Some Classes of New Quantum MDS and Synchronizable Codes Constructed From Repeated-Root Cyclic Codes of Length 6ps

In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$6p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{p^{m}}$ </tex-math></inline-formula>. We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$6p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{p^{m}}$ </tex-math></inline-formula>. Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.


I. INTRODUCTION
An [n, k] linear code C over F p m is a k-dimensional subspace of F n p m , where p is a prime number and F p m is a finite field. Let C be a linear code of length n over F p m . Then C is called a λ-constacyclic code if it is an ideal of x n −λ . If λ = 1, −1, those λ-constacyclic codes are called cyclic codes, negacyclic codes, respectively.
The associate editor coordinating the review of this manuscript and approving it for publication was A. Taufiq Asyhari .
He also discussed about dual constacyclic codes of these lengths. In 2014, [18] determined the structure of codes of length lp s over F p m .
Let C = [n, k, d H ] q be a code. Then [60] showed that n, k, d H must satisfy k ≤ n − d H +1 (the Singleton bound). If k = n − d H +1, then C is called a maximum-distanceseparable (briefly, MDS) code. The problem of constructing MDS codes is a hot topic because an MDS code has the greatest detecting and error-correcting capabilities.
An [[n, k]] QEC code encodes k logical qubits into n physical qubits. An (a b , a e ) − [[n, k]] QSC is an [[n, k]] quantum error-correcting code that corrects not only bit errors and phase errors but also misalignment to the left by a b qubits and to the right by a e qubits for some non-negative integers a b and a e . Block synchronization is an important problem in classical digital communications which was studied in [4], [33], [55], [63], [65], [74]. However, in quantum information, the methods in [4], [33], [55], [63], [65], [74] don't apply. Therefore, Fujiwara [32] first proposed QSCs to correct both quantum noise and block synchronization errors. After that, [58] proposed a class of QSCs from repeated-root codes using the CSS construction.
In [27], we studied qMDS codes from negacyclic and cyclic codes of length 2p s over F p m . We also gave some QSCs constructed from cyclic and negacyclic codes of length 2p s over F p m . However, in [27], we did not find some QEC codes using the CSS and Steane's constructions. In this paper, we construct some new QEC codes from cyclic codes of length 6p s using the CSS and Steane's constructions and some new QSCs from cyclic codes of length 6p s . By applying the CSS construction, we also provide all qMDS codes built from cyclic codes of length 6p s . Note that the structure of codes of length 6p s is much more complicated than cyclic and negacyclic codes of length 2p s . Repeated-root cyclic codes of length 6p s over F p m form a very interesting class of constacyclic codes. Their algebraic structures in term of generator polynomials were provided in 2014 in [26]. Recently, these structures were used in [28] to completely determine the Hamming distances of all such cyclic codes.
Motivated by these, in this research, we construct QEC codes from cyclic codes of length 6p s over F p m using CSS and Steane's constructions. Especially, we compare our QEC codes with all previous QEC codes to show that some our QEC codes are new in the sense that their parameters are different from all the previous results. We also provide all qMDS codes from cyclic codes of length 6p s over F p m using the CSS construction. Furthermore, we also construct QSCs from cyclic codes of length 6p s over F p m . This paper is organized as follows. Section 2 gives some basic results. Section 3 constructs QEC codes from cyclic codes of length 6p s over F p m using the CSS and Steane's constructions. Section 4 studies qMDS codes from cyclic codes of length 6p s over F p m using the CSS construction. Section 5 constructs QSCs from cyclic codes of length 6p s over F p m . Section 6 gives some examples to illustrate our results in Sections 3, 4 and 5, where we present numerous qMDS and QSCs codes. Section 7 concludes our paper with some possible open direction for future studies.

