From Skew-Cyclic Codes to Asymmetric Quantum Codes

We introduce an additive but not $\mathbb{F}_4$-linear map $S$ from $\mathbb{F}_4^{n}$ to $\mathbb{F}_4^{2n}$ and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]_4$-code, then $S(C)$ is an additive $(2n,2^{2k},2d)_4$-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)_4$-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.


1.
Introduction. The class of skew-cyclic codes was introduced in [2]. These linear codes have the property of being invariant under the operation of cyclic shift composed with overall conjugation. Demanding an ideal structure on the codes forces us, over F 4 , to work in even lengths only. By relaxing this structure to that of a module [3], it is now possible to deal with skew-cyclic codes of any lengths.
In the present work, a mapping S is introduced to map any skew-cyclic codes of length n over F 4 into codes of length 2n which are invariant under a coordinate permutation denoted by σ. The permutation σ is a cyclic permutation for n odd and a product of two cycles of equal length for n even.
Besides these structural properties, the mapping S has interesting duality properties and preserves nestedness. These allow us to construct asymmetric quantum codes following the method given in [6].
The material is organized as follows. In Section 2, we state some basic definitions and properties of linear and additive codes. More specifically, the two families, 4 H and 4 H + , of codes over F 4 are formally defined. Their respective dualities and weight enumerators are stated.
Section 3 introduces the mapping S and its basic properties. The definition of and some algebraic background on module θ-cyclic codes are discussed in Section 4. The study of the images of these codes under the mapping S is also given. A very brief introduction to asymmetric quantum codes follows in Section 5.
In Section 6, an analysis of the weight enumerators is performed. This is important in understanding the parameters of the asymmetric quantum codes that can be obtained under the mapping S. Two systematic constructions of asymmetric quantum codes are given in Section 7. The one based on best-known linear codes is presented in Subsection 7.1 while the other one, based on concatenated Reed-Solomon codes, is given in Subsection 7.2. The last section contains conclusions and open problems.

2.
Preliminaries. Let p be a prime and q = p m for some positive integer m. An [n, k, d] q -linear code C of length n, dimension k, and minimum distance d is a subspace of dimension k of the vector space F n q over the finite field F q = GF (q) with q elements. For a general, not necessarily linear, code C, the notation (n, M = |C|, d) q is commonly used.
Let n be a positive integer and let 1 ≤ l < n be a divisor of n. A linear [n, k, d] qcode C is quasi-cyclic of index l or l-quasi-cyclic if In particular, a 1-quasi-cyclic code is a cyclic code.
As is the case for linear codes, we define the notions of an additive cyclic code and an additive quasi-cyclic code similarly by requiring the code to be additive, instead of linear.
The Hamming weight of a vector or a codeword v in a code C, denoted by wt H (v), is the number of its nonzero entries. Given two elements u, v ∈ C, the number of positions where their respective entries disagree, written as dist H (u, v), is called the Hamming distance of u and v. For any code C, the minimum distance If C is additive, then its additive closure property implies that d(C) is given by the minimum Hamming weight of the nonzero vectors in C.
Definition 2.1. Let F 4 := 0, 1, ω, ω 2 = ω . For x ∈ F 4 , set x = x 2 , the conjugate of x. Let n be a positive integer and u = (u 0 , u 1 , . . . , Definition 2.2. A code C of length n over F 4 is said to be an additive F 4 -code if C belongs to the family 4 H + . Let C be a code. Under a chosen inner product * , the dual code C ⊥ * of C is given by C ⊥ * := u ∈ F n q : u, v * = 0 for all v ∈ C . A code is said to be self-orthogonal if it is contained in its dual and is said to be self-dual if its dual is itself. We say that a family of codes is closed if (C ⊥ * ) ⊥ * = C for each C in that family. It has been established [14,Ch. 3] that both families of codes in Definition 2.1 are closed.
The weight distribution of a code and that of its dual are important in the studies of their properties.
where A i is the number of codewords of weight i in the code C.
The weight enumerator of the Hermitian dual code C ⊥ H of an [n, k, d] 4 -code C is connected to the weight enumerator of the code C via the MacWilliams Equation 3. The Mapping S on Codes over F 4 . Codes belonging to the family 4 H + have been studied primarily in connection to designs (e.g. [11]) and to stabilizer quantum codes (e.g. [10, Sec. 9.10]). It is well known that if C is an additive (n, 2 k ) 4 -code, then C ⊥tr is an additive (n, 2 2n−k ) 4 -code. Note that if the code C is F 4 -linear with parameters [n, k, d] 4 , then C ⊥ H = C ⊥tr . This is because C ⊥ H ⊆ C ⊥tr and C ⊥ H is of size 4 n−k = 2 2n−2k which is also the size of C ⊥tr .
Applying S yields By definition, S(C) is an additive 2-quasi-cyclic code. Proof. Let v = (v 0 , v 1 , . . . , v n−1 ), u = (u 0 , u 1 , . . . , u n−1 ) ∈ C. Then 4. Module θ-Cyclic Codes over F 4 . The motivation for our definition of module θ-cyclic codes comes from [2] and [3]. Given F q and an automorphism θ of F q , we can define a ring structure on the set In R, the addition operation is the usual polynomial addition and the multiplication is defined by the extension to all elements of R, by associativity and distributivity, the basic rule Xa = θ(a)X for all a ∈ F q .
The ring R is a left and right Euclidean ring whose left and right ideals are principal. Right division means that for nonzero f (X), g(X) ∈ R, there exist unique polynomials Q r (X), R r (X) ∈ R such that If the codewords of C are identified with the list of the coefficients of the remainder of a right division by f (X) in R, then the elements of Rg(X)/Rf (X) are all of the left multiples of g(X) = g r X r + . . . + g 1 X + g 0 .
Thus, a generator matrix G of the corresponding module θ-code of length n = deg(f (X)) is given by (4.1) depending only on g(X) and n.
Theorem 4.2. A module θ-cyclic code C θ has the following property Proof. The proof of this property for an ideal θ-cyclic code C is established in [2, Theorem 1]. The same proof works when we replace ideal by module.
Since a module θ-cyclic code C θ has a representation in the skew polynomial ring R = F q [X, θ] (see [3]), when θ is fixed, we call C θ a skew-cyclic code.
We consider, for the rest of the paper, the Frobenius automorphism defined in where T is the cyclic shift module 2n and τ = (12)(34) . . . (2n − 1, 2n). Since T 2 and τ commute, σ can be written as T 2 • τ as well. We denote the identity permutation by (1).

