Hermitian dual-containing constacyclic BCH codes and related quantum codes of length q 2m −1 q+1
aa r X i v : . [ c s . I T ] J u l Noname manuscript No. (will be inserted by the editor)
Hermitian dual-containing constacyclic BCH codes andrelated quantum codes of length q m − q + Xubo Zhao , , · Xiaoping Li , , · Qiang Wang · Tongjiang Yan , , Received: date / Accepted: date
Abstract
In this paper, we study a family of constacyclic BCH codes over F q of length n = q m − q + , where q is a prime power, and m ≥ n isdetermined. Furthermore, the exact dimension of the constacyclic BCH codes with givendesign distance is computed. As a consequence, we are able to derive the parameters ofquantum codes as a function of their design parameters of the associated constacyclic BCHcodes. This improves the result by Yuan et al. (Des Codes Cryptogr 85(1): 179-190, 2017),showing that with the same lengths, except for three trivial cases ( q = , , n = q m − q + , some constructed quantum codeshave better parameters or are beneficial complements compared with some known results(Aly et al., IEEE Trans Inf Theory 53(3): 1183-1188, 2007, Li et al., Quantum Inf Process18(5): 127, 2019, Wang et al., Quantum Inf Process 18(8): 323, 2019, Song et al., QuantumInf Process 17(10): 1-24, 2018.). Keywords
Constacyclic codes · Cyclotomic cosets · Hermitian construction · Quantumcodes
Mathematics Subject Classification (2010) · · Xubo ZhaoE-mail: [email protected] LiE-mail: [email protected] WangE-mail: [email protected] YanE-mail: [email protected] College of Science, China University of Petroleum (East China), Shandong, Qingdao 266580, China2 Key Laboratory of Applied Mathematics(Putian University), Fujian Province University, Fujian Putian,351100, China3 Shandong Provincial Key Laboratory of Computer Networks, Shandong, Jinan, 250014, China4 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S5B6, Canada Xubo Zhao , , et al. Quantum error correction codes play an important role in protecting quantum informationfrom decoherence in quantum computations and quantum communications. One of the prin-cipal problems in quantum error correction is to construct quantum codes with good param-eters. After the establishment of the connections between quantum codes and classical codesby Calderbank et al. [3, 7], construction of quantum codes can be deduced to find classical“dual-containing” or “self-orthogonal” linear codes with respect to certain inner products.Among these inner products, the Hermitian inner product has been widely used, and the cor-responding Hermitian dual-containing or Hermitian self-orthogonal codes can yield manygood quantum codes [8–15, 18–20, 34, 40].Denote F q be a finite field with q elements and F ∗ q = F q \{ } , where q is a prime power.For any α ∈ F q , the conjugate of α is defined by ¯ α = α q . Let F nq be the F q -vector spaceof n -tuples. A q -ary linear code C of length n is an F q -subspace of F nq . A linear code C iscalled an [ n , k , d ] q linear code if its dimension is k and minimum Hamming distance is d .Given two vectors u = ( u , u , · · · , u n ) and v = ( v , v , · · · , v n ) ∈ F nq , their Hermitian innerproduct h u, v i H is defined as h u, v i H = X ni = u i ¯ v i . (1)The vectors u, v are called orthogonal with respect to the Hermitian inner product if h u, v i H =
0. For a q -ary linear code C , its Hermitian dual code is defined as C ⊥ H = { u ∈ F nq | h u, v i H = , ∀ v ∈ C} . (2)If C ⊥ H ⊆ C and C , F nq , C is called a Hermitian dual-containing code, and C ⊥ H is called aHermitian self-orthogonal code. One of the most frequently-used and e ff ective constructionmethods of quantum codes is as follows (see [3]). Theorem 1 If C is an [ n , k , d ] q linear code such that C ⊥ H ⊆ C , then there exists an [[ n , k − n , ≥ d ]] q quantum code. We refer to the construction in Theorem 1 as Hermitian construction, which suggests that q -ary quantum codes can be obtained from classical Hermitian dual-containing linear codesover F q .Bose-Chaudhuri-Hocquenghem (BCH) codes are a significant class of good cyclic codes,which have e ffi cient encoding and decoding algorithm and strong error-correcting capabil-ity. They have been widely employed in data storage systems and satellite communications.Another important application of BCH codes is to construct quantum codes. Aly et.al [1, 2]studied Euclidean or Hermitian dual-containing BCH codes of general length, they derived aformula for the dimension of narrow-sense BCH codes with small design distance, and thenspecified the parameters of quantum BCH codes in terms of the design distance. Thereafter,a lot of good quantum codes have been obtained by BCH codes [9–11, 18, 26, 39, 40].Constacyclic BCH codes [27] as a generalization version of the well-known BCH codes,are naturally considered to construct quantum codes. Generally speaking, the parameter ofquantum codes constructed from constacyclic BCH codes [21–24, 28, 30, 35, 36] might bebetter than the ones derived from BCH codes. Based on Hermitian construction, in orderto construct quantum codes, it is necessary to know the exactly dimension and minimumdistance