1 Introduction
In [4], Galindo et al. give two constructions of quantum errorcorrecting codes defined from Cartesian product point sets, and the resulting codes have good parameters. In their work, they consider asymmetric quantum codes, meaning that phaseshift and bitflip errors are not treated equally and both of the corresponding minimal distances are of interest. Another viewpoint commonly seen in the literature is that no distinction should be made between the two types of errors when assessing the errorcorrecting capabilities of a quantum errorcorrecting code. In this setting, only one distance is associated to the quantum code, and the code is called symmetric. Clearly, the parameters in the asymmetric setting can be translated into the symmetric setting by ignoring the highest distance. Thus, someone interested in symmetric codes could use the results from [4], but discarding the highest distance essentially wastes coding space which could instead be used to increase the dimensions of the codes. In this work, we take an alternative approach and apply Steaneenlargement to that family of codes in order to produce symmetric codes directly. We thereby produce quantum errorcorrecting codes with good – sometimes even optimal – parameters.
The classical codes considered in this work are special cases of what is called monomial Cartesian codes in a recent work [13]. In that paper, the authors derive a way to determine if a monomial Cartesian code is selforthogonal, and use this to construct quantum codes via the CSSconstruction. The classical codes used in their construction are, however, different from the ones used in the current paper. In particular, the improved codes considered in this work have the best possible dimension given any designed distance.
2 Preliminaries
In this section, we recall two results on the CSSconstruction and Steaneenlargement that allow construction of quantum codes from classical codes. Then we give a description of a family of codes and the corresponding improved codes, both of which were previously considered in [4]. In our analysis, we will rely on the notion of relative distances of nested pairs of classical linear codes. Thus, recall that for codes their relative distance is defined as
where denotes the usual Hamming weight. In general, however, the relative distance is difficult to determine, and the bound is commonly used instead.
2.1 The CSSconstruction and Steaneenlargement
One way to construct quantum errorcorrecting codes is by using the socalled CSSconstruction [1, 16] named after Calderbank, Shor, and Steane. The original construction uses a selforthogonal classical linear code to construct a symmetric quantum errorcorrecting code.
Theorem 1.
If the linear code contains its Euclidean dual, then an
symmetric quantum code exists.
Steane [17] proposed a variation on this procedure which in some cases allows an increase in dimension compared to the corresponding CSScode but without reducing the minimal distance. Below, we state the ary generalization of this procedure, which may be found in [9, 12].
Theorem 2.
Consider a linear code that contains its Euclidean dual . If is an code such that and , then an
quantum code exists with and .
Here we note that if and are codes that satisfy the conditions of Theorem 2, then the inclusions hold, which implies . In particular, this means that whenever , it must be the case that . For the specific enlargements considered in Section 3, it turns out that this observation allows us to use the usual minimal distances rather than the relative distances while still obtaining the same parameters of the quantum codes.
2.2 Codes from Cartesian product point sets
Let where is a prime number, and let be positive integers such that . Then we have the inclusions , and it is possible to consider the Cartesian product . Now, define the polynomials
and consider the ring where
is the vanishing ideal of the ’s. Letting and
, we obtain a vector space homomorphism
given byas described in [4]. Adopting a vectorized version of their notation, we define for the set
where we use the multiindex notation . For a subset , define the code
(1) 
which clearly has length . To describe the distance of , we use the map given by
Proposition 3.
Proof.
The claim about the dimension is for instance shown in the proof of [4, Thm. 16]. The inequality (2) can be proved by using the footprint bound as done in [6, Prop. 1].
To see the equality, write , let , and observe that the expansion of the polynomial
contains only monomials with as described in the proposition, meaning that . Moreover, it possesses exactly nonzeros. ∎
