From primary to dual affine variety codes over the Klein quartic

07/19/2021
by   Olav Geil, et al.
0

In [17] a novel method was established to estimate the minimum distance of primary affine variety codes and a thorough treatment of the Klein quartic led to the discovery of a family of primary codes with good parameters, the duals of which were originally treated in [23][Ex. 3.2, Ex. 4.1]. In the present work we translate the method from [17] into a method for also dealing with dual codes and we demonstrate that for the considered family of dual affine variety codes from the Klein quartic our method produces much more accurate information than what was found in [23]. Combining then our knowledge on both primary and dual codes we determine asymmetric quantum codes with desirable parameters.

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1 Introduction

In [17] the authors studied a family of primary affine variety codes over , defined from the Klein quartic. The length of these codes is , and the dimensions are easy to establish, but to lower bound the minimum distances they introduced a new method where the footprint bound from Gröbner basis theory is applied in a novel manner. The resulting codes have good parameters, either similar to the best known codes according to [18] or in a few cases with a defect in the minimum distance of only one.

In the present paper we show, using simple arguments, how to translate the findings from [17] into information on the corresponding dual affine variety codes. Besides giving us a new family of good classical linear codes this allows us to construct good asymmetric quantum codes, the handling of which requires detailed information on a set of nested classical linear codes as well as on the set of nested dual codes . The dual classical codes from the Klein quartic were originally treated in [23][Ex. 3.2, Ex. 4.1] and in [10] which are among the seminal papers on Feng-Rao theory. One way of viewing our method is to consider it as a way of exhuming what (in our understanding) is the most basic principle that makes the Feng-Rao bound work, and to employ this principle in a novel manner. Doing so, for the dual codes related to the Klein quartic we derive much sharper bounds on Hamming weights, and thereby minimum distances, than have previously been reported. In addition to the above we present a universal procedure for establishing primary descriptions of dual affine variety codes and vice versa. Our procedure being universal means that given a polynomial ideal over and a corresponding monomial ordering it returns a primary description for all related dual affine variety codes and vice versa. Thereby it provides a relevant alternative to the newly presented method in [24].

The paper is organized as follows. In Section 2 we recall the method from [17] for handling primary affine variety codes and list results for the case of the Klein quartic that will be needed throughout the paper, including a small refinement which does not change the overall analysis of [17], but which shall prove important in connection with our treatment of dual codes. Furthermore, we enhance the analysis to also treat the relative distance between two nested codes, the information of which is of importance when constructing asymmetric quantum codes. Then in Section 3 we introduce fundamental, yet simple, results which allow us to employ the findings for primary codes to establish bounds on dual codes. This involves descriptions both at a linear code level as well as using the language of affine variety codes. From that we then in Section 4 establish extensive information on a family of dual codes from the Klein quartic, and we make the comparison with [23][Ex. 3.2, Ex. 4.1], demonstrating the advantage of our method. We then in Section 5 establish primary descriptions of dual codes as well as dual descriptions of primary codes, meaning that for any code considered in [17] as well as any code of the present paper we know both a generator matrix and a parity-check matrix. From this in Section 6 we are able to demonstrate tightness of the minimum distance estimates from Section 4 in a considerable amount of cases and to improve upon one of them. Finally, in Section 7 we construct asymmetric quantum codes through the use of the CSS construction, and we demonstrate that they have desirable parameters. This includes examples of impure codes.

2 Affine variety codes and the results from [17]

The concept of affine variety codes was originally coined by Fitzgerald and Lax in [11]. Our exposition on the topic relies on the footprint of an ideal

Definition 1.

Given a field , an ideal and a monomial ordering on the set of monomials in the variables , the corresponding footprint is given by

From [5][Prop. 4, Sec. 5.3] we have:

Theorem 2.

The set is a basis for

as a vector space over

.