MARTIANUS FREDERIC EZERMAN, SAN LING, PATRICK SOLÉ AND OLFA YEMEN
Since C θ is a skew-cyclic code, we have Hence, Lemma 4.4. The order of σ is 2n if n is odd and n if n is even.
Proof. For 1 ≤ i ≤ 2n, σ follows the following rule With computation done modulo 2n, observe that if i is odd, then σ(i) = i + 3 and Hence, σ 2 = T 4 . Now, let n = 2l for some positive integer l. We have since σ k sends 1 to 4i + 1 = 1. Consequently, the order of σ is n.
In the case where n = 2l + 1, we have To show minimality, we first note that σ n = τ since Consider the following two subcases. For 1 ≤ k < n, the same argument as in the even case above shows that σ k = (1). For n + 1 ≤ k < 2n, We conclude that the order of σ is 2n.
Definition 5.1. Let d x and d z be positive integers. A quantum code Q in V n = C q n with dimension K ≥ 2 is called an asymmetric quantum code with parameters ((n, digits of X-errors and, at the same time, d z − 1 quantum digits of Z-errors.
The following result has been shown recently in [6].
As explained in [2] and in [3], there are two major gains of using module θ-codes. First, there is more flexibility and generality in constructing (linear) codes without increasing the complexity of the encoding and decoding process. The notion of qcyclic codes, introduced in [8], for instance, covers ideal θ-cyclic codes with θ limited to the Frobenius automorphism only.
More important to the agenda of constructing asymmetric quantum codes is the second gain, which is the minimum distance improvement. Exhaustive search on module θ-codes up to certain length has yielded linear codes with better parameters. More systematically, the BCH approach of constructing codes with a prescribed lower bound on the minimum distance can be extended to module θ-codes as well. Section 3 of [3] contains the construction details. The resulting improvements have been added to the database of best-known linear codes (BKLC) of MAGMA [1].
For the remaining of the paper, we will concentrate on constructing asymmetric quantum codes with d z ≥ d x = 2 based on Theorem 5.2. We will see how the mapping S can be used as an aid in construction. All computations are done in MAGMA V2. 16-5. 6. Analysis on the Weight Enumerators. In this section, the weight enumerators of S(C) and of S(C) ⊥tr are analyzed. This analysis will be useful in determining d x .
Let A i be the number of codewords of weight i in an additive (n, M, d) 4 -code C. Then the weight enumerators of S(C) and S(C) ⊥tr can be written in terms of the weight enumerator of C with the help of Equation (2.5) More explicitly, where L i is given by Denote the number of codewords of weight i in the code C ⊥tr by A ⊥tr i . By using the Pless power moments with q = 4 (see [10, p. 259] for the linear version), we have If we further assume that A ⊥tr 1 = A ⊥tr 2 = 0, then the following statements hold for Equation (6.2). 1 by Equations (6.6) and (6.7). If we rewrite  The parameters of the resulting code Q based on the construction in Proposition 6.1 are not so good. Fortunately, the mapping S preserves nestedness. This fact can be used to derive asymmetric quantum codes with better parameters.
Henceforth, any asymmetric quantum code Q satisfying k = n − d x − d z + 2 is printed in boldface. We call such a code an asymmetric quantum MDS code.   Table III].
The next section presents two systematic constructions of asymmetric quantum codes with d z ≥ d x = 2 by using the database of BKLC and by applying the mapping S on concatenated Reed-Solomon codes, respectively. 7. Two Constructions. Under the mapping S, Theorem 6.2 says that while we cannot improve on d x = 2, we can relax the condition on the inner code C to possibly improve on the size of Q as well as on d z . Our aim, then, is to choose the smallest possible subcode C of D such that d(C ⊥tr ) ≥ 2 while keeping the size and the minimum distance of D relatively large.