of the classical codes. However, for general code length, the dimension and mini-mum distance of BCH or constacyclic BCH codes have limited knowledge because of their ermitian dual-containing constacyclic BCH codes and related quantum codes of length q m − q + intricate structure [37, 40]. Therefore, it is very hard to obtain the precise dimension andminimum distance of quantum codes in general, a summary of some known constructionsof quantum constacyclic codes with special code lengths is provided in Table 3 of litera-ture [36].In [23], Zhu et al. studied the dimension, the minimum distance, and the weight distri-bution of certain negacyclic BCH codes (a subclass of constacyclic BCH codes) of length n = q m − q − over F q with odd q and any positive integer m . From this class of negacyclicBCH codes, they acquired some quantum codes with good parameters. In [28] and [30],Kai and Guo et al. investigated negacyclic BCH codes of length n = q m − over F q withodd q , or odd q and odd m , respectively. Later, Wang et al. [36] discussed narrow-senseand non-narrow-sense negacyclic BCH codes of length n = q m − a , where a | ( q m −
1) is even,both q and m are odd. Then they constructed some quantum codes with good parametersfrom those dual-containing negacyclic BCH codes. By investigation of suitable propertieson q -ary cyclotomic cosets, Liu et al. [22] determined a class of Hermitian dual-containingconstacyclic BCH codes of length n = q m + F q with odd q and any integer m ≥ d > q . Wang etal. [35] focused on a class of q -ary constacyclic BCH codes of length n = q m +
1) withodd q and any m ≥
3, via Hermitian construction, they gained many new quantum codes,which are not covered in the literature.As shown above, for the case of odd q , many q -ary constacyclic BCH codes with specialcode lengths have been well studied and have been used to construct quantum codes withdesirable parameters via Hermitian construction. While for the case of even q , especiallywith even m , Hermitian dual-containing constacyclic BCH codes are much less investigateddue to the di ffi culty of computing the exact dimension of this class of codes. Based on clas-sical quaternary constacyclic BCH codes, Lin [21] got a class of binary quantum codes oflength m − ( m ≥ q -ary case with length n = q m − q + by applying classical q -ary constacyclic BCH codes, where q is a prime powerand m ≥
2. Moreover, compared with the corresponding ones in [21], the narrow-senseconstacyclic BCH codes in [24] have relatively large design distance. To determine the di-mension of the Hermitian dual-containing constacyclic BCH codes, Yuan et al. followed adirectly generalized formula (see Eq.(3) in [24]) of [21]. However, the dimension of narrow-sense constacyclic BCH codes presented in [24] are just the lower bound of their actualdimension. The main reason is that the union of q -cyclotomic cosets for the defining setof constacyclic BCH codes will involve more repeated ingredients as the size of finite fieldand the design distance become larger. Very recently, Wang et al. [35] study Hermitian dual-containing narrow-sense constacyclic BCH codes of length q m − ρ and aim at constructingnew quantum codes with the corresponding constacyclic BCH codes, where ρ | ( q + m ≥
3. By employing the technique of [38], they determine all coset leaders of cyclotomiccosets in the defining set and discuss the q -adic expansion about the design distance to com-pute the dimension of constacyclic BCH codes. This method is e ffi cient for the case of odd m (see Theorem 3 in [35]), however, the derived parameters of constacyclic BCH codes andthe related quantum codes are not concise enough for the case of even m ≥ n = q m − q + with special values of q and m (seeRemark 2 in Subsection 3.2). In addition, some quantum codes constructed from narrow-sense constacyclic BCH codes of length n = q m − q + have better parameters compared to some Xubo Zhao , , et al. known results studied in the literature [2,17,18,24,25,35]. Thus, this family of constacyclicBCH codes deserve to be investigated further. In this article, we focus on determining theparameters of Hermitian dual-containing constacyclic BCH codes and the related quantumcodes of length n = q m − q + for the case of even m , and the results about the case of odd m canbe found in Subsection 2.2 of [24] or Theorem 3 in [35]. Note that in the case of even m ,it is more challenging to derive the parameters of this family of Hermitian dual-containingconstacyclic BCH codes, since the defining set of the considered codes involves a moreintricate structure compared with the case of odd m . We provide necessary and su ffi cientconditions for the existence of narrow-sense Hermitian dual-containing constacyclic BCHcodes (see Theorem 3), which allows one to identify all constacyclic BCH codes that canbe used in quantum codes construction. Furthermore, di ff erent from the idea of finding outall coset leaders of the union of cyclotomic cosets in the defining set, we counting the jointcyclotomic cosets via solving the associated congruent equations. Moreover, based on adetailed study