II. PRELIMINARIES
The following lemma is given in [60].
Lemma 1 (cf. [60]): Let C be a linear code of length n over F p m . Then C is λ-constacyclic over F p m if and only if C is an ideal of x n −λ . Given n-tuples Dual code of a linear code C over F p m , denoted by C ⊥ , is defined as follows: The result on the dual of a -constacyclic code is provided in [23] as follows.
In [26], Dinh studied cyclic codes of length 6p s over F p m . We recall the structure of cyclic codes of length 6p s over F p m when p m ≡ 2 (mod 3).
We recall the definition of QEC codes appeared in [67]. Definition 4 [67]: Let q = p m and H q (C) be a q-dimensional Hilbert vector space. Denote H n q (C) = H q (C)⊗· · ·⊗H q (C) (n times). A quantum code of length n and dimension k over F q is defined to be a q k dimensional subspace of H n q (C) and simply denoted by [ We give a small lemma. Lemma 5: Let 0 < t ∈ N. Then there are (t+2)(t+1) 2 pairs of non-negative integers x, y such that x + y ≤ t.
Proof: If x = 0, then we have t+1 options for y. If x = 1, then we have t options for y. In general, for any x = j, where 0 ≤ j ≤ t, there are t − j + 1 options for y. That means y can be any integer from 0 to t − j. It implies that there are 1 + 2 + 3 + · · · + t + (t + 1) = (t+2)(t+1) 2 pairs of non-negative integers x, y such that x + y ≤ t.

III. QUANTUM CODES FROM CYCLIC CODES OF LENGTH 6p s OVER F p m
In 1995, QEC codes were first introduced by Shor [72]. After that, in 1996, by using the structure of classical codes over GF(4), [6] found some QEC codes. In 1998, [7] gave a new method to construct QEC codes from classical codes. Recently, [2], [7], [20], [37], [53] constructed some QEC codes over finite fields and some classes of finite rings. However, QEC codes constructed from cyclic codes of length 6p s over F p m using the CSS and Steane's constructions have not been studied in the past.
We recall a construction of QEC codes, the so-called CSS construction.
Theorem 6 (CSS Construction [6]): Let C 1 = [n, k 1 , d 1 ] q and C 2 = [n, k 2 , d 2 ] q be two linear codes satisfying C 2 ⊆ C 1 . Then there exists a QEC code with the parameters [[n, Throughout this paper, p m ≡ 2 (mod 3). Recall that C is dual-containing if C ⊥ ⊆ C. We give the condition of a cyclic code of length 6p s over F p m to be dual-containing to construct QEC codes.
Proposition 7: Let C be a cyclic code of length 6p s over F p m which is of the form h 0 ( In addition, the number of dual-containing codes is ( p s +1 2 ) 4 . Proof: By Theorem 3, it is easy to see that In this case, the number of QEC codes constructed from cyclic codes of length 6p s over F p m using the CSS construction is ] p m . Using Proposition 7, the number of dual-containing codes is ( p s +1 2 ) 4 . Hence, the number of QEC codes constructed from cyclic codes of length 6p s over F p m using the CSS construction is ( p s +1 2 ) 4 . A construction which links between linear codes and QEC codes is the Steane's construction.
Theorem 9: (Steane's Construction [77]): Let C 1 and C 2 be two linear codes over F p m with parameters [n, Combining Proposition 7 and Theorem 9, we construct QEC codes from the class of cyclic codes of length 6p s over F p m using the Steane's construction.
Theorem 10: Let C be a cyclic code of length 6p s over

IV. QUANTUM MDS CODES
In 1992, The Singleton bound is given in [75] as follows: . The case of binary codes was first proved in [51]. Motivated by this, [48] also considered this problem. In 1974, the proof for general q-ary case is given by Denes and Keedwell [21]. A code C satisfying |C| = p m(n−d H (C)+1) which is called an MDS code. In 1952, Bush gave some results on MDS codes. After that, [36], [73] and [61] also provided several interesting results on MDS codes. The problem of the weight enumerator for such codes was considered by many researchers (for examples, [29], [60], [78]).
In 2020, [28] investigated the Hamming distances of cyclic codes of length 6p s over F p m and provided all MDS constacyclic codes of length 6p s over F p m as follows.
Theorem 11: Let C = f (x) be a cyclic code of length 6p s . Then C is an MDS code if and only if Using Theorem 11, we give all MDS cyclic codes of length 6p s over F p m .
Theorem 12: Let C be a cyclic code of length 6p s over F p m which is of the

3). Then C is an MDS cyclic code if and only if
In the next part, we construct qMDS codes from cyclic codes of length 6p s over F p m using the CSS construction. To do so, we recall the quantum Singleton bound for all classes of codes over finite fields as follows.
. Then the following statements hold: We consider 2 cases as follows:

V. QUANTUM SYNCHRONIZABLE CODES
QSCs are used for correcting the extract the Pauli errors on qubits and preventing the destruction of qubits in the quantum states. Therefore, several QSCs are provided to use in quantum synchronizable codes (for examples, [32], [34], [35], [79], [58], [59], [80]). Let be an integer satisfying gcd( , p) = 1, where ≥ 2. Assume that C t, is the cyclotomic coset of t modulo over F q and denote by T the set of representatives of all q-ary cyclotomic cosets. Let f t (x) = i∈C t, (x − ξ i ) be the minimal polynomial of ξ t over F q , where ξ is a primitive -th root of unity in F q . Then the polynomial x p s − 1 over F q can be factored as In 2015, by using the class of cyclic codes of length p s over F q , [58] constructed some QSCs.