In the following we concentrate on finite fields and extend any given ideal to . Clearly, the variety is finite and from its elements we obtain the map

This map is obviously a vector space homomorphism and it is well-known that it is in fact an isomorphism [11]. For simplicity in the following we shall always write rather than .
For any set of monomials we now define a linear code

the dimension of which equals due to Theorem 2 and ev being an isomorphism. Such a code is called a primary affine variety code and its dual is said to be a dual affine variety code. To estimate the minimum distance of we may apply the below corollary of Theorem 2, known as the footprint bound [20].

Corollary 3.

Given an ideal the variety is of size .

Consider namely a codeword (i.e.  is a linear combination of monomials in ). Applying the above corollary to the ideal we see that the Hamming weight of equals

(1)

where lm denotes the leading monomial and . Knowing a Gröbner basis for would provide us with , assuming we know , but as we shall need to consider classes of polynomials , rather than individual ones, such a basis cannot be specified. However, if we calculate a Gröbner basis for with respect to we obtain full information on and the task then is to estimate how many monomials inside can be found as a leading monomial of a polynomial of the form

(2)

where are arbitrary polynomials.

To estimate the minimum distance of a primary affine variety code the most common approach (e.g. [21, 13, 4, 3, 12]) is to establish information on using only information on and paying in the analysis no attention to the coefficients of lower terms. I.e. for each , one detects a set of monomials which is a subset of for any having as leading monomial. Using in [15] the concept of one-way well-behaving pairs the authors took the first step in the direction of employing information on the coefficients of the non-leading monomials in the (possible) support of . The method from [17] can be seen as a further development in this direction where for each class of polynomials with a given leading monomial , starting from , one applies a series of calculations involving a mix of multiplication by monomials and polynomial divisions, modulo polynomials in , the result being in each step a polynomial of the form (2). Writing where the enumeration is done according to the ordering they consider , . Whenever during the process it is possible to establish conditions on the coefficients for which a substantial amount of monomials in can be demonstrated to be leading monomials of expressions of the form (2) this is recorded and in the following calculations the conditions are assumed not to hold. The process stops when all possible combinations of coefficients have been covered. We should mention that in continuation of [17] the procedure has also been successfully implemented in [28] to treat a family of codes defined from a particular hyperelliptic curve.

We now recall how the above procedure was applied to give a thorough treatment of a family of primary affine variety codes related to the Klein quartic . The reason for recalling such findings is two-fold. Firstly, our method for treating dual codes relies on our findings regarding primary codes, and secondly for the application of asymmetric quantum codes we will need information on both primary and dual codes. Furthermore, for the mentioned application we will need to enhance previous findings on minimum distances to results on relative distances.

The monomial ordering that we apply is the weighted graded ordering defined by if either holds or if , but . The Gröbner basis for

becomes from which the footprint can be seen to equal

corresponding to the fact that the number of affine roots of the Klein curve is 22.

Recall that given the task is for

to consider an exhaustive series of cases of different combinations of the coefficients . In each case we determine monomials in for which a polynomial of the form (2) exists having that monomial as leading monomial. It is straightforward to see that among such monomials we have those that are divisible by . However, for particular choices of additional monomials are determined in [17]. Table 1 through Table 8 explain the results for those cases. Here we use the notation

and using this notation for each choice of coefficients we specify in column two monomials which can be found as leading monomial of a polynomial in (2). I.e. . We stress that the last row in each of the tables is a conclusion that we entirely make for the purpose of treating dual codes in the present paper. What is listed here is the intersection of all the established sets of leading monomials , the information of which being not relevant for the treatment of primary codes. Moreover, in the second column of the tables some entries are marked in bold. These are entries that we are able to add in addition to those established in [17]. The reason for the entries in bold not to be established in [17] is that these entries do not change the analysis for the primary codes, but we will need them to analyze the dual codes. We illustrate our remarks in an example.

Example 1.