of properties of the q -cyclotomic cosets (see Lemmas 2, 3), we computethe exact dimensions of this family constacyclic BCH codes for all design distance δ in therange 2 ≤ δ ≤ δ max (see Theorem 4). These results facilitate us to determine the parametersof quantum codes as a function of their design parameters n , q , and δ of the related narrow-sense constacyclic BCH codes (see Theorem 5).The remainder of this paper is organized as follows. In Section 2, some basic definitionsand properties about constacyclic codes are reviewed. In Section 3, the Hermitian dual-containing conditions, as well as properties of corresponding cyclotomic cosets of narrow-sense constacyclic BCH codes are investigated. Some new quantum codes are constructedfrom the underlying constacyclic BCH codes, and the parameters of resultant quantum codesare compared with previous results. The conclusion of this paper is given in Section 4. In this section, we review some basic notation and results about quantum codes and consta-cyclic codes. Throughout this paper, F q denotes the finite field with q elements, where q is a prime power, Z represents the ring of integers, ⌊ X ⌋ means the maximal integer, whichis not more than X , and ⌈ X ⌉ is the minimal integer, which is not less than X . The bracketnotation [statement] takes the value 1 if statement is true, and 0 otherwise, for instance, wehave [ q even] = q − η ∈ F ∗ q , a linear code C of length n over F q is called η - constacyclic code [5, 33]provided that for each codeword c = ( c , c , · · · , c n − ) in C , ( η c n − , c , · · · , c n − ) is also acodeword in C . Some of the most important classes of codes can be realized as special casesof constacyclic codes. For example, the case η = cyclic codes , and η = − negacyclic codes . Customarily, each codeword c = ( c , c , · · · , c n − ) ∈ C is identified withits polynomial representation c ( x ) = c + c x + · · · + c n − x n − . It is easy to check that an η -constacyclic code C of length n over F q corresponds to a principal ideal h g ( x ) i of thequotient ring F q [ x ] / h x n − η i , where g ( x ) is a monic divisor of x n − η . In this case, C isgenerated uniquely by g ( x ), i.e., C = h g ( x ) i , the polynomial g ( x ) is called the generatorpolynomial of the code C , and the dimension of C is n − deg( g ( x )).We assume that n and q are relatively prime, i.e., gcd( n , q ) =
1, so that the polynomial x n − η over F q does not involve repeated roots. We denote the order of η in the multiplicativegroup F ∗ q by ord( η ). Assume that ord( η ) = r , then η is called a primitive r th root of unity .Suppose that the multiplicative order of q modulo nr is m (the smallest positive m such that ermitian dual-containing constacyclic BCH codes and related quantum codes of length q m − q + nr divides q m − nr ( q ) = m . Then there exists a primitive nr th root β of unityin F q m such that β n = η . Let ξ = β r , then ξ is a primitive n th root of unity. Thus, the roots of x n − η are βξ i = β + ri , i = , , · · · , n −
1, i.e., x n − η = Q n − i = ( x − β + ri ). Let O = { + ri | ≤ i ≤ n − } (mod nr ) . (3)The defining set of the η -constacyclic code C = h g ( x ) i is defined as T = { j ∈ O | g ( β j ) = } . (4)Let C s be the q -cyclotomic cosets modulo nr , which contains s . That is C s = { sq l mod nr | l ∈ Z , ≤ l ≤ m s − } , where m s is the smallest positive integer such that sq m s ≡ s mod nr ,and s is not necessarily the smallest number in the coset C s . Recall that for each integer τ ,with 0 ≤ τ < nr −
1, the minimal polynomial of β τ over F q is M τ ( x ) = Q i ∈ C τ ( x − β i ). And x nr − = Q τ ∈ Ω M τ ( x ) is the factorization of x nr − F q , where Ω is the set of representatives of all the distinct q -cyclotomic cosets modulo nr (see [4, 6]).Obviously, ( x n − η ) | ( x nr − x n − η = Q τ ∈O∩ Ω M τ ( x ), thereby the defining set of code C = h g ( x ) i must be a union of some q -cyclotomic cosets modulo nr . Definition 1 [25, 27] Assume that gcd( n , q ) =
1. Let C = h g ( x ) i be an η -constacyclic codeof length n over F q , where η is a primitive r th root of unity. Let β be a primitive nr th rootof unity in some extension field of F q , such that β n = η . An η -constacyclic BCH code oflength n with design distance δ is an η -constacyclic code with defining set T = ∪ δ − i = C b + ri , (5)where b = + jr , j ∈ Z . Such a code is called primitive if n = q m −
1, and non-primitive,otherwise. If b = C is called a narrow-sense code, and non-narrow-sense, otherwise.For an η -constacyclic BCH code C , the following BCH bound (see [4] Theorem 2.2or [8] Theorem 2.8) shows the relationship between the minimum distance of C and thedesign distance δ of C . Theorem 2
Let C be an η -constacyclic BCH code with defining set T ⊆ O . If { + ri | a ≤ i ≤ a + δ − } ⊆ T , where a ∈ Z . Then the minimum distance of C is at least δ . As mentioned in Theorem 1, q -ary quantum codes can be obtained from classical Her-mitian dual-containing linear codes over F q by Hermitian construction. Normally, the con-dition of Hermitian dual-containing C ⊥ H ⊆ C can be guaranteed by the popular defining setcriterion in the following (for example, see [15] Lemma 2.2). Lemma 1
