Quantum Codes Obtained from Constacyclic Codes

Abstract

In this article, by using Hermitian construction, we obtain several new quantum codes and quantum MDS codes compare to the known codes from the constacyclic codes with only one q²-cyclotomic coset containing at least two consecutive integers with arbitrary difference r but fixed. This work is the generalization of the recent study on quantum codes from cyclic codes by La Guardia (Int. J. Theor. Phys. 56(8): 2479–2484, 2017) and from negacyclic codes by Gao and Wang (Int. J. Theor. Phys. 57(3): 682–686, 2018), respectively.

Full text

International Journal of Theoretical Physics
https://doi.org/10.1007/s10773-019-04260-y
Quantum Codes Obtained from Constacyclic Codes
Habibul Islam1·Om Prakash1·Dipak Kumar Bhunia1
Received: 15 May 2019 / Accepted: 22 August 2019 /
©Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
In this article, by using Hermitian construction, we obtain several new quantum codes and
quantum MDS codes compare to the known codes from the constacyclic codes with only one
q2-cyclotomic coset containing at least two consecutive integers with arbitrary difference r
but fixed. This work is the generalization of the recent study on quantum codes from cyclic
codes by La Guardia (Int. J. Theor. Phys. 56(8): 2479–2484, 2017) and from negacyclic
codes by Gao and Wang (Int. J. Theor. Phys. 57(3): 682–686, 2018), respectively.
Keywords Constacyclic code ·Cyclotomic coset ·Quantum code ·MDS code
1 Introduction
The quantum error-correcting codes have been growing interest among coding analyst after
the significant works of Shor [19] and Calderbank et al. [2] in the last decade of the
twentieth century. For last 15 years, quantum codes have been constructed using classical
error-correcting codes [8,11–14]. It is well understood that the cyclic code is one of the fas-
cinating sources to obtain new quantum codes [1,4,5,9,15,18], while the several articles
also certify the importance of the constacyclic codes to produce the new (better) quantum
codes [3,6,10,16,17,20–22]. In 2017, La Guardia [15] constructed some new quantum
codes from the cyclic codes with only one cyclotomic coset and consequently, Gao and
Wang [7] studied negacyclic codes to get new quantum codes under the similar approach of
La Guardia.
Recall that a q-ary quantum code is a K-dimensional subspace of qn-dimensional Hilbert
space (Cq)⊗n,whereqis a prime power integer. The standard notation for a quantum code
is [[n, k, d ]]q,wherenis the length, dis the minimum distance and K=qk.Theq-ary
quantum code [[n, k, d ]]qsatisfies the quantum singleton bound (QSB), k+2d≤n+2.
Om Prakash
om@iitp.ac.in
Habibul Islam
habibul.pma17@iitp.ac.in
Dipak Kumar Bhunia
dipak.mtmc17@iitp.ac.in
1Department of Mathematics, Indian Institute of Technology Patna, Patna 801 106, India

