 } , where α is a r o ot of f ( x ) = x 2 + x + 2. It is ea sy to chec k that α is a primitive element of GF (9), α 2 = 2 α + 1, α 3 = 2 α + 2, α 4 = 2, α 5 = 2 α , α 6 = α + 2, and α 7 = α + 1. It is obvious that α 4+ i = − α i for 0 ≤ i ≤ 7. Construct H 2 , 4 =  1 1 1 0 0 1 − 1 1  , H 2 , 5 =  1 1 α α 0 0 1 α 2 α 3 α  , H 2 , 6 =  1 α α α α 0 0 1 α 2 α 4 α 6 α  , H 2 , 7 =  1 1 1 1 1 1 0 0 1 α 1 α 2 α 5 α 7 1  , H 2 , 8 =  1 1 1 1 1 α α 0 0 1 2 α 1 α 5 1 2 1  , H 2 , 9 =  1 1 1 ... 1 0 1 α 1 ... α 7  , H 2 , 10 =  1 1 1 1 1 1 α α α 0 0 1 α α 4 α 6 α 7 α 3 α 4 α 6 1  , It is easy to chec k tha t: F or 4 ≤ n ≤ 10 , the co de C 2 ,n generated by H 2 ,n is a se lf- orthogo nal co de over GF (9) and d ⊥ = 3, he nc e there is an [[ n, n − 4 , 3]] 3 quantum MDS co des. Second, we discuss the co ns truction of [[ n, n − 4 , 3 ]] q quantum co de for q = 3 r ≥ 9. T o ac hieve this, we consider four cases separa tely . Case 3.1 Let 4 ≤ n ≤ q 2 − 4 and n ≡ 0( mod 2 ). Let n − 4 = 2 k 1 , u = 2Σ k 1 i =1 ( x i ) q +1 . Cho ose γ , δ, ǫ ∈ GF ( q 2 ) s uch that δ q +1 ∈ GF ( q ) \ GF (3 ) and 2 δ q +1 + u 6 = 0 , γ q +1 = − ( n − 4 + 2 δ q +1 ), ǫ q +1 = − ( u + 2 δ q +1 ). Construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ − δ ǫ  . Case 3.2 Let 4 ≤ n ≤ q 2 − 4 and n ≡ 1( mod 2 ). Let n − 5 = 2 k 1 , u = 2Σ k 1 i =1 ( x i ) q +1 . C ho ose γ , δ, ǫ ∈ GF ( q 2 ), s uch that δ q +1 ∈ GF ( q ) \ GF (3) a nd u + δ q +1 ( α q − 1 + 1) q +1 6 = 0 , γ q +1 = − ( n − 5 + δ q +1 ), ǫ q +1 = 4 R UIHU LI (1) (2) , ZONGBEN XU (1) − ( u + δ q +1 ( α q − 1 + 1) q +1 ). Construct A 2 ,n = γ 1 1 · · · 1 1 δ α q − 1 2 δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ α q − 1 2 − δ α q − 1 δ ( α q − 1 + 1) ǫ ! . Case 3.3 Let q 2 − 3 ≤ n ≤ q 2 − 1. Sub case 3. 3.1 If n = q 2 − 1, choo se γ , δ, ǫ ∈ GF ( q 2 ) such that δ q +1 ∈ GF ( q ) \ GF (3 ) and 1 − δ q +1 − ( α q − 1 + 1) q +1 6 = 0, γ p +1 = 5 − 2 δ q +1 , ǫ q +1 = 2( δ q +1 + ( α q − 1 2 + 1) q +1 − 1). Construct A 2 ,n =  γ 1 1 · · · 1 1 1 1 δ δ 0 0 x 1 − x 1 · · · x k − x k 1 − 1 δ α q − 1 2 − δ α q − 1 2 ǫ  . Sub case 3.3.2 If n = q 2 − 2, ch o ose ǫ ∈ GF ( q 2 ) such that ǫ = α q − 1 2 + 1, and construct A 2 ,n =  1 1 1 · · · 1 1 1 1 1 0 0 x 1 − x 1 · · · x k − x k 1 α q − 1 2 − α q − 1 2 − 1 ǫ  . Sub case 3. 