Let C = h g ( x ) i be an η -constacyclic code of length n over F q with defining setT . Then C contains its Hermitian dual code, namely, C ⊥ H ⊆ C , if and only if T ∩ T − q = ∅ ,where T − q = {− q j mod nr | j ∈ T } . Chen et.al in [8] have shown that if an η -constacyclic BCH code over F q is a Hermitiandual-containing code, then we have ord( η ) | ( q + n = q m − q + , where m ≥ q is a prime power. We take η = α q − asin [24], where α is a primitive element of F q . Thus, r = ord( η ) = q + nr = q m −
1, and T − q = {− q j mod q m − | j ∈ T } . Xubo Zhao , , et al. In this section, we first determine the corresponding parameters for which C ⊥ H ⊆ C , andthen derive the dimensions and bound the minimum distances of C , thereby we can constructrelated quantum codes from constacyclic BCH codes C .3.1 Hermitian dual-containing conditions and the dimension of narrow-sense constacyclicBCH codes of length n = q m − q + Yuan et.al in [24] gave a su ffi cient condition on the design distances for which a narrow-sense constacyclic BCH code is Hermitian dual-containing code. In the following Theorem,we can derive a necessary and su ffi cient condition on the design distance for which a narrow-sense constacyclic BCH code is Hermitian dual-containing code. The significance of thisresult is that it allows us to identify all constacyclic BCH codes that can be used for constructquantum codes. Theorem 3
Let n = q m − q + , where q is a prime power, and m ≥ is an even. Denote η ∈ F ∗ q ,and ord ( η ) = r = q + . Let C = h g ( x ) i be a narrow-sense constacyclic BCH code withdefining set T = S δ − i = C + ri , where C + ri , ≤ i ≤ δ − , are q -cyclotomic cosets modulo nr,and δ ∈ Z . Then C ⊥ H ⊆ C , if and only if δ is in the range of ≤ δ ≤ δ max , (6) where δ max = q − q + q + q + + , m = and q ≥ , q m + − q + q + q + , otherwise . (7) Proof
Firstly, we prove the su ffi ciency. In terms of Lemma1, C ⊥ H ⊆ C if and only if T ∩ T − q = ∅ . Assume that T ∩ T − q , ∅ , then there exist two integers i , j , 0 ≤ i , j ≤ δ max − + r j ≡ − q (1 + ri ) q l (mod nr ) , (8)or equivalently, (1 + r j ) q m − l ) ≡ − q (1 + ri ) (mod nr ) , (9)where l ∈ { , , · · · , m − } . We note that m ≥ m ≤ l ≤ m −
1, then0 ≤ m − l − ≤ m −
1. Thus we just need to consider the following equation,(1 + ri ) q l + + + r j ≡ nr ) , l ∈ { , , · · · , m − } . (10)If m > , or m = q ≤
4, then 1 ≤ + r j ≤ q m + − q − q + < q m + − q ≤ (1 + ri ) q l + ≤ q m − ( q m + − q − q + ≤ q m − q m + (0 ≤ l ≤ m − m = q ≥ ≤ + r j ≤ q − q + q ≤ (1 + ri ) q l + ≤ q ( q − q + = q − q + q ( l = + q ≤ (1 + ri ) q l + + + r j < q m − = nr , hence(1 + ri ) q l + + + r j . nr ). This contradicts to the assumption T ∩ T − q , ∅ . So weconclude that if 0 ≤ i , j ≤ δ max −
2, then T ∩ T − q = ∅ , namely, C ⊥ H ⊆ C . ermitian dual-containing constacyclic BCH codes and related quantum codes of length q m − q + Now we will show the necessity, namely, if δ exceeds δ max , then C ⊥ H * C . If m > , or m = q ≤
4, assume that δ = δ max +
1, i.e., δ − = q m + − q + q + , then we have − q [1 + ( q + q m + − q + q + q m − ≡ q m + − q m − − ≡ + ( q + i (mod nr ) , (11)where i = q m + − q m − − q + . If m = q ≥
5, assume that δ = δ max +
1, i.e., δ − = q − q + q + q + ,then we have − q [1 + ( q + q − q + q + q + ≡ q − q − q − ≡ + ( q + i (mod nr ) , (12)where i = q − q − q − q + . Note that for both cases, 0 ≤ i ≤ δ −
2, so 1 + ri ∈ T ∩ T − q . Thus, T ∩ T − q is not empty, so C ⊥ H * C . The desired result follows. ⊓⊔ Some counting properties about q -cyclotomic cosets C + ri are provided in the followinglemma, which is essential to our subsequent arguments. Lemma 2
We keep the notation of Theorem 3. The cardinalities of q -cyclotomic cosetsC + ri modulo nr, with i in the range ≤ i ≤ q m + − q + , can be computed as follows, | C + ri | = m / , q > is even , and i = i ∗ , m , other w ise , (13) where i ∗ = ⌈ q + ⌉ ( q m + − [ q e v en ] q + .Proof Note that ord nr ( q ) = m , so we have | C + ri | (cid:12)(cid:12)(cid:12) m , 0 ≤ i ≤ q m + − q + . Suppose that for some i , | C + ri | < m , where 0 ≤ i ≤ q m + − q + . Then there exists a positive integer l , 1 ≤ l < m , suchthat (1 + ri ) q l ≡ + ri (mod nr ), namely,(1 + ri )( q l − ≡ nr ) . (14)Since m ≥ | C + ri | ≤ m , and 1 ≤ l ≤ m in Eq.(14). We first consider the caseof 1 ≤ l ≤ m − m ≥ ≤ + ri ≤ + ( q + q m + − q + = q m + − + < q m + , q − ≤ q l − ≤ q m − −
1, we have q − ≤ (1 + ri )( q l − < q m + ( q m − − < q m − < nr = q m −
1. Hence (1 + ri )( q l − . nr ), contradicting the assumption | C + ri | < m . And then we consider the case of l = m ( m ≥ + ri )( q m − ≡ nr ), which is equivalent to( q + i ≡ − q m + . (15)Note that gcd( q + , q m + = gcd( q + , q − q + , q m + = q is even, andgcd( q + , q m + = q is odd. In terms of [31] Proposition 3.2.7, when q is odd, Eq.(15)has no solution, when q is even, Eq.(15) has exactly one solution, and the unique solutioncan be given by i = i ∗ , ⌈ q + ⌉ ( q m + − [ q e v en ] q + . (16) Xubo Zhao , , et al. It is easy to see that i ∗ < q m + − q + for even q , q >