International Journal of Theoretical Physics
Those codes which attain the QSB are known as MDS (maximum-distance-separable)
codes.
In this article, we obtain some new quantum codes with respect to quantum singleton
bound from the constacyclic code whose defining set is the only one q2-cyclotomic coset.
To present the results, we follow the same line of arguments as given in [7,15]. Also, we
obtain several new and MDS quantum codes which are not available in the literature yet.
2 Preliminary
Let Fq2be the Galois field of q2elements where q=pm,for an odd prime p, and ma
positive integer. Recall that a linear code Cof length nis a subspace of the vector space
Fn
q2and each element of Cis known as codeword. Let ξbe a non-zero element of Fq2.
Then the linear code Cis said to be ξ-constacyclic if (ξcn−1,c
0,··· ,c
n−2)∈Cwhenever
(c0,c
1,··· ,c
n−1)∈C. By identifying each codeword c=(c0,c
1,··· ,c
n−1)∈Cto a
polynomial c(x) =c0+c1x+···+cn−1xn−1in Fq2[x]/xn−ξunder the correspondence
c=(c0,c
1,··· ,c
n−1)−→ c(x) =c0+c1x+··· +cn−1xn−1(mod xn−ξ),
we can say that Cis ξ-constacyclic if and only if it is an ideal of the ring Fq2[x]/xn−ξ.
Moreover, Cis principally generated by the monic polynomial f(x)where f(x) |(xn−ξ)
and dimension of Cis n−deg(f (x)). For any two codewords c=(c0,c
1,··· ,c
n−1), d =
(d0,d
1,··· ,d
n−1), the Hermitian inner product is defined by c, d=n−1
i=0cidq
i.Also,
the dual (Hermitian) code of Cis C⊥h={c∈Fn
q2|c, d=0,∀d∈C}.Itiswellknown
that for ξ-constacyclic code Cof length n, the dual C⊥his an ξ−q-constacyclic code and
hence an ideal of Fq2[x]/xn−ξ−q.
Throughout the article, rrepresents an arbitrary chosen positive integer but fixed. Let
gcd(n,q ) =1andξ∈Fq2be the r-th primitive root of unity. Also, let αbe the rn-th prim-
itive root of unity in some extension of Fq2such that αn=ξ. Then the set of roots of xn−ξ
is {α1+ri |0≤i≤n−1}.LetΩ={1+ri |0≤i≤n−1}. Then for each t∈Ω,letCtbe
the q2-cyclotomic coset modulo rn containing t, defined by Ct={t, tq2,··· ,tq2(mt−1)},
where mtis the smallest positive integer such that tq2mt≡t(modrn). Now, the polynomial
f(x) =i∈Ct(x −αi)is a factor of xn−ξand C=f(x)is the ξ-constacyclic code of
length nover Fq2where Ctis the defining set of Cand dimension of C=n−|Ct|= n−mt.
Proposition 1 [22](BCH bound for constacyclic codes) Let ξ∈Fq2be the r-th primitive
root of unity and C,an ξ-constacyclic code of length nover Fq2.Letαbe the rn-th primitive
root of unity in some extension of Fq2such that αn=ξ. If the set of roots of the generator
polynomial of Ccontains {α1+ir |i1≤i≤i1+δ}, then the minimum distance of Cis
d(C)≥δ+2.
3 Main Results
The following theorem is the key result of the article which guarantees the existence of
constacyclic code with one q2-cyclotomic coset as the defining set that contains at least two
consecutive integers of specific type.

International Journal of Theoretical Physics
Theorem 1 Suppose q≥3is a prime power integer, n>m,are integers such that
gcd(q, rn) =1,gcdq2ai−1
r,n
=1for i=1,2,··· ,δ where m=ordrn q2(multi-
plicative order of q2modulo rn), δ≥1and 1≤a1,a
2,··· ,a
δ≤mare integers. Further, if
n|gcd(t2,t
3,··· ,t
δ)where tj=j−(j −1)q2ajq2aj−1
r−1
−q2a1−1
r−1
(calcula-
tions are done under modulo n), then there exists a q2-ary [n, n −m∗,≥δ+2]constacyclic
code, where m∗is the size of q2-cyclotomic coset containing (δ +1)consecutive integers of
the form (1+ri).
Proof First, we consider the system of congruences
xq2a1≡(x +r) (mod rn)
(x +r)q2a2≡(x +2r) (mod rn)
(x +2r)q2a3≡(x +3r) (mod rn)
.
.
.
[x+(δ −1)r]q2aδ≡(x +δr) (mod rn),
where δ≥1and1≤a1,a
2,··· ,a
δ≤m.Sincegcd q2ai−1
r,n
=1, the above system is
converted into the following:
x≡q2a1−1
r−1
(mod n)
x≡(2−q2a2)q2a2−1
r−1
(mod n)
x≡(3−2q2a3)q2a3−1
r−1
(mod n)
.
.
.
x≡δ−(δ −1)q2aδq2a3−1
r−1
(mod n).
Therefore, the system has a solution if and only if
i−(i −1)q2aiq2ai−1
r−1
=j−(j −1)q2ajq2aj−1
r−1
(mod n)
∀i, j =2,3,··· ,δ,
and
q2a1−1
r−1
=i−(i −1)q2aiq2ai−1
r−1
(mod n)
∀i=2,3,··· ,δ.
Hence, n|tjfor all j=2,3,··· ,δ where tj=j−(j −1)q2ajq2aj−1
r−1
−
q2a1−1
r−1
.
Now, if Cis the constacyclic code of length nwhose defining set is a q2-cyclotomic coset