3.3 If n = q 2 − 3, choo se γ , δ, ǫ ∈ GF ( q 2 ) s uch that δ q +1 ∈ GF ( q ) \ GF (3 ), γ p +1 = 1 − 2 δ q +1 , ǫ q +1 = (2 − 2 δ q +1 )( α q − 1 2 + 1) q +1 . Construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ 0 0 x 1 − x 1 · · · x k − x k δ ( α p − 1 2 + 1) − δ ( α p − 1 2 + 1) ǫ  . Case 3.4 Let q 2 ≤ n ≤ q 2 + 1. If n = q 2 , construct A 2 ,n =  1 1 1 · · · 1 1 0 1 α · · · α q 2 − 3 α q 2 − 2  . If n = q 2 + 1, let a = q − 1 2 . C ho ose δ, ǫ ∈ GF ( q 2 ), such that δ q +1 ∈ GF ( q ) \ GF (3), ǫ q +1 = (1 − δ q +1 )( α q − 1 2 + 1) q +1 , and construct A 2 ,n =  1 1 1 · · · 1 1 1 1 1 δ δ δ 0 0 x 1 − x 1 · · · x k − x k 1 α a − ( α a + 1) − δ − δ α a δ ( α a + 1) ǫ  . It is e a sy to chec k that: In the ab ov e four case s , the co de gener ated by A 2 ,n is a n [ n, 2 , n − 1 ] self-o rthogona l co de over GF ( q 2 ), and its dual distance is 3. Hence, there are [[ n, n − 4 , 3]] q quantum MDS codes for 4 ≤ n ≤ q 2 + 1, where q = 3 r . Summarizing the ab ov e discussion, Theorem 1.1 holds for q = 3 r . 4. [[ n, n − 4 , 3]] q f or q = p r and prime p ≥ 5 In this section, we will prove Theorem 1.1 holds for q = p r , and we alwa ys assume that p ≥ 5 is a prime and α is a primitive elemen t of GF ( q 2 ). T o give the construction of quantum [[ n, n − 4 , 3]] q co des, we cons ider four cases separ ately . Case 4.1 Let 4 ≤ n ≤ q 2 − 4 and n = mp + 2 . Sub case 4.1. 1 If n ≡ 0( mod 2), let n − 4 = 2 k 1 . If v = 2Σ k 1 i =1 ( x i ) q +1 + 4 6 = 0, choos e γ , δ, ǫ ∈ GF ( q 2 ) such that γ q +1 = − 2, δ q +1 = 2, ǫ q +1 = − v . Construct 5 A 2 ,n =  γ 1 1 · · · 1 1 δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ − δ ǫ  . If v = 2Σ k 1 i =1 ( x i ) p +1 + 4 = 0, choose γ , δ, ǫ ∈ GF ( q 2 ), such that γ q +1 = − 2, δ q +1 = 2, ǫ q +1 = 8, a nd construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ α q − 1 2 − δ α q − 1 2 ǫ  . Sub case 4.1. 2 If n ≡ 1( mod 2), let n − 5 = 2 k 1 . If v ′ = 2Σ k 1 i =1 ( x i ) q +1 + 2( α q − 1 2 + 1) q +1 6 = 0, cho ose γ , δ, ǫ ∈ GF ( q 2 ) s uch that γ q +1 = − 3, δ q +1 = 2 and ǫ q +1 = − v ′ . Construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ δ α q − 1 2 − δ ( α q − 1 2 + 1) ǫ  . If v ′ = 2Σ k 1 i =1 ( x i ) q +1 + 2( α q − 1 2 + 1) p +1 = 0, cho ose γ , δ, ǫ ∈ GF ( q 2 ) s uch that γ q +1 = − 6, δ q +1 = 3 and ǫ q +1 = − ( α q − 1 2 + 1) q +1 . Construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ δ α q − 1 2 − δ ( α q − 1 2 + 1) ǫ  . Case 4. 2 Let 4 ≤ n ≤ q 2 − 4 and n − 2 6 = 0 ( mod p ). Sub case 4.2. 1 If n ≡ 0( mod 2), let n − 4 = 2 k 1 . If w = 2Σ k 1 i =1 ( x i ) q +1 + 2 6 = 0, cho ose γ , ǫ ∈ GF ( q 2 ) such that γ q +1 = − n + 2 and ǫ q +1 = − w . Cons truct A 2 ,n =  γ 1 1 · · · 1 1 1 1 0 0 x 1 − x 1 · · · x k 1 − x k 1 1 − 1 ǫ  . If w = 2Σ k 1 i =1 ( x i ) q +1 + 2 = 0, cho ose γ , ǫ ∈ GF ( q 2 ) such that γ q +1 = − n + 2 and ǫ q +1 = 4. Co nstruct A 2 ,n =  γ 1 1 · · · 1 1 1 1 0 0 x 1 − x 1 · · · x k 1 − x k 1 α q − 1 2 − α q − 1 2 ǫ  . Sub case 4.2. 