2, and i ∗ > q m + − q + for q =
2. Thus, the proofis complete. ⊓⊔ Remark 1.
When i in the range 0 ≤ i ≤ δ max −
2, then for even q , if q =
2, or q = m =
2, we have i ∗ > δ max −
2. Therefore, if 0 ≤ i ≤ δ max −
2, then | C + ri ∗ | = m /
2, only if q = , m >
2, or q > S δ − i = C + ri , 2 ≤ δ ≤ δ max , which con-tributes to computing the precise dimensions of narrow-sense constacyclic BCH codes withdefining set T = S δ − i = C + ri . To compute | S δ − i = C + ri | , we entail the following tasks:(1). count every cardinality of q -cyclotomic coset C + ri , 0 ≤ i ≤ δ −
2, which has done inLemma 2,(2). find out all disjoint or joint q -cyclotomic cosets in S δ − i = C + ri .Generally, compared with the case of odd m , task (2) is more challenging in the case of even m , especially q is also even, since the disjoint or joint q -cyclotomic cosets are di ffi cult todetermine, and the sizes of q -cyclotomic cosets are not always the same. Di ff erent fromthe idea of determining all coset leaders of the union of cyclotomic cosets in the definingset, we counting the joint cyclotomic cosets via solving the associated congruent equations.Consequently, | S δ − i = C + ri | is a function of n , q , and δ . Lemma 3
We keep the notation of Theorem 3, then | δ − [ i = C + ri | , N ( q , m , δ ) (17) = ( δ − N − m , ≤ δ < i ∗ + , ( δ − N − − [ q ≥ e v en ]) m , i ∗ + ≤ δ < i ∗ + + q m − q + , ( δ − N − − [ q ≥ e v en ] − N ) m , i ∗ + + q m − q + ≤ δ ≤ δ max , (18) where N = ⌊ δ − ( q + q ⌋ + , andN = ⌊ ( q + δ − q m − −⌈ q + ⌉⌋ , i ∗ + + q m − q + ≤ δ ≤ δ max , q ≥ , , other w ise . (19) Proof
Assume that C + ri = C + r ¯ i , where i , ¯ i ∈ Z , and 0 ≤ i < ¯ i ≤ δ max −
2. Then there existsan integer l , 1 ≤ l ≤ m −
1, such that1 + r ¯ i ≡ (1 + ri ) q l (mod q m − . (20)Since gcd( q m , q m − =
1, Eq.(20) is equivalent to(1 + r ¯ i ) q m − l ) ≡ + ri (mod q m − . (21)Thus, if m ≥ + r ¯ i ≡ (1 + ri ) q l (mod q m − , ≤ l ≤ m / , (22)and (1 + r ¯ i ) q l ′ ≡ + ri (mod q m − , ≤ l ′ ≤ m / − . (23)Note that for m =
2, Eq.(20) can be reduced to a special case of Eq.(22), namely, thevalue of l takes m / m ≥
4, for Eq.(23), we have 1 + ( q + ≤ ermitian dual-containing constacyclic BCH codes and related quantum codes of length q m − q + + r ¯ i ≤ + r ( δ max − < q m + −
1, and q ( q + ≤ (1 + r ¯ i ) q l ′ < q m −
1. Thus, by Eq.(23), onecan deduce (1 + r ¯ i ) q l ′ = + ri for 1 ≤ l ′ ≤ m / −
1, but this contradicts to the assumption i < ¯ i . Therefore, for Eq.(20), we just need to consider the formulas in Eq.(22).For Eq.(22), if 1 ≤ l ≤ m / − m ≥ ≤ + ri ≤ + r ( δ max − < q m + − q ,and q ≤ (1 + ri ) q l < ( q m + − q ) q m − = q m − − q m < q m −
1. Thus, in the case of1 ≤ l ≤ m / − m ≥ + r ¯ i = (1 + ri ) q l , (24)namely, ¯ i = q l − q + + iq l = ( q − q l − + q l − + · · · + q + + iq l , (25)which implies ¯ i ≡ q − q ) . (26)For the case of l = m /
2, Eq.(22) becomes1 + r ¯ i ≡ (1 + ri ) q m (mod q m − , (27)or equivalently, r ¯ i ≡ ri + ( q m − + ri ) (mod q m − , (28) ⇐⇒ ¯ i ≡ i + q m − q + + ri ) (mod q m − q + . (29)Let i = q m − q + w + θ, (30)where w ∈ Z , w ≥
0, and 0 ≤ θ < q m − q + . Then substituting Eq.(30) into Eq.(29), we have¯ i ≡ q m θ + (1 − w ) q m − q + t ) , (31)where t = q m − q + . Since 0 ≤ i < ¯ i ≤ δ max −
2, there exists a positive integer w , such that q m − q + w ≤ δ max − < q m − q + ( w + ≤ w ≤ w ≤ q −
1. We now consider threecases w = w =
1, and 1 < w ≤ q − q ≥ w =
0, we have i = θ , ¯ i ≡ q m θ + q m − q + (mod t ), where 0 ≤ θ < q m − q + . It iseasy to get that < ¯ i = q m − q + ≤ δ max − < t , θ = , < δ max − < ¯ i = q m θ + q m − q + < t , < θ < q m − q + . (32)This case means that for Eq.(27), if 0 < i = θ < q m − q + , then δ max − < ¯ i < t . And if i = < ¯ i = q m − q + ≤ δ max −
2, which also implies ¯ i ≡ q − q ). , , et al. (2). For the case w =