Theorem 15 ([58, Theorem 3]): Let
Then the following conditions hold: In such cases, if there exists an integer r ∈ T with gcd(r, ) = 1 satisfying either i r − j r > p s−1 or i r − j r > 0 and i r − j r > p s−1 for some r = r ∈ T , then for any pair of non-negative integers a b , a e satisfying a b + a e < p s , there exists an (a b , a e )-[[ p s + a b + a e , p s − 2 t∈T u t| C t, |]] q QSC.
Using Theorem 15, we construct QSCs from cyclic codes of length 6p s over F p m as follows.
Theorem 16: Let C be a cyclic code of length 6p s over Then the following conditions hold: In such cases, if there exists an integer r ∈ T 6 satisfying either u r − j r > p s−1 or u r − j r > 0 and u r − j r > p s−1 for some r = r ∈ T 6 , then for any pair of non-negative integers a b , a e satisfying a b + a e < 6p s , there exists an (a b , a e ) − [[6p s + a b + a e , 6p s − 2u 0 − 2u 1 − 4u 2 − 4u 3 ]] q QSC. If we fix u t , j t , r, where t = 0, 1, 2, 3 and r ∈ T 6 , then there are 3p s · (6p s + 1) such QSCs. Proof: where t = 0, 1, 2, 3, i.e., 0 ≤ u t ≤ p s 2 and 0 ≤ j t ≤ p s 2 , showing (i) and (ii). From C 1 ⊆ C 2 , it implies that 0 ≤ j t < u t ≤ p s , proving (iii). Since C ⊥ 1 ⊆ C 1 , C ⊥ 2 ⊆ C 2 , and C 1 ⊆ C 2 , by using Theorem 15, if there is an integer r ∈ T 6 such that either u r − j r > p s−1 or u r − j r > 0 and u r − j r > p s−1 for some r = r ∈ T 6 , then for any pair of non-negative integers a b , a e satisfying a b + a e < 6p s , there exists an Assume that u t , j t , r are fixed, where t = 0, 1, 2, 3 and r ∈ T 6 . Using Lemma 5 for n = 6p s −1, there are 3p s ·(6p s +1) pairs of non-negative integers a b , a e satisfying a b +a e < 6p s . It means that there are 3p s · (6p s + 1) such QSCs. BCH codes are used in coding theory since they have useful in encoding and decoding algorithms. Let n be a divisor of p m − 1 and γ be an element of F p m with multiplicative order n. A BCH code of length n is a cyclic code such that its generator polynomial has a set of α − 1 consecutive roots γ e , γ e+1 , · · · , γ e+α−2 , where e ∈ N . Applying the BCH bound, we see that the minimum distance of the BCH code is at least α. Therefore, the designed distance of the BCH code is α. If C is a BCH code satisfying the length n = p m − 1, then C is called primitive. If e = 1, i.e., the α − 1 consecutive roots start from γ , then C is called narrow-sense.
Remark 17: In 2015, [58, Table 2] gave some parameters of non-primitive, narrow-sense BCH codes C over F q in Table 2. Some parameters of cyclic codes of length 6p s over F p are listed in Table 3 to show that the code lengths of cyclic codes of length 6p s over F p are smaller than BCH codes given in Table 2 but the Hamming distances of repeated-root cyclic codes of length 6p s over F p are better than γ max , where γ max  is a precise lower bound for the largest minimum distance of a dual-containing BCH code. This is the reason why QSCs constructed from repeated-root cyclic codes of length 6p s over F p are better than QSCs constructed from non-primitive, narrow-sense BCH codes. Put Then we have the following table.
We compare the QEC code and online table [42] to see that the QEC code with parameters [ [66,41,4]] 11 is new in the sense that the parameters are different from all the previous constructions.
(ii) Let C 3 = h 0 (x) 2 g 1 (x)h 3 (x) and C 4 = g 0 (x)h 1 (x) . It is easy to see that C 3 ⊆ C 4 . We have k C 3 = 61 and k C 4 = 64. Using Theorem 3.13 in [28], d H (C 3 ) = 3 and d H (C 4 ) = 2. By Theorem 10, we see that there exists a QEC code with parameters [[66, 59, min{3, 24 11 }]] 11 = [[66, 59, 3]] 11 . We compare the QEC code and online table [42] 11 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 3 and C 4 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42].
(ii) Let C 5 = h 0 (x) 3 g 1 (x)h 3 (x) and C 6 = g 0 (x)g 1 (x)h 3 (x) . It is easy to see that C 5 ⊆ C 6 . We have k C 5 = 60 and k C 6 = 61. Using Theorem 3.13 in [28], d H (C 5 ) = 4 and d H (C 6 ) = 3. By Theorem 11 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 5 and C 6 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42].
