Quantum codes over rings 1. Introduction Quantum ...

International Journal of Quantum Information Vol. 12, No. 4 (2014) 1450020 (11 pages) # .c World Scienti¯c Publishing Company DOI: 10.1142/S0219749914500208

Quantum codes over rings

Kenza Guenda Faculty of Mathematics, USTHB, BP. 32 El Alia Bab Ezzouar, 16111 Algiers, Algeria [email protected] T. Aaron Gulliver Department of Electrical and Computer Engineering, University of Victoria, P. O. Box 1700, STN CSS, Victoria, BC, Canada V8W2Y2 [email protected] Received 29 April 2013 Revised 5 May 2014 Accepted 10 May 2014 Published 10 June 2014 This paper considers the construction of quantum error correcting codes from linear codes over ¯nite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over Fp k are also obtained from linear codes over rings using the generalized Gray map. Keywords: CSS construction; codes over rings; Frobenius rings; Gray map; matrix product codes.

1. Introduction Quantum error-correction plays a signi¯cant role in quantum computation and quantum information. An important class of quantum codes is the class of stabilizer codes. Stabilizer codes over ¯nite ¯elds and in particular the so-called Calderbank– Shor–Steane (CSS) codes have been extensively studied.1–3 More recently, Nadella and Klappenecker4 extended the concept of stabilizer codes to codes over Frobenius rings. This provides the possibility of constructing improved stabilizer codes since it is known that in general codes over rings can have better parameters than codes over ¯elds. Ling and Sole5 and Yin and Ma6 have constructed binary quantum codes from linear codes over Z4 and F2 þ uF2 þ u 2 F2 , respectively, using a Gray map. In this paper, we ¯rst extend the CSS construction to codes over Frobenius rings. As a generalization of a result in Ref. 4, we prove that free stabilizer codes over ¯nite commutative Frobenius rings cannot be better than stabilizer codes over ¯nite ¯elds. 1450020-1

K. Guenda & T. A. Gulliver

An advantage of these codes is that several stabilizer codes over ¯elds can be obtained with parameters depending on the local component rings. Quantum codes are also extended to matrix product codes. This allows for the construction of new quantum codes from existing codes. Further, quantum codes over Fp k are obtained from linear codes over rings using the generalized Gray map. This allows us to take advantage of the homogeneous weight to construct quantum codes over ¯elds with good parameters. 2. Finite Commutative Frobenius Rings In this section, we summarize some results concerning codes over ¯nite commutative Frobenius rings. Further details can be found in Refs. 7–9. We also prove some results concerning the minimum Hamming distance of codes over these rings. A commutative ring is called Artinian if every descending chain of ideals is ¯nite. Let R be an Artinian ring and fmi ; 1 i sg be the set of maximal ideals of this ring. For 1 i s we denote Ri ¼ R=mi with residue ¯elds ki ¼ Ri =mi . Note that Ri is a ¯nite local commutative ring which we call a local component ring of R. The annihilator set of mi is the ideal of R de¯ned by Annðmi Þ ¼ fa 2 R; an ¼ 0; 8 n 2 mi g: An Artinian ring R is called a Frobenius ring if and only if dimki Annðmi Þ ¼ 1

for all 1 i s:

ð1Þ

We have a canonical epimorphism i : R ! Ri . Furthermore, the following result holds. Lemma 2.1. Let R be a ¯nite commutative Frobenius ring. Then we have the ring isomorphism Rﬃ

s Y

Ri ;

i¼1

de¯ned by ðvÞ ¼ ð1 ðvÞ; 2 ðvÞ; . . . ; s ðvÞÞ. Clearly the isomorphism given in Lemma 2.1 can be extended to R n . A linear code over a commutative ring R is de¯ned to be an R-module. If the code is isomorphic to R k then it is said to be free, and we refer to its rank as the free rank of the code. Here, all codes are assumed to be linear. Let R be a ¯nite commutative Frobenius ring and fRi g; 1 i s, the set of local component rings of R. Then for any linear code C over R there exist linear codes Ci , 1 i s, called local component codes of C , such that C ¼ 1 ðC1 Cs Þ ¼ f 1 ðx1 ; x2 ; . . . ; xs Þ; xi 2 Ci g: We refer to C as the Chinese product of codes C1 ; C2 ; . . . ; Cs . 1450020-2