1, we have i = q m − q + + θ , ¯ i ≡ q m θ (mod t ), where 0 ≤ θ < q m − q + . It iseasy to get that ¯ i = , θ = , < δ max − < ¯ i = q m θ < t , < θ < q m − q + . (33)This case implies that for Eq.(27), if i = q m − q + , then ¯ i = < i , and if i = q m − q + + θ , 0 ≤ θ < q m − q + ,then δ max − < ¯ i < t .(3). For the case 1 < w ≤ q − q ≥ i = q m − q + w + θ , where 0 ≤ θ < q m − q + . And ¯ i ≡ t − ( w − q m − q + (mod t ) , θ = , ¯ i ≡ q m θ − ( w − q m − q + (mod t ) , < θ < q m − q + . (34)It is easy to get that δ max − < t − ( w − q m − q + ≤ ¯ i = t − ( w − q m − q + ≤ t − q m − q + < t , θ = ,δ max − < q m − ( w − q m − q + ≤ ¯ i = q m θ − ( w − q m − q + < q m ( q m − q + − q m − q + < t , ≤ θ < q m − q + . (35)This case implies that for Eq.(27), if i = q m − q + w + θ , where 1 < w ≤ q − q ≥ ≤ θ < q m − q + and θ ,
1, then δ max − < ¯ i < t . For θ =
1, we have i = q m − q + w +
1, and ¯ i = q m − ( w − q m − q + ,where 1 < w ≤ q − q ≥ i , ¯ i satisfying C + ri = C + r ¯ i should hold¯ i ≡ q − q ) , (36)or i = q m − q + w + , and ¯ i = q m − ( w − q m − q + , (37)where 0 ≤ i < ¯ i ≤ δ − ≤ δ max − , < w ≤ q − q ≥ i ≡ q − q ), namely, ¯ i = q − + kq , where k ∈ Z , k ≥
0, and¯ i ≤ δ max − + r ¯ i ≡ q (1 + rk ) (mod q m − i = k , it is easy toget that 0 ≤ i < ¯ i , and C + ri = C + r ¯ i . If ¯ i = q m − ( w − q m − q + , i = q m − q + w +
1, where0 ≤ i < ¯ i ≤ δ − ≤ δ max − , < w ≤ q − q ≥ + r ¯ i ≡ q m (1 + ri ) (mod q m − C + ri = C + r ¯ i . Therefore, C + ri = C + r ¯ i , 0 ≤ i < ¯ i ≤ δ max −
2, if and only if Eq.(36)and Eq.(37) hold.For Eq.(36), the number of the eligible ¯ i , denoted by N , can be easily calculated by N = ⌊ δ − − ( q − q ⌋ + = ⌊ δ − ( q + q ⌋ +
1. Let N be the the number of the eligible ¯ i in Eq.(37). Interms of Eq.(37), we have q m − q + w + < q m − ( w − q m − q + , q m − ( w − q m − q + ≤ δ − , ≤ w ≤ q − , ≤ δ ≤ δ max , (38)namely, q + + ( q + − δ ) q m − ≤ w ≤ ⌊ q + ⌋ , ≤ δ ≤ δ max . (39) ermitian dual-containing constacyclic BCH codes and related quantum codes of length q m − q + It is not di ffi cult to derive that by Eq.(39), w has integer solutions if and only if δ ≥ + ( q m − ⌈ q + ⌉ q + = i ∗ + + q m − q + , and q ≥
5. The number of the solutions of w in Eq.(39) can becalculated by ⌊⌊ q + ⌋ − ( q + − ( q + − δ ) q m − ⌋ + = ⌊ ( q + δ − q m − − ⌈ q + ⌉⌋ + = ⌊ ( q + δ − q m − − ⌈ q + ⌉⌋ .Therefore, N can be given by N = ⌊ ( q + δ − q m − − ⌈ q + ⌉⌋ , i ∗ + + q m − q + ≤ δ ≤ δ max , q ≥ , , other w ise . (40)Combined with Lemma 2 and Remark 1, we obtain the desired conclusions. ⊓⊔ By lemma 3, we can easily provide values of N ( q , m , δ ) with specific q and m as follows. Corollary 1
Let the symbols be defined as above. Then we have1. N ( q ≤ , m = , δ ) = δ − N − , where ≤ δ ≤ δ max = q − q + q + q + .2. N ( q ≤ , m > , δ ) = ( δ − N − m , ≤ δ < i ∗ + , ( δ − N − − [ q = m , i ∗ + ≤ δ ≤ δ max = q m + − q + q + q + . N ( q ≥ odd , m , δ ) = ( δ − N − m , ≤ δ < i ∗ + + q m − q + , ( δ − N − N − m , i ∗ + + q m − q + ≤ δ ≤ δ max . N ( q ≥ e v en , m , δ ) = ( δ − N − m , ≤ δ < i ∗ + , ( δ − N − ) m , i ∗ + ≤ δ < i ∗ + + q m − q + , ( δ − N − N − ) m , i ∗ + + q m − q + ≤ δ ≤ δ max . Proof
Applying Lemma 3, the result follows. ⊓⊔ Theorem 4
Let n = q m − q + , where q is a prime power, and m ≥ is even. Denote η ∈ F ∗ q ,and ord ( η ) = r = q + . Let C be a narrow-sense constacyclic BCH code with defining setT = S δ − i = C + ri , where ≤ δ ≤ δ max . Then C is an [ n , n − N ( q , m , δ ) , ≥ δ ] q Hermitiandual-containing constacyclic BCH code, where N ( q , m , δ ) is given as in Lemma 3.Proof Applying Theorem 2 and Lemma 3, the result follows. ⊓⊔ Remark 2 . We call a linear code with parameters [ n , k , d ] q optimal (almost optimal), ifthe code (the code with parameters [ n , k , d + q ) reaches some bound. Notice that some ofconstacyclic BCH codes constructed in Theorem 4 are optimal or almost optimal, they reachor almost reach the lower (upper) bound maintained by Markus Grassl [16]. For example,by the Database [16], the constacyclic BCH codes [5 , , ≥ , [20 , , ≥ , [20 , , ≥ , [20 , , ≥ , [85 , , ≥ are optimal if the equalities of their minimum distanceparameters are achieved. And the constacyclic BCH codes [20 , , ≥ , [85 , , ≥ ,[85 , , ≥ , [85 , , ≥ , [85 , , ≥ are almost optimal if the equalities of theirminimum distance parameters are achieved, otherwise, they are optimal.Now we construct quantum codes. , , et al. Theorem 5