2 If n ≡ 1( mod 2), let n − 5 = 2 k 1 . If w ′ = 2 Σ k 1 i =1 ( x i ) q +1 + ( α q − 1 2 + 1) q +1 6 = 0, choose γ , ǫ ∈ GF ( q 2 ) suc h that γ q +1 = − n + 2, ǫ q +1 = − w 1 . Construct A 2 ,n =  γ 1 1 · · · 1 1 1 1 1 0 0 x 1 − x 1 · · · x k 1 − x k 1 1 α q − 1 2 − α q − 1 2 − 1 ǫ  . If w ′ = 2Σ k 1 i =1 ( x i ) q +1 + ( α q − 1 2 + 1) q +1 = 0 and n + 1 6 = 0( mod p ), choose γ , δ, ǫ ∈ GF ( q 2 ) s uch that γ q +1 = − n − 1, δ q +1 = 2 and ǫ q +1 = − ( α q − 1 2 + 1) q +1 . Construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ δ α q − 1 2 − δ α q − 1 2 − δ ǫ  . If w ′ = 2Σ k 1 i =1 ( x i ) q +1 + ( α q − 1 2 + 1) q +1 = 0 and n + 1 ≡ 0( mod p ), choose γ , δ, ǫ ∈ GF ( q 2 ) such that γ q +1 = − n − 4, δ q +1 = 3 and ǫ q +1 = − 2 ( α q − 1 2 + 1) q +1 . 6 R UIHU LI (1) (2) , ZONGBEN XU (1) Construct A 2 ,n =  γ 1 1 · · · 1 1 δ δ δ 0 0 x 1 − x 1 · · · x k 1 − x k 1 δ δ α q − 1 2 − δ α q − 1 2 − δ ǫ  . Case 4.3 Let q 2 − 3 ≤ n ≤ q 2 − 1. Sub case 4. 3.1 If n = q 2 − 1, choose γ , ǫ ∈ GF ( q 2 ) such that γ q +1 = 3, ǫ q +1 = 2( α q − 1 2 + 1) q +1 , and construct A 2 ,n =  γ 1 1 · · · 1 1 1 1 1 1 0 0 x 1 − x 1 · · · x k − x k 1 − 1 α q − 1 2 − α q − 1 2 ǫ  . Sub case 4. 3.2 If n = q 2 − 2, choose γ , ǫ ∈ GF ( q 2 ) such that γ q +1 = 4, ǫ q +1 = ( α q − 1 2 + 1) q +1 . Construct A 2 ,n =  γ 1 1 · · · 1 1 1 1 1 0 0 x 1 − x 1 · · · x k − x k 1 α q − 1 2 − α q − 1 2 − 1 ǫ  . Sub case 4.3.3 Let n = q 2 − 3. Choose γ , δ, ǫ ∈ GF ( q 2 ) s uch that γ q +1 = 3, δ q +1 = 2, ǫ q +1 = − 2 ( α q − 1 2 + 1) q +1 , and constr uct A 2 ,n =  γ 1 1 · · · 1 1 δ δ 0 0 x 1 − x 1 · · · x k − x k δ ( α q − 1 2 + 1) − δ ( α q − 1 2 + 1) ǫ  . Case 4.4 Let q 2 ≤ n ≤ q 2 + 1. If n = q 2 , construct A 2 ,n =  1 1 1 1 · · · 1 1 0 1 α α 2 · · · α p 2 − 3 α p 2 − 2  . If n = q 2 + 1, let a = q − 1 2 . Cho ose γ , δ, ǫ ∈ GF ( q 2 ) such that γ q +1 = − 2, δ q +1 = 2 and ǫ q +1 = − ( α q − 1 2 + 1) q +1 . Construct A 2 ,n =  γ 1 1 · · · 1 1 1 1 1 δ δ δ 0 0 x 1 − x 1 · · · x k − x k 1 α a − ( α a + 1) − δ − δ α a δ ( α a + 1) ǫ  . It is easy to chec k that: In the ab ov e four cases, the co de gener ated by A 2 ,n is an [ n, 2 , n − 1] self-orthog onal co de ov er GF ( q 2 ) with dual distance is 3. Hence, there are [[ n, n − 4 , 3]] q quantum MDS co des for 4 ≤ n ≤ q 2 + 1, where q = p r and p ≥ 5 . Summarizing the ab ov e discussion, Theorem 1.1 holds for q = p r and p ≥ 5 is a prime. 5. 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