ð2Þ


We attach the standard inner product to the ambient space, i.e. ½v; w ¼ The dual code C ? of C is de¯ned as C ? ¼ fv 2 R n ; ½v; w ¼ 0 for all w 2 Cg:

P

vi wi . ð3Þ

If C C ? , we say that the code is self-orthogonal, and if C ¼ C ? , we say that the code is self-dual. A simple yet very useful characterization for ¯nite commutative Frobenius rings is the following property of any submodule C of R n (Ref. 9) jCjjC ? j ¼ jRj n

and

ðC ? Þ ? ¼ C:

ð4Þ

The following result was proven in Ref. 7. Proposition 2.2. Let R be a ¯nite commutative Frobenius ring and C ¼ CRT ðC1 ; C2 ; . . . ; Cs Þ a code over R of length n with local component codes C1 ; C2 ; . . . ; Cs . Then we have the following. Q (i) jCj ¼ si¼1 jCi j. (ii) C is a free code if and only if each Ci is free and has same free rank. For the code C ¼ 1 ðC1 Cs Þ, de¯ne ResðCi Þ as follows ResðCi Þ ¼ fv 2 ki ; 9 u 2 Ri ; v þ mi u 2 Ci g: For 1 i s, the code ResðCi Þ is a linear code over ki . We now prove the following result. Proposition 2.3. Let C be a linear code over a ¯nite commutative Frobenius ring R with local component codes Ci , 1 i s. Then the following results hold. (i) dH ðCÞ ¼ minfdH ðCi Þg. (ii) dH ðResðCi ÞÞ dH ðCi Þ. Proof. Let v 2 C so that i ðvÞ ¼ vi 2 Ci and then wtH ðvÞ wtH ðvi Þ. Consider the following word ð0; . . . ; 0; vi ; 0; . . . ; 0Þ 2 Ci . Since 1 ð0; . . . ; 0; vi ; 0; . . . ; 0Þ 2 C and has weight less than or equal to the minimum weight of Ci , we obtain that dH ðCÞ minfdH ðCi Þg. Hence the equality. For (ii), consider v 2 ResðCi Þ, then mi v 2 Ci , which gives the result. 2.1. Finite chain rings A ¯nite commutative Frobenius ring with a unique local component ring which is also principal is called a ¯nite chain ring. Hence a ¯nite chain ring is a commutative ring R with 1 6¼ 0 such that its ideals are ordered by inclusion h0i ¼ h e i ( h e1 i ( ( hi ( R: The integer e is called the index of nilpotency of R. 1450020-3


Lemma 2.4 (Ref. 10). Let R be a ¯nite chain ring with maximal ideal hi, nilpotency index e, and residue ¯eld R=hi ¼ Fp k . Further, let V R be a set of representatives for the equivalence classes of R under congruence modulo . Then the following results hold. P i (i) For all r 2 R there exist unique a0 ðrÞ; . . . ; ae1 ðrÞ 2 V such that r ¼ e1 i¼0 ai ðrÞ . (ii) jV j ¼ jFp k j. (iii) jh j ij ¼ Fp kðejÞ for 0 j e 1. It can be deduced from Lemma 2.4 that jRj ¼ p ke ¼ q, where p k is the cardinality of the residue ¯eld Fp k . For codes over rings, several weight and metrics can be de¯ned.11 These can be used to di®erentiate the elements of the rings. Given a ¯nite chain ring R with nilpotency index e and residue ¯eld Fp k , a homogenous weight is de¯ned as follows 8 k kðe2Þ if r 2 Rnh e1 i; > : 0 otherwise: The homogeneous minimum distance dhom of the code C is the minimum homogenous weight of C . It has been proven by Heise, Honold and Nechaev11 that a homogeneous weight can be de¯ned for any code over a ¯nite chain ring. To take advantage of the homogenous weight, an isometry is needed. One isometry is the generalized Gray map de¯ned by Greferath and Schmidt12 and Jitman and Udomkavanich.10 Note that from Lemma 2.4, we have that any r 2 R n can be written uniquely as r¼