Example 19: Let p = 17, s = 1, m = 1. We have . Hence, C 1 ⊆ C 2 and k C 1 = 96, k C 2 = 97. Applying Theorem 3.13 in [28], d H (C 1 ) = 4 and d H (C 2 ) = 3. From Proposition 7, it is easy to see that C ⊥ 1 ⊆ C 1 . Using Theorem 8 for C 1 17 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 1 and C 2 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42]. 3 and C 4 = h 0 (x) 6 g 1 (x) 3 g 3 (x) . Hence, C 3 ⊆ C 4 and k C 3 = 84, k C 4 = 85. Applying Theorem 3.13 in [28], d H (C 3 ) = 8 and d H (C 4 ) = 7. From Proposition 7, it is easy to see that C ⊥ 3 ⊆ C 3 . Using Theorem 8 for C 3 17 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 3 and C 4 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42]. (iii) Let C 5 = h 0 (x) 8 h 1 (x) 4 g 2 (x) 2 g 3 (x) 4 and C 6 = h 0 (x) 7 g 1 (x) 3 g 2 (x)h 3 (x) 3 . Hence, C 5 ⊆ C 6 and k C 5 = 78, k C 6 = 84. Applying Theorem 3.13 in [28], d H (C 5 ) = 9 and d H (C 6 ) = 8. From Proposition 7, it is easy to see that C ⊥ 5 ⊆ C 5 . Using Theorem 8 for C 5 , there exists a QEC code with parameters [[102, 54,9]] 17 . We compare the QEC code and online table [42] to see that the QEC code with parameters [[102, 54,9]] 17 is coincided with a QEC code listed in [42], i.e., it is not new in the sense that the parameters are different from all the previous constructions. However, by Theorem 10, we see that there exists a QEC code with parameters [[102, 60, min{9, 144 17 }]] 17 = [[102, 60,9]] 17 . We compare the QEC code and online table [42] to see that the QEC code with parameters [[102, 60,9]] 17 is new in the sense that the parameters are different from all the previous constructions. Moreover, the QEC code with parameters [[102, 60,9]] 17 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 5 and C 6 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42].
Example 20: Let p = 5, s = 2 and m = 1. We have  17 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 1 using CSS construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42]. We see that the number of QEC codes constructed from all cyclic codes of length 150 over F 5 using the CSS construction is 11325.
(ii) Let C 2 = h 0 (x) 2 f 1 (x)h 3 (x). Using Theorem 3.13 in [28], d H (C 2 ) = 3. It is easy to see that k C 1 = 144 and k C 2 = 145. By Theorem 10, we see that there exists a QEC code with parameters [[150, 139, min{4, 18 5 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 1 and C 2 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42].
Example 21: Let p = 23, s = 1 and m = 1. We have . Using Theorem 3.13 in [28], d H (C 1 ) = 4 and d H (C 2 ) = 4. It is easy to see that k C 1 = 127 and k C 2 = 129. By Theorem 23 . We compare the QEC code and online table [42] to see that the QEC code with parameters [[138, 118, 4]] 23 is coincided with a QEC code listed in [42], i.e., it is not new in the sense that the parameters are different from all the previous constructions.
(ii) Let C 3 = h 0 (x) 3 g 1 (x) 2 g 3 (x) and C 4 = h 0 (x) 2 g 1 (x)h 3 (x) . It is easy to see that C 3 ⊆ C 4 . We have k C 3 = 131 and k C 4 = 133. Using Theorem 3.13 in [28], d H (C 3 ) = 4 and d H (C 4 ) = 3. By Theorem 10 23 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 3 and C 4 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42].
Example 22: Let p = 29, s = 1 and m = 1. We have  29 . We compare the QEC code and online table [42] to see that the QEC code with parameters [[174, 149, 4]] 29 is coincided with a QEC code listed in [42], i.e., it is not new in the sense that the parameters are different from all the previous constructions. ( We have k C 3 = 167 and k C 4 = 169. Using Theorem 3.13 in [28], d H (C 3 ) = 4 and d H (C 4 ) = 3. By Theorem 10 29 . We compare the QEC code and online table [42] to see that the QEC code with parameters [[174, 162, 4]] 29 is new in the sense that the parameters are different from all the previous constructions. Moreover, the QEC code with parameters [[174, 162, 4]] 29 is better than all QEC codes with same length and Hamming distance listed in [42], i.e., the QEC code constructed from cyclic code C 3 and C 4 using the Steane's construction has the dimension that is larger than the dimension of all QEC codes with same length and Hamming distance listed in [42].
Example 23: Let p = 11, s = 1, m = 1. We see that Example 24: Let p = 11, s = 2, m = 1. We see that  11 . Example 25: Let p = 29, s = 1, m = 1. We have   29 . Remark 27: We can compare our qMDS codes and known families of qMDS codes (Table 1) and [42] to see that our qMDS codes are new in the sense that their parameters are different from all the known ones.
Example 30: Let p = 23, s = 1, m = 1. We see that  4 . We see that u 1 − j 1 > 1. Applying Theorem 16, for any pair a b , a e of non-negative integers satisfying a b + a e < 138, there exists an (a b , a e ) − [[138 + a b + a e , 88]] 23 QSC. By using Lemma 5, we see that there are 9591 such QSCs.