Let n = q m − q + , where q is a prime power, and m ≥ is even. Then there exists aquantum code with parameters [[ n , n − N ( q , m , δ ) , ≥ δ ]] q , where N ( q , m , δ ) is given as inLemma 3.Proof Applying the Hermitian construction to the constacyclic code C in Theorem 4, weimmediately obtain a q -ary quantum code with parameters [[ n , n − N ( q , m , δ ) , ≥ δ ]] q asdescribed in Theorem 5. ⊓⊔ Yuan et al. [24] constructed many quantum codes from a class of narrow-sense Hermi-tian dual containing constacyclic BCH codes of length n = q m − q + . Aly et al. in [2] yieldedquantum codes of general lengths by narrow-sense Hermitian dual containing BCH codes.Yves Edel provided code tables of some good quantum twisted codes for q ≤ n = ( q − q + b , where b | q −
1. Fixingthe code lengths n = q m − q + ( m ≥ Lemma 4 ( [24], Theorem 3) Let n = q m − q + , where q is a prime power, and m ≥ is aneven. Let α = + ( q + j α ∈ { + ( q + i | ≤ i ≤ δ Y − } , where δ Y = ⌊ qq + ( q m − q + ⌋ + .Let λ α = α + qq + − ⌊ α + q q + q ⌋ , then there exists a quantum code with parameters [[ n , n − λ α m , ≥ j α + q , q is odd , or q = , [[ n , n − λ α m , ≥ j α + q , q = s with s ≥ , α < (1 + s − )(1 + sm ) , [[ n , n − λ α m + m , ≥ j α + q , q = , m > , or q = s with s ≥ , α ≥ (1 + s − )(1 + sm ) . (41) Lemma 5 ( [2], Theorem 21) Let n = q m − q + , where q is a prime power, m = ord n ( q ) ≥ even, and ≤ δ ≤ δ A = q m − q + , then there exists a quantum code with parameters [[ n , n − m ⌈ ( δ − − q − ) ⌉ , ≥ δ ]] q . Lemma 6 ( [25], Theorems 3 and 4) Let n = q − q + , where q ≥ is an odd prime power. For ≤ δ ≤ δ L = q − q + , denote | T ( δ ) | = ⌈ ( δ − − q − ) ⌉ + . Then there exists aquantum code with parameters [[ n , n − | T ( δ ) | , ≥ δ ]] q . Lemma 7 ( [25], Theorem 5) Let n = q − q + , where q ≥ is an even prime power. For ≤ δ ≤ δ L = q − q + , denote | T ( δ ) | = ⌈ ( δ − − q − ) ⌉ + , ≤ δ ≤ q − q + , δ − − q − ) + , q − q + ≤ δ ≤ q − q + , (42) Then there exists a quantum code with parameters [[ n , n − | T ( δ ) | , ≥ δ ]] q . By comparison, it shows that for n = q m − q + ( m is even), quantum codes constructed byTheorem 5 have the same parameters with those in [24] with q = , ,
4. Whereas, with thesame lengths, if q ≥
5, for all design distance δ in the range i ∗ + + q m − q + ≤ δ ≤ δ max ,our quantum codes can always yield strict dimension or design distance gains than the onespresented in [24]. Tables 1, 2 provide some code comparisons between them. From thesetables, we can see that our quantum codes have better performance. Moreover, for the max-imum design distance δ max in this paper, δ A in [2], δ Y in [24], and δ L in [25], it is easy to ermitian dual-containing constacyclic BCH codes and related quantum codes of length q m − q + infer that δ max > ( q − δ A , δ max = δ Y or δ Y +
1, and δ max = δ L or δ L +
1. It means that forlength n = q m − q + ( m is even), the Hermitian dual containing constacyclic BCH codes studiedin this paper have relatively large design distance. We observe that quantum codes obtainedby Theorem 5 can yield better parameters than those known constructions of quantum BCHcodes or good quantum twisted codes [2, 17]. We also notice that non-narrow sense consta-cyclic BCH codes or BCH codes investigated in [25] can construct abundant quantum codeswith good parameters. Nonetheless, with the same length, parameters of quantum codesconstructed by non-narrow sense codes are not always superior to the narrow-sense ones,some of quantum codes constructed from narrow-sense constacyclic BCH codes