This proposition not only allows us to determine the exact minimal distance of the codes considered in the following section, but more importantly it also enables us to determine certain relative distances when combined with the observations below Theorem 2.
2.3 Improved codes
The information on the minimal distance provided by leads to improved code constructions in a straightforward manner. By defining
(3) 
the code has designed distance by Proposition 3. In addition, this is the true minimal distance since if . The dual of can be described by studying the map defined as
In particular, by letting we obtain the following result.
Proposition 4.
Let be defined as in (3). Then .
Proof.
First, note that for . This implies that the number of monomials with a given value is exactly the number of monomials with value . As a consequence,
Hence, it suffices to show that , and we do so by proving that the evaluation of any with must be in .
Using contraposition, assume that . Then some satisfies . As shown in [5, Prop. 1], this happens if and only if^{1}^{1}1In their notation, the situation in consideration has and for each and holds true for each index . In other words, we have or . In each case, this implies . In combination with the fact that since , we obtain the inequalities
In conclusion, if , we have which proves the proposition by the observations in the beginning of the proof. ∎
3 Steaneenlargement of improved codes
We are now ready to apply Steaneenlargement to the codes defined in Section 2.3. Our results rely on a simple, but crucial, observation: for each index , contains an ‘edge’ with values . This is illustrated in Figures 1 and 2. This means that we can easily give a lower bound on the dimension increase when enlarging the code . To ease the notation in the following, we will order the exponents such that .
Proposition 5.
Let , and let be a vector such that for each and . Additionally, let , and let be the largest index such that . Then if is a selforthogonal code, there exists a quantum errorcorrecting code with parameters
(4) 
Proof.
Write , and let . Since , the observation at the start of this section implies that there are at least monomials such that . Thus, has dimension . As described in Section 2.3, and have minimal distances and , respectively. Thus the observation below Theorem 2 ensures that , and we obtain
where the last equality stems from the assumption that . The claim now follows by applying Theorem 2 to and , and by using the bound . ∎
A few additional remarks can be made about the Steaneenlargement described in Proposition 3. First of all, the observation that leads to Proposition 3 does not help in the case since we require . This does not mean that Steaneenlargement is impossible for , but merely that we cannot guarantee that enlargement is possible.
Secondly, the increase in dimension when applying Steaneenlargement to the code may be greater than the specified in Proposition 5 since this is determined by considering monomials along the ‘edges’ as in Figure 2. There may be several other monomials that have value , yielding a quantum errorcorrecting code with even better parameters. In Section 3.1, we characterize the situations where this may happen, and give an improved bound in such cases. First, however, we illustrate the result through an example.
Example 1.
Let and . The classical code has parameters , whence the CSSconstruction, Theorem 1, gives a quantum code. Since , Proposition 5 ensures that Steaneenlargement will instead provide a quantum code with parameters . In this case, the true dimension is in fact .
Using the same and , the code is a classical code, yielding a quantum code via the CSSconstruction. This time, Proposition 5 only guarantees a dimension increase of when applying Steaneenlargement, but the true parameters of the enlarged code are , meaning that the dimension has been increased by .
3.1 Determining the exact dimension increase
As previously mentioned, the dimension of an enlarged code may be greater than predicted in (4). In this section, we will generalize the map from [2] to provide an algorithm for computing the exact dimension increase when applying Steaneenlargement to the code . This generalization will also aid in characterizing those values of where Proposition 5 underestimates the dimension.
Definition 6.
For and , we let denote the number of tuples such that for every , and such that .
Proposition 7.
Let and be as in Definition 6, and assume that . Let be the largest index such that . Then if is…