VII. CONCLUSION
In this paper, we use the CSS and Steane's constructions to establish QEC codes from cyclic codes of length 6p s over F p m (Theorems 8 and 10). We get some new QEC codes in the sense that the parameters are different from all the previous constructions (Examples 3.6 and 3.7). Applying the quantum Singleton bound, all qMDS cyclic codes of length 6p s over F p m using the CSS construction are determined in Theorem 14. As in Section 5, we construct QSCs from cyclic codes of length 6p s over F p m (Theorem 16) and such codes are applicable in quantum synchoronizable. Remark 17 shows that QSCs constructed from repeated-root cyclic codes of length 6p s over F p m are better than QSCs constructed from non-primitive, narrow-sense BCH codes. In Section 6, we provide some examples to illustrate our work in Sections 3, 4 and 5.
Although we only consider the case p m ≡ 2 (mod 3) in this paper, the situation of p m ≡ 1 (mod 3) can be studied in a similar fashion. When p m ≡ 1 (mod 3), from [26], all cyclic codes of length 6p s have the form C = (x −1) u 0 (x +1) u 1 (x − ξ p m −1 where 0 ≤ u t ≤ p s (t = 0, 1, 2, 3, 4, 5) and ξ ∈ F p m is a primitive (p m − 1)th root of unity. Applying the method used in [28], we can determine the Hamming distances of all such cyclic codes. We also compute all Hamming distances of negacyclic codes of length 6p s over F p m . Similar to Theorems 8 and 10, we can construct new QEC codes from cyclic and negacyclic codes of length 6p s over F p m using the CSS and Steane's constructions.
Let q = p m and F q 2 be a finite field of q 2 elements. If e = (e 0 , e 1 , . . . , e n−1 ), t = (t 0 , t 1 , · · · , t n−1 ) are two vectors of F q 2 , then Hermitian inner product of e and t is e • F q 2 t = e 0t0 + e 1t1 + · · · + e n−1tn−1 , wheret i = t q i . The Hermitian dual code of C is defined as If C ⊥ H ⊆ C, then C is said to be Hermitian dualcontaining.
The Hermitian construction is also an important construction appeared in [1].
By giving the condition of a cyclic and negacyclic code of length 6p s over F q 2 to construct QEC codes, similar to Theorem 14, we can construct new QEC codes from cyclic and negacyclic codes of length 6p s over F q 2 using the Hermitian construction.
We also investigate the QSCs constructed from negacyclic codes of length 6p s over F p m , or more generally 2 m p s , for any non-negative integer m in near future. We believe that these lengths can provide good and new QEC codes and QSCs.