e1 X

ai ðrÞ i ;

i¼0

where ai ðrÞ ¼ ðri;0 ; ri;1 ; . . . ; ri;n1 Þ 2 V n for 0 i e 1. Now let v 2 Zp k . Then v has a unique p-adic representation given by v ¼ v0 ðvÞ þ v1 ðvÞp þ þ vk1 ðvÞp k1 ; where vi ðvÞ 2 f0; 1; . . . ; p 1g for 0 i k 1. If is a primitive element of Fp k , then there is a unique v 2 F kp corresponding to v in Fp k given by v ¼ v0 ðvÞ þ v1 ðvÞ þ þ vk1 ðvÞ k1 : Furthermore, w 2 Zp kðe1Þ has p k -adic representation w ¼ w0 ðwÞ þ w1 ðwÞp k þ þ we2 ðwÞp kðe2Þ ; where wi ðwÞ 2 f0; 1; . . . ; p k 1g for 0 i e 2. Then for any r ¼ R n , the generalized Gray map is the map : R n !

kðe1Þn F pp k

ðrÞ ¼ ðb0 ; b1 ; . . . ; bp kðe1Þ1 Þ; 1450020-4

P e1

de¯ned by

i¼0

ai ðrÞ i 2


where bwp k þv ¼ v f a0 ðrÞ þ

e2 X l¼1

wl1 ðwÞ f al ðrÞ þ f av ðrÞ;

for all 0 w p kðe2Þ 1 and 0 v p k 1, with f ai ðrÞ ¼ ðri;0 ; . . . ; ri;n1 Þ, and where ri;j denotes the image of ri;j by the canonical projection from R to Fp k . Lemma 2.5 (Ref. 10). The mapping is an isometric injection from ðR n ; dhom Þ to kðe1Þ ðF pp k n ; dH Þ. 3. Stabilizer Quantum Codes Over Rings Let q ¼ p r be a prime power. A q-ary quantum code of length n is a subspace of C q . Stabilizer codes have been characterized by nice error bases. In this section, we recall some results from Ref. 4 and introduce the CSS construction for codes over rings. Assume that R is a ¯nite commutative Frobenius ring with cardinality q. For a and b in R, de¯ne over C q the following operators: n

XðaÞ : C q ! C q

ZðbÞ : C q ! C q ;

and

by XðaÞjxi ¼ jx þ ai and

ZðbÞjxi ¼ ðbxÞjxi;

respectively, where x 2 R and is an irreducible character of ðR; þÞ. We can easily deduce the following lemma from Corollary 3 of Ref. 4. Lemma 3.1. Let R be a ¯nite commutative Frobenius ring. Then the set E ¼ fXðaÞZðbÞ; a; b 2 R n g; is a nice error basis on C q . n

For ðujvÞ in R 2 n with u ¼ ðu1 ; . . . ; un Þ and v ¼ ðv1 ; . . . ; vn Þ 2 R n , the symplectic weight is de¯ned as swtðujvÞ ¼ jfi : ui 6¼ 0 or vi 6¼ 0gj: For ðujvÞ and ðu 0 jv 0 Þ in R 2n , de¯ne the symplectic inner product h; is by hðujvÞ; ðu 0 jv 0 Þis ¼ v u 0 v 0 u; where \" denotes the standard inner product. Let ! ¼ expð2i=mÞ and de¯ne the error group as Gn ¼ h! c XðaÞZðbÞ j a; b 2 R n ; c 2 Zi: Gn is the group generated by an element of E and has exponent m, which is the exponent of ðR; þÞ. Let S be a subgroup of Gn . The stabilizer code FixðSÞ associated 1450020-5


with S is given by FixðSÞ ¼ fv 2 C q j Ev ¼ v for all E 2 Sg: n

Since S is a group, the set FixðSÞ is a subspace of C q . Thus it is a quantum code called a stabilizer code over the ring R (since S depends on R). The following result gives a necessary and su±cient condition on the existence of stabilizer codes over R. n