International Journal of Theoretical Physics
Cx,thenCxcontains x, x +r, x +2r, ··· ,x+δr consecutive (δ +1)integers of the form
(1+ri). Therefore, by BCH bound for constacyclic codes (Proposition 1), we conclude
that the minimum distance of Cis d(C)≥δ+2. Also, since |C|= m∗, then dimension of
C=n−m∗. Thus, we have the q2-ary [n, n −m∗,≥δ+2]constacyclic code.
Proposition 2 [11](Hermitian Construction) Let Cbe a q2-ary [n, k, d ]linear code. If
C⊥h⊆C, then there exists an [[n, 2k−n, ≥d]]qquantum code.
Corollary 1 Assuming all criterion of the Theorem 1 and Cis a constacyclic code whose
defining set is q2-cyclotomic coset Cxwhich contains the set {1+ri |0≤i≤δ}.If
Cx= C−x, then there exists an [[n, n −2m∗,≥δ+2]]qquantum code.
Proof By Theorem 1, we have the constacyclic code Cwith parameter [n, n −m∗,≥δ+2].
Since Cx= C−x,thenC⊥h⊆C. Therefore, by Proposition 2, there exists an [[n, n−2m∗,≥
δ+2]]qquantum code.
4 New Codes and Comparison
In this section, we present several new quantum codes in the sense of quantum singleton
bound (QSB) and also compare them with the codes which are previously known in the
literature.
Example 1 Let q=5,r =3andn=13. Then m=or drn(q2)=ord39(25)=2. Now,
in 25-cyclotomic cosets, we have C31 ={31,34}.LetCbe a constacyclic code of length
13 over F25 with defining set C31. Then Cis a 25-ary [13,11,≥3]linear code. Therefore,
we have the associated quantum code [[13,9,≥3]]5. Here, the quantum code [[13,9,≥
3]]5attains the quantum singleton bound k+2d≤n+2 and thus, it is a quantum MDS
code.
Example 2 Let q=23,r =3andn=53. Then m=ordrn(q2)=ord159(529)=2.
Now, one of 529-cyclotomic cosets is C25 ={25,28}. Consider the constacyclic code Cof
length 53 over F529 whose defining set is C25. Therefore, we have 529-ary [53,51,≥3]
linear code and corresponding quantum code [[53,49,≥3]]23, which is an MDS code as it
attains the quantum singleton bound k+2d≤n+2.
Example 3 Let q=11,r =3andn=57. Then m=ordrn(q2)=ord171(121)=3.
Hence, in 121-cyclotomic cosets, we have C10 ={10,13,34}.LetCbe a constacyclic code
of length 57 over F121 with defining set C10. Then Chas the parameter [57,54,≥3]121
and the associated quantum code is [[57,51,≥3]]11. Here, we observed that the obtained
quantum code have the same dimension and distance as of [[63,51,≥3]]11 shown in [15]
but our code has larger code rate.
Example 4 Let q=23,r =3andCbe a constacyclic code of length nover F529 with
defining set Cx(529-cyclotomic coset modulo rn).
1. Let n=13. Then m=ordrn(q 2)=ord39(529)=3. Further, let x=7, i.e., C7=
{7,34,37}. Then the quantum code [[13,7,≥3]]23 which has the larger code rate than
[[16,8,≥3]]23 given in [7].