Consider . Table 3 lists information on for different cases which together cover all possible situations. For the first case the entry in the second column reads meaning that

As the number of elements in the listed set is we conclude that whenever . Ignoring in the next lines the entries in bold, we obtain for the considered coefficients information on the corresponding set as established in [17]. Observe, that the subsets of that we establish are not identical, but that all subsets are of size at least from which we conclude that for all polynomials with it holds that . The entries in bold (which we add) have no implication for the treatment of the primary code, as for instance we cannot add anything to the first row of Table 3, and therefore we cannot increase the estimate on the Hamming weight for general polynomial having as leading monomial beyond . However, for the dual codes we shall need information on the intersection of all sets listed in column 2 which is the reason for adding the entries in bold. The intersection is what we list in the last line of the table. For the considered choice of leading monomial the intersection is of size so we would not want to use that as the estimate on the Hamming weight of . Finally, to understand why we can add the values in bold observe for instance that whenever we know that is the leading monomial of a polynomial as in (2) then from the last line of Table 2 we can conclude that also is (and monomials divisible by it). Similarly when is included in column 2, then from Table 1 we see that also (and monomials divisible by it) can be added.

Coefficients Subset of
Intersection
Table 1:
Coefficients Subset of
Intersection
Table 2:
Coefficients Subset of
Intersection
Table 3:
Coefficients Subset of
Intersection
Table 4:
Coefficients Subset of
Intersection
Table 5:
Coefficients Subset of
An exhaustive set of ten different cases
Intersection
Table 6:
Coefficients Subset of
Intersection
Table 7:
Coefficients Subset of
Intersection
Table 8:

We conclude this section by collecting the established information on primary codes. First we introduce some notation for the general situation of primary affine variety codes. For each let be the minimal number of monomials in which by some given method have been shown to be leading monomials of expressions of the form (2), the minimum being taken over all polynomials having as leading monomial. Recall, that the relative distance between a pair of nested linear codes is given by

We then have the following theorem, the last part of which was not treated in [17], but which is included here due to its importance in connection with the CSS construction of asymmetric quantum codes (Section 7).

Theorem 4.

The minimum distance of is at least

Let and define

Then the relative distance is greater than or equal to

Proof.

To see the last part note that if then for some with support in and with at least one monomial in the support not belonging to . The last property implies that . ∎

For later reference, for we define and . The latter code is said to be of designed minimum distance and as far as our analysis goes these codes have at least as good parameters as the first mentioned codes.

Our treatment above of the Klein quartic immediately translates into the estimates , in Figure 1, from which it is straightforward to determine the dimension and to estimate the minimum distance of any corresponding code and .

Figure 1: As a conclusion of Table 1 through Table 8 plus the observation prior to them, the figure contains in the lower part the estimates for all

3 From bounds on primary codes to bounds on dual codes

In this section we start by enhancing Theorem 4 to cover the general situation of primary linear codes. From that we then devise a result for general dual linear codes which is finally translated to the language of affine variety codes to obtain the counter part of Theorem 4 for dual affine variety codes. Theorem 5 below in our opinion captures the very essence of Feng-Rao theory for primary codes, although to the best of our knowledge it has not been reported in this general version before. Recall that given vectors and the componentwise product is given by .

Theorem 5.

The following three statements are equivalent for a word :

  1. is the maximal integer for which there exists a vector space of dimension such that for any it holds that .

  2. is the maximal integer for which there exists a linearly independent set
    and a corresponding set of vectors such that .

Proof.

1. 2.:   It is enough to show that if and only if there exists a space of dimension at least satisfying the conditions in 2. We first observe that if then a space spanned by pairwise different standard vectors with a in a position where the corresponding entry of is non-zero satisfies the conditions in 2. Next, let be a vector space of dimension at least satisfying the conditions of 2. Aiming for a contradiction assume . But then there exist two different vectors in which have identical entries in those positions where is non-zero, and therefore the componentwise product with are the same. The difference between the vectors belongs to , but satisfies that the componentwise product with equals which is a contradiction.
2. 3.:   It is enough to prove that the conditions of 2. are satisfied for a number if and only if the conditions of 3. are satisfied for the same number. First assume that the conditions of 2. hold for a given . The set is a vector space, but as by assumption for different vectors , this vector space is of dimension implying that the conditions of 3 hold. Next assume that the conditions of 3. hold for a given . We then have

when not all coefficients equal . In particular must be a linearly independent set as