in this pa-per have better performance than the ones in [25]. Since Yves Edel’s table [17] includesgood quantum twisted codes with lengths up to 1000, we just compare our quantum codesin Theorem 5 with those obtained in Refs. [2, 17, 25] in the case of n < − ” means that thereis no quantum code with given length or design distance. Remark 3 . After we finish this paper, we find that Wang et al. in [32] deal with a familyof narrow-sense constacyclic BCH codes of length q m − ρ ( ρ is a positive divisor of ( q + m ≥ n = q m − q + to more general case. Compared with thecodes of length n = q m − q + in [32], our results have the following merits. (1). For even m ,we enlarge the range of m . Wang et al. in [32] discuss the cases m ≥
4, we include the case m =
2. So we can provide many short lengths Hermitian dual containing constacyclic BCHcodes and related quantum codes, which will be interesting from a practical point of view.(2). We simplify the dimension expressions of constacyclic BCH codes or related quantumcodes. In [32], for given design distance δ , one should first get the q -adic expansion of1 + ( q + δ −
2) by Σ mi = δ i q i , and then discuss the values of δ i (0 ≤ i ≤ m ), thereby δ isthe function of q , m , δ i (0 ≤ i ≤ m ), and the dimension of the Hermitian dual-containingconstacyclic BCH code or related quantum code is the function of q , m , δ, δ , and δ m (seeLemma 5 and Theorem 4 in [32]). While by our Theorems 4 and 5, the dimension of theHermitian dual-containing constacyclic BCH codes is a function of their design parameters,namely, n , q , and the given design distance δ , as well as the related quantum codes. Quantumcodes of length n = q m − q + also have been studied in [18] using non-narrow-sense BCH codes,and many quantum codes in their paper have parameters better than our constructions fromnarrow-sense constacyclic BCH codes. However, we relax the values of even m from m ≥ m ≥
2, and remove the condition q ≡ m ). Thus, the obtained constacyclic BCHcodes or resultant quantum codes in this paper are beneficial complements for the know oneswith length n = q m − q + . In this paper, the maximum design distance and properties of the corresponding cyclotomiccosets for a family of narrow-sense Hermitian dual-containing constacyclic BCH codes C with length q m − q + ( m ≥ C with design distance δ in the rang 2 ≤ δ ≤ δ max was completely determined. Conse-quently, applying Hermitian construction to these underlying constacyclic BCH codes, manyquantum codes with desirable parameters were constructed. , , et al. Table 1
Quantum codes(QC) of length n = q m − q + with m = q = , , , m , q , δ QC in Theorem 5 QC in [24] QC in Theorem 5 QC in [24] m = , q = · · · δ =
19 [[104 , , ≥ [[104 , , ≥ m = , q = · · · ≤ δ ≤
39 [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ [[300 , , ≥ m = , q = · · · ≤ δ ≤
52 [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ [[455 , , ≥ m = , q = · · · ≤ δ ≤
67 [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ [[656 , , ≥ · · · · · · · · · · · · · · · Table 2
Quantum codes(QC) of length n = q m − q + with m = q = , m , q , δ QC in Theorem 5 QC in [24] m = , q = , , ≥ [[65104 , , ≥ ≤ δ ≤
518 [[65104 , , ≥ [[65104 , , ≥ · · · · · · [[65104 , , ≥ [[65104 , , ≥ m = , q = , , ≥ [[720600 , , ≥ ≤ δ ≤ , , ≥ [[720600 , , ≥ · · · · · · [[720600 , , ≥ [[720600 , , ≥ · · · · · · · · · Acknowledgements
This work is supported by the National Natural Science Foundation of China un-der Grant No.61902429, No.11775306, the Shandong Provincial Natural Science Foundation of China un-der Grants No.ZR2019MF070, the Key Laboratory of Applied Mathematics of Fujian Province University(Putian University) under Grants No.SX201806, the Open Research Fund from Shandong provincial KeyLaboratory of Computer Network under Grant No.SDKLCN-2018-02, Fundamental Research Funds for theCentral Universities No.17CX02030A.
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