International Journal of Theoretical Physics
2. Let n=21. Then m=ordrn(q 2)=ord63(529)=3. Again, let x=10, i.e., C10 =
{10,61,13}. Then, we have the quantum code [[21,15,≥3]]23 which has the larger
dimension and code rate than the quantum code [[20,12,≥3]]23,obtained in [7].
3. Let n=39. Then m=ordrn(q 2)=ord117(529)=3. Further, let x=34, i.e.,
C34 ={34,85,37}. Then, we have the quantum code [[39,33,≥3]]23 which has the
larger code rate than [[40,32,≥3]]23 shown in [7].
Example 5 Let q=27,n =73 and r=4. Then m=ordrn(q2)=or d292 (729)=2. In
729-cyclotomic cosets, C217 ={217,221}.LetCbe a constacyclic code of length 73 over
F729 with defining set C217.ThenChas the parameter [73,71,≥3]729 and the associated
quantum code is [[73,69,≥3]]27,which is an MDS code attains the quantum singleton
bound k+2d≤n+2.
In Table 1, we present some new q-ary quantum codes based on the constacyclic codes.
First column denotes the length of the constacyclic codes, third column is the defining q2-
cyclotomic coset, fourth and fifth column are parameters of constacyclic and associated
quantum codes, respectively. In addition, sixth column is given from different references to
Table 1 Some new quantum codes
nrC
x[n, k, d ]q2[[n, k, d]]q[[n,k
,d]]q
72C3={3,5,13}[7,4,≥3]25 [[7,1,≥3]]5
21 3 C10 ={10,61,13}[21,18,≥3]25 [[21,15,≥3]]5
31 3 C4={4,7,42}[31,28,≥3]25 [[31,25,≥3]]5[[30,20,3]]5([17])
53C1={1,4}[5,3,≥3]49 [[5,1,≥3]]7MDS
11 3 C7={7,13,10,28,19}[11,6,≥4]49 [[11,1,≥4]]7
19 3 C10 ={10,34,13}[19,16,≥3]49 [[19,13,≥3]]7[[18,12,3]]7([17])
25 3 C61 ={61,64}[25,23,≥3]49 [[25,21,≥3]]7MDS
43 3 C13 ={13,121,124}[43,40,≥3]49 [[43,37,≥3]]7[[42,36,3]]7([6])
57 3 C4={4,25,28}[57,54,≥3]49 [[57,51,≥3]]7[[60,24,3]]7([5])
13 2 C17 ={17,25,23}[13,10,≥3]81 [[13,7,≥3]]9
17 2 C3={3,5,31,29}[17,13,≥3]81 [[17,9,≥3]]9
19 3 C4={4,28,25}[19,16,≥3]121 [[19,13,≥3]]11 [[24,16,≥3]]11([7])
37 3 C25 ={25,28,58}[37,34,≥3]121 [[37,31,≥3]]11
11 2 C1={1,15,5,9,3}[11,6,≥4]169 [[11,1,≥4]]13
54C13 ={13,17}[5,3,≥3]169 [[5,1,≥3]]13 MDS
17 2 C7={7,10}[17,15,≥3]169 [[17,13,≥3]]13 MDS
61 4 C25 ={25,77,81}[61,58,≥3]169 [[61,55,≥3]]13
72C1={1,9,11}[7,4,≥3]289 [[7,1,≥3]]17
13 2 C17 ={17,25,23}[13,10,≥3]289 [[13,7,≥3]]17
29 3 C13 ={13,16}[29,27,≥3]289 [[29,25,≥3]]17 MDS
54C13 ={13,17}[5,3,≥3]729 [[5,1,≥3]]27 MDS
19 4 C1={1,45,49}[19,16,≥3]729 [[19,13,≥3]]27
37 4 C21 ={21,65,25}[37,34,≥3]729 [[37,31,≥3]]27 [[35,27,≥3]]27([15])
41 4 C21 ={21,57,61,25}[41,37,≥3]729 [[41,33,≥3]]27 [[40,32,≥3]]27([7])