Proposition 3.2 (Ref. 4). An ððn; K; dÞÞR stabilizer code exists if and only if there exists an additive code C R 2n of size jCj ¼ jRj n =K such that C C ?s and swtðC ? nCÞ ¼ d if K > 1 and swtðC ? f0gÞ ¼ d if K ¼ 1. We now present the following results. The proof is quite similar to the CSS construction for codes over ¯nite ¯elds given in Ref. 3. Theorem 3.3. Let R be a ¯nite commutative Frobenius ring. Further, let C1 and C2 denote two classical linear codes over R with parameters ðn; K1 ; d1 Þ and ðn; K2 ; d2 Þ such that C 2? C1 . Then there exists an ððn; K1 K2 =jRj n ; dÞÞR stabilizer code with minimum distance d ¼ minfwtðcÞ j c 2 ðC1 nC 2? [ C2 nC 1? Þg that is pure to minfd1 ; d2 g. Proof. Let C ¼ C 1? C 2? R 2 n . Hence from (4) we obtain that jCj ¼ jRj 2n = ðK1 K2 ). Furthermore, if ðc1 jc2 Þ and ðc 01 jc 02 Þ are two elements of C . Then c2 c 01 c 02 c1 ¼ 0. Hence we have that ðhðc1 jc2 Þjðc 01 jc 02 ÞiÞ ¼ 1. Hence we obtain C C ?s . Thus from Proposition 3.2, there exists an ððn; K1 K2 =jRj n ; dÞÞR stabilizer code with minimum distance d ¼ minfwtðcÞ j c 2 ðC1 nC 2? [ C2 nC 1? Þg that is pure to minfd1 ; d2 g. Corollary 3.4. Let R be a ¯nite commutative Frobenius ring and C be an ðn; K; dÞ linear code over R. If C ? C, then there exists an ððn; K 2 =jRj n ; dðC nC ? ÞÞÞ stabilizer quantum code over R. Proof. It su±ces to consider C1 ¼ C2 ¼ C in Theorem 3.3. Table 1 presents some ððn; q; dÞÞR quantum stabilizer codes derived from codes over R ¼ Zm using Theorem 3.3. Table 2 presents some ððn; q; dÞÞq quantum stabilizer codes derived from self-dual codes over ¯nite chain rings of cardinality q using Corollary 3.4. Theorem 3.5. Let R be a ¯nite commutative Frobenius ring such that for 1 i s, the corresponding local component rings Ri have residue ¯elds ki . Assume there exists a free stabilizer code ððn; jRj m ; dÞÞR over the ring R. Hence an ððn; jRi j m ; dÞÞ free stabilizer code exists over the ¯eld ki for 1 i s. Proof. Let C be a free code over a Frobenius ring R with local components Ri , 1 i s. Note that C ? ¼ CRT ðC 1? ; C 2? ; . . . ; C s? Þ. Then C C ? if and only if for all 1 i s we have Ci C i? . Hence from Lemma 2.2 and Propositions 2.3 and 3.2, we have the existence of an ððn; jRi j m ; dÞÞRi free stabilizer code for 1 i s. Then from 1450020-6

Quantum codes over rings Table 1. ððn; q; dÞÞ quantum stabilizer codes derived from Codes over Zm using Theorem 3.3. Ring

ððn; K; dÞÞm

Z6 Z6 Z6 Z6 Z6 Z6 Z10 Z10 Z10 Z10

ðð4; 6; 2ÞÞ ðð6; 36; 2ÞÞ ðð9; 6; 3ÞÞ ðð6; 1; 6ÞÞ ðð10; 6; 3ÞÞ ðð10; 216; 2ÞÞ ðð4; 10; 2ÞÞ ðð6; 100; 2ÞÞ ðð7; 50; 2ÞÞ ðð8; 100; 2ÞÞ