Note that there is no additive (n, 2, d) 4 -code with d(C ⊥tr ) ≥ 2. The smallest additive code with d(C ⊥tr ) = 2 is an (n, 4, n) 4 = [n, 1, n] 4 -code C consisting of the scalar multiples of a codeword v of weight n. Since this code C is MDS, its dual C ⊥tr = C ⊥ H is of parameters [n, n − 1, 2] 4 . 7.1. Construction from best-known linear codes (BKLC). Let n, k be fixed with 2 ≤ k ≤ n − 1. The strategy here is to consider the best-known linear code D of length n and dimension k stored in the MAGMA database and check if the code contains codewords of weight n and put them in a set R. If R is non-empty, we choose an arbitrary codeword v ∈ R and construct a subcode C ⊂ D of parameters [n, 1, n] 4 whose elements are the four scalar multiples of v.
Based on the codes C and D, two asymmetric quantum codes can be derived, one from Theorem 5.2 directly without the mapping S by letting C ⊥tr 1 = C and C 2 = D and another from Theorem 6.2 under the mapping S. We label the first quantum code Q while the second one Q S . Proof. A general proof for the existence of an [[n, n − 2, 2/2]] q -asymmetric quantum MDS code is already given in [17,Cor. 3.4]. Here we present a simple constructive proof for q = 4. A cyclic code D with parameters [n, n − 1, 2] 4 can be constructed by using X + 1 as its generator polynomial. Its minimum distance is two since the check polynomial is 1 + X + . . . + X n−1 . By [5, Th. 1], D has codewords of length n. One such codeword can be chosen to form an [n, 1, n] 4 -code C. Applying Theorem 5.2 with C ⊥tr 1 = C and C 2 = D brings us to the conclusion.
For a fixed n, it is not guaranteed that for all k ∈ {2, . . . , n − 2}, the best-known linear code with parameters [n, k, d] 4 has codewords of weight n. For example, there is no codeword of weight 6 in the best-known [6, 4, 2] 4 -code stored in the database of MAGMA that we use here. Table 1 lists down the resulting quantum codes for n = 4 to n = 20 based on the list of best-known linear codes with parameters [n, k] 4 invoked under the command BKLC in MAGMA. We exclude the case of k = n − 1 in light of Theorem 7.1 and the case of k = 1 due to [6,Ex. 8.2]. The process can of course be done for larger values of n if so desired. Interested readers may contact the first author for the complete list of codes Q and Q S with d z ≥ d x = 2 which are derived from the best-known linear codes for up to n = 46.
Remark 7.2. Aside from its nice structural property, the advantage of using the mapping S can be seen, for instance, from the fact that we have the [

7.2.
Construction from concatenated Reed-Solomon codes. Let m be a positive integer. Concatenation is used to obtain codes over F q from codes over an extension F q m of F q . A general method of performing concatenation is presented in [12,Sec. 6.3] and in [13,Ch. 10].
Our strategy here is to construct nested codes C ⊂ D over F 4 from nested codes A ⊂ B over F 4 m . We then use the codes C and D and the mapping S to get a quantum code Q.
Note that we can choose an F 4 -basis {β 1 . . . , β m } of F q such that a generator matrix of C ′ = φ * (A) is given by the m × mq matrix G = (I m |I m | . . . |I m ) where I m is the m × m identity matrix. Hence, C ′ is of parameters [mq, m, q] 4 . Define C to be the [mq, 1, mq] 4 -repetition code subset of C ′ . This is valid since we know that 1 = (1, . . . , 1) ∈ C ′ . The code D = φ * (B) is an [mq, mk, d ′ ≥ (q − k + 1)] 4 -code that contains C. Repeating the proof of Theorem 6.2 yields the following result.   Table 2.