International Journal of Theoretical Physics
compare the obtained quantum codes with previously known quantum codes. Our obtained
codes in fifth column have the larger code rate than the codes shown in sixth column.
Remark 1 To the best of our knowledge, all given codes in Table 1 are new. It is noted
that the codes [[11,1,≥4]]7,[[11,1,≥4]]13,[[17,9,≥3]]9and [[41,33,≥3]]27 satisfy
n+2−k−2d=4,i.e., these codes have singleton defect 4 while remaining codes in Table
1 (except MDS codes) satisfy the condition n+2−k−2d=2 with singleton defect 2.
5 Conclusion
In this article, we obtained several MDS and new q-ary quantum codes from constacylic
codes whose defining set is a q2-cyclotomic coset containing at least two consecutive
integers of specific type. We belief that more new quantum codes can be constructed by
computation in this set up.
Acknowledgments The authors are thankful to the University Grants Commission (UGC), Govt. of India
for financial support and Indian Institute of Technology Patna for providing research facilities. The authors
would like to thank the anonymous referee(s) for their careful reading and valuable comments, which helped
to improve the presentation of the manuscript.
References
1. Ashraf, M., Mohammad, G.: Quantum codes over Fpfrom cyclic codes over Fp[u, v]/u2−1,v
3−
v, uv −vu. Cryptogr. Commun. 11(2), 325–335 (2019)
2. Calderbank, A.R., Rains, E.M., Shor, P.M., Sloane, N.J.A.: Quantum error correction via codes over
GF (4). IEEE Trans. Inform. Theory. 44, 1369–1387 (1998)
3. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans.
Inform. Theory. 61, 1474–1484 (2015)
4. Dertli, A., Cengellenmis, Y., Eren, S.: On quantum codes obtained from cyclic codes over A2.Int.J.
Quantum Inf. 13(3), 1550031 (2015)
5. Gao, J.: Quantum codes from cyclic codes over Fq+vFq+v2Fq+v3Fq. Int. J. Quantum Inf. 13(8),
1550063(1-8) (2015)
6. Gao, J., Wang, Y.: u-Constacyclic codes over Fp+uFpand their applications of constructing new
non-binary quantum codes. Quantum Inf. Process. 17(1), 9 (2018)
7. Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57(3), 682–686
(2018)
8. Grassl, M., Beth, T.: On optimal quantum codes. Int. J. Quantum Inf. 2, 55–64 (2004)
9. Islam, H., Prakash, O.: Quantum codes from the cyclic codes over Fp[u, v, w ]/u2−1,v
2−1,w
2−
1,uv −vu, vw −wv , wu −uw. J. Appl. Math. Comput. 60(1–2), 625–635 (2019)
10. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inform.
Theory. 60(4), 2080–2085 (2014)
11. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.: Nonbinary stabilizer codes over finite fields.
IEEE Trans. Inform. Theory. 52(11), 4892–4914 (2006)
12. La Guardia, G.G., Palazzo, R. Jr.: Constructions of new families of nonbinary CSS codes. Discrete Math.
310, 2935–2945 (2010)
13. La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inform. Theory. 57(8), 5551–5554
(2011)
14. La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inform. Theory.
60(3), 1528–1535 (2014)
15. La Guardia, G.G.: Quantum codes derived from cyclic codes. Int. J. Theor. Phys. 56(8), 2479–2484
(2017)
16. Liu, Y., Li, R., Lv, L., Ma, M.: A class of constacyclic BCH codes and new quantum codes. Quantum
Inf Process. 16(3), 16 (2017)

International Journal of Theoretical Physics
17. Ma, F., Gao, J., Fu, F.W.: Constacyclic codes over the ring Fp+vFp+v2Fpand their applications of
constructing new non-binary quantum codes. Quantum Inf Process. 17(6), 19 (2018)
18. Qian, J., Ma, W., Gou, W.: Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7,
1277–1283 (2009)
19. Shor, P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A. 52, 2493–2496 (1995)
20. Taneja, D., Gupta, M., Narula, R., Bhullar, J.: Construction of new quantum MDS codes derived from
constacyclic codes. Int. J. Quantum Inf. 15(1), 1750008 (2017)
21. Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf Process.
14(3), 881–889 (2015)
22. Yang, Y., Cai, W.: On self-dual constacyclic codes over finite fields. Des. Codes Cryptogr. 74(2), 355–
364 (2013)
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps
and institutional affiliations.