Table 2. ððn; q; dÞÞq quantum stabilizer codes derived from self-dual codes over ¯nite chain rings of cardinality q using Corollary 3.4. ððn; 1; dÞÞq ðð5; 1; 3ÞÞ25 ðð6; 1; 4ÞÞ25 ðð7; 1; 3ÞÞ25 ðð2m; 1; 2ÞÞ4 , m > 1 ðð8; 1; 4ÞÞ4 ðð32; 1; 16ÞÞ4

Ring

References

Z25 Z25 Z25 F2 þ uF2 Z4 Z4

Table 5 of Ref. 13 Table 5 of Ref. 13 Table 5 of Ref. 13 Proposition 5.4 of Ref. 14 15 16

Theorem 16 of Ref. 4, for the residue ¯eld ki of the local component ring Ri of R there exists an ððn; jRi j m ; dÞÞ free stabilizer code over the ¯eld ki for 1 i s. Example 3.6. The codes given in Table 2 are all free, so they correspond to quantum codes over the residue ¯elds with the same parameters. In addition, the minimum distances of the residue ¯eld codes are greater than or equal to the minimum distances of the original quantum codes. 4. Matrix Product Codes Matrix product codes over Frobenius rings17,18 were introduced by Blackmore and Norton19 as a generalization of matrix product codes over ¯elds. This construction includes the Plotkin and Turyn constructions as special cases. In this section, we extend this concept to obtain quantum codes. De¯nition 4.1. Let C1 ; . . . ; Cm be m linear codes of the same length n over a ¯nite Frobenius ring, and A an m l matrix with entries in R. The matrix product codes are de¯ned as follows ½C1 ; . . . ; Cm A ¼ fðc1 ; . . . ; cm ÞA; ; c1 2 C1 ; . . . ; cm 2 Cm g: For simplicity, we restrict ourselves to the case where A is an invertible matrix in R. 1450020-7


Proposition 4.2 (Theorems 3.1 and 3.3 of Ref. 17). Let Cj be a linear ðn; Mj Þ code over R, 1 j m, and let A be an m m invertible matrix over R. Then Q C ¼ ½C1 ; . . . ; Cm A is an ðnm; m j¼1 Mj ; dÞ linear code over R with minimum distance d minðminfdH ðUA ðkÞÞdH ðCk Þ; 1 k mg; minfdH ðLA ðkÞÞdH ðCk Þ; 1 k mgÞ; where UA ðkÞ is the linear code formed by the ¯rst k rows of A and LA ðkÞ is the linear code formed by the last m k rows of A. Furthermore, we have ð½C1 ; . . . ; Cm AÞ ? ¼ ½C1 ? ; . . . ; Cm ? ðA 1 Þ t : Now we prove the following result which will be useful for constructing good codes. Lemma 4.3. Assume that A is an m m invertible matrix such that its nonzero entries are units in R. Let Ci and C i0 be linear codes such that Ci C i0 for all 1 i m. Hence we have 0 ½C1 ; . . . Cm A ½C 10 ; . . . C m A:

Proof. If A is invertible then the rows of A are linearly independent. Since the nonzero entries of A are units, then ½C1 ; . . . ; Cm A ¼ ðA1 ; . . . ; Am Þ, where each Ai is a 0 linear combination of some Ci for 1 i m. This also holds for ½C 10 ; . . . ; C m A. Since 0 Ci C i , the result is obtained. Theorem 4.4. Assume there exist m quantum codes ððn; Ki ; di ÞÞR each obtained from a pair of linear codes Ci and C i0 over R with parameters ððn; Ki ; di ÞÞR and ððn; K i0 ; d 0i ÞÞR , respectively, such that Ci C i0 for 1 i m. Assume also that A is an m m nonsingular matrix such that its nonzero entries are units in R. Then there Q 0