…prime, we have

…square, we have

…nonprime and nonsquare, we have
Proof.
Assume first that is prime. Then any tuple with must have for some and for . Hence, in this case is the number of indices such that , which is exactly .
If is nonprime, there are still tuples with a single entry greater than as in the prime case. But we may also split in two factors such that . Now, for any distinct indices , the tuple with , , and for is one of the tuples counted by . The number of ways to choose the indices is . If is not a square number, and are distinct, and each of the choices of leads to a distinct tuple. Is is a square, we may have , and the number of distinct tuples is instead . In both cases, we obtain the claimed inequality by adding . ∎
Proposition 8.
Let . Then the number of monomials that have is .
Proof.
We have if and only if . Since , this is equivalent to for , proving the proposition. ∎
Corollary 9.
Example 2.
Since it may not be obvious how to compute , we give the following recursive algorithm. Its correctness can be shown by a simple inductive argument.
Algorithm 1.
On input and , this algorithm computes :

Check if is a single value . If this is the case, return if , and otherwise.

Initialize a counter variable

For each integer with , do the following:

Let and compute .

Update to be


Return
3.2 Examples of parameters
To conclude our exposition, we give concrete parameters of Steaneenlarged codes in several examples. For each code presented here, we will compare it to the GilbertVarshamov bound from [3].
Theorem 11.
Let with , and let . Then there exists a pure stabilizer quantum code if the inequality
(5) 
is satisfied.
In the same way as [14], we will use the notation in the following to indicate that the parameters exceed the GilbertVarshamov bound – i.e. that (5) is not satisfied – and we will write if satisfies (5), but does not. This is only possible for , which is always the case for CSScodes from selforthogonal codes, but not necessarily for Steaneenlarged codes. Thus, for code parameters with , we will use the same notation, albeit with the bound applied to the parameters . There is another bound, [10, Cor. 4.3], which covers all values of and . For the parameters presented in the current work, however, that bound is weaker than (5), and several of the codes in the examples below exceed [10, Cor. 4.3] but not Theorem 11. For this reason, we shall use Theorem 11 throughout.
In addition to the GilbertVarshamov bound, we will refer to the quantum Singleton bound in some cases. This bound is
(6) 
Example 3.
Example 4.
Construction  Dimension increase  

Thm. 1  Thm. 2  Prop. 5  Cor. 9  Prop. 10 
Construction  Dimension increase  

Thm. 1  Thm. 2  Prop. 5  Cor. 9  Prop. 10 
Example 5.
In Tables 1–4, we list parameters of quantum codes in various cases where Proposition 5 guarantees that enlargement is possible. The tables contain both the original CSScode and its Steaneenlarged code along with the predicted dimension increases from Proposition 5 and Corollary 9.
In these tables, the first column shows the parameters of quantum codes obtained by applying Theorem 1 to selforthogonal codes of the form . The second column shows the results of enlarging the codes in the first column using in Theorem 2. The third column gives the dimension increase guaranteed by Proposition 5, and the fourth shows the bound provided by Corollary 9. Any number marked with an asterisk is known to be the true value since is a prime. The final column shows the actual increase as computed by Algorithm 1.
The codes in Table 1 have better parameters than those in [7, Tables 1 and 2] for small values of . More concretely, [7] lists codes with parameters , , , and . For larger values of , however, [7] outperforms the codes in Table 1. Likewise, for the parameters of the codes in Table 2 surpass those presented in [8, Tables 1 and 2]. There, codes with parameters and are given. As before, [8] also contains codes with higher minimal distances that have better parameters than the corresponding codes obtained in this work. All the codes in Tables 1 and 2 have with and , which are in fact special cases of hyperbolic codes. It seems to be a general pattern for such codes, that the Steaneenlargements with small distances outperform the codes in [7, 8], but that this relation is reversed for larger distances.
Additionally, the codes in Table 1 have favourable parameters compared to the codes from [14, Ex. 5] defined from the Suzuki curve. Specifically, the codes in [14] have parameters , , , , , and , which are all worse than those in Table 1 except the one with distance . As a final remark, the code with parameters meets the quantum Singleton bound (6).
The codes in Tables 3 and 4 have parameters that cannot be achieved using the method from [7, 8] since those codes all have lengths for some , where is the field size. Studying the tables, it is also evident that Corollary 9 provides a better bound for the dimension than Proposition 5, but that the actual increase in dimension may be significantly higher. In any case, however, Proposition 10 ensures that the true increase can be computed using Algorithm 1.
Among the codes presented here, two were MDScodes: and . From recent work [13, Cor. 3.10] the same lengths, dimensions, and minimal distances can be achieved, but the field size is much larger. In particular, they require so the corresponding field sizes are at least and , respectively.
Acknowledgements
The authors express their gratitude to Diego Ruano for delightful discussions in relation to this work.
References
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