? exists an ððnm; m i¼1 K i ; d minfd i ; 1 i mg minfd i ; di ; 1 i mgÞÞR quantum code. Proof. Since Ci C i0 , then the ððn; Ki ; di ÞÞR codes are such that K i ¼ K i0 =Ki and di minfwtðC i0 nCi Þ; wtðC i? nC i0? Þg. From Lemma 4.3, we obtain that ½C1 ; . . . ; Q 0 C A ½C 10 ; . . . ; C m A, with parameters ½nm; m i¼1 Ki ; d min di and ½nm; Qmm 0 0 0 i¼1 K i ; d min d i , respectively. Hence by applying Theorem 3.3 to each of Q 0 0 Q ½C1 ; . . . ; Cm A and ½C 10 ; . . . ; C m A, we obtain ððnm; m Ki ; dÞÞ stabilizer i¼1 K i = codes. From Proposition 4.2, the dual of ½C1 ; . . . ; Cm A is ½C1 ? ; . . . ; Cm ? ðA 1 Þ t , and 0 0? the dual of ½C 10 ; . . . ; C m A is ½C 10? ; . . . ; C m ðA 1 Þ t . Hence dC 0? min dC i0? and dC ? min dC i? , and for each i we have d i minfwtðC i0 nCi Þ; wtðC i? nC i0? Þg minfd 0i ; di ? g. Example 4.5. Consider C1 , the ¯rst quantum code in Table 1. This code has parameters ðð5; 1; 3ÞÞ. The matrix A ¼ 10 11 is a nonsingular matrix with unit nonzero entries. Hence for two quantum codes C1 and C2 over R, we obtain ½C1 ; C1 A ¼ ðC1 jC1 þ C1 Þ which is a quantum code over Z25 with parameters ðð10; 1; 3ÞÞ. 1450020-8


5. Quantum Codes from Linear Codes Over Finite Chain Rings Theorem 3.5 shows that each free stabilizer code over a ¯nite commutative Frobenius ring R is associated with m free stabilizer codes over ¯nite ¯elds. The minimum distances of these codes are at least equal to the minimum distance of the code over R. In this section, a construction is given for stabilizer codes over ¯elds from codes over ¯nite chain rings. This construction can lead to codes with better parameters than with the direct construction using CSS codes over rings given in Sec. 4. This is because the homogeneous weight of codes over rings is employed, which is a key property. We require the following characterization of quantum codes given in Ref. 20. Theorem 5.1. There exists a quantum ððn; K; dÞÞq code with K 2 if and only if there exist K nonzero mappings i : Fq ! C, 1 i K, which satisfy the following condition. For each, partition f1; . . . ; ng ¼ A [ B with jAj ¼ d 1 and jBj ¼ n d þ 1, and any cA ; c 0A 2 F d1 q , 1 i; j n X 0 ð5Þ i ðcA ;cB Þ j ðc A ; cB Þ ¼ i;j f; c B 2F qndþ1

where f is independent of i and depends only on cA and c 0A , and is the Kronecker symbol. Lemma 5.2. Let R be a ¯nite chain ring and C a linear code over R of length n and cardinality K such that dhom ? d. Then for each partition f1; 2; . . . ; p kðe1Þ ng ¼ A [ B with jAj ¼ d 1, jBj ¼ p kðe1Þ n d þ 1, and any cA 2 F pd1 k , we have that kðe1Þ ndþ1

jfcB 2 F pp k

: 1 ðcA ; cB Þ 2 v þ Cgj ¼ K=p kðd1Þ :

Proof. The Gray image of C is a code over Fp k (not necessarily linear), of length p kðe1Þ n and cardinality K with formal homogeneous dual distance d ? hom . By the equivalence between codes and orthogonal arrays (Theorem 4.9 of Ref. 21), any translation of this Gray image is an orthogonal array of level p k and strength dhom ? 1. Theorem 5.3. Let C be a linear code of length n over a ¯nite chain ring R. Assume that d ? ¼ dhom ðC ? Þ, and let V ¼ fvi : 1 i Kg be a set of distinct vectors in R. Let dv ¼ minfwhom ðvi vj þ cÞ : 1 i 6¼ j K; c 2 Cg and d ¼ minfdv ; d ? g. If d > 0, then there exists a quantum code with parameters ððp kðe1Þ n; K; dÞÞp k . kðe1Þ n

Proof. For each 1 i K, de¯ne a mapping i : F pp k 1 if u 2 ðvi þ CÞ; u 7! 0 if u 62 ðvi þ CÞ:

! C given by

We need to verify that condition (5) of Theorem 5.1 is satis¯ed. 1450020-9


For each partition f1; 2; . . . ; p kðe1Þ ng ¼ A [ B, jBj ¼ p kðe1Þ n d þ 1, and any cA ; c 0A 2 F d1 p k , we have 0 i ðcA ;cB Þ j ðc A ; cB Þ

with

jAj ¼ d 1

and

6¼ 0;

if and only if ðcA ; cB Þ 2 ðvi þ cÞ

and

ðc 0A ; cB Þ 2 ðvi þ CÞ:

and

1 ðc 0A ; cB Þ 2 vi þ C;

This is equivalent to 1 ðcA ; cB Þ 2 vi þ C which in turn is equivalent to 1 ðcA ; cB Þ 2 vi þ C and 1 ðcA ; cB Þ 1 ðc 0A ; cB Þ ¼ 1 ðcA ; 0Þ 1 ðc 0A ; 0Þ 2 C: Hence from Lemma 5.2 we obtain that ( X 0 if 1 ðcA ; 0Þ 1 ðc 0A ; 0Þ 62 C; 0 i ðcA ;cB Þ j ðc A ; cB Þ ¼ K=p kðd1Þ if 1 ðcA ; 0Þ 1 ðc 0A ; 0Þ 2 C; p kðe1Þ ndþ1 cB 2F

pk

for 1 i 6¼ j K. Since whom ð 1 ðcA ; 0Þ 1 ðc 0A ; 0ÞÞ ¼ wH ðcA c 0A ; 0Þ jAj ¼ d 1 whom ðvi vj þ CÞ, then ð 1 ðcA ; 0Þ 1 ðc 0A ; 0ÞÞ 62 vj vi þ C. Thus i ðcA ;cB Þ

j ðcA ; cB Þ

¼ 0, and therefore X

i ðcA ;cB Þ

0 j ðc A ; cB Þ

¼ 0:

kðe1Þ ndþ1 cB 2F p k p

Corollary 5.4. Let C1 and C2 be two linear codes over R of length n such that C1 C2 with jC1 j ¼ K1 and jC2 j ¼ K2 . Then there exists a ððp kðe1Þ n; K2 =K1 ; fdhom ðC2 nC1 Þ; dhom ðC 1? ÞgÞÞp k quantum code. Proof. Let v1 ; . . . ; vK be elements of C2 such that v1 þ C1 ; . . . ; vK þ C1 are K distinct cosets of C1 . By taking C ¼ C1 in Theorem 5.3, the result follows. Example 5.5. Tables 3 and 4 present examples of quantum codes over F2 obtained by applying Corollary 5.4 to self-orthogonal or self-dual codes over Z4 from Table 2 of Ref. 22 and Ref. 23, respectively. Table 3. ððn; K; dÞÞ2 quantum codes obtained by applying Corollary 5.4 to the codes in Table 2.22 ðð8; 2 5 ; 2ÞÞ2 ðð16; 2; 6ÞÞ2

ðð12; 2 7 ; 2ÞÞ2 ðð16; 2; 4ÞÞ2 1450020-10

Quantum codes over rings Table 4. ððn; K; dÞÞ2 quantum codes obtained by applying Corollary 5.4 to the codes in.23 ðð16; 1; 6ÞÞ2 ðð64; 1; 14ÞÞ2

ðð48; 1; 12ÞÞ2 ðð160; 1; 26ÞÞ2

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