Quantum codes from a new construction of self-orthogonal algebraic geometry codes

by   Fernando Hernando, et al.

We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes, which greatly extends the class of a previous paper due to Munuera, Tenório and Torres. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.



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1. Introduction

Polynomial time algorithms on quantum computers for integer prime factorization and discrete logarithms were given by Shor [37]

. This justifies the great importance of quantum computation and, specifically, the relevance of quantum error-correcting codes because they protect quantum information from decoherence and quantum noise. Over the last twenty-five years, error-correction has proved to be one of the main obstacles to scaling up quantum computing and quantum information processing. One of the first examples of a quantum error-correcting code is Shor’s 9-qubit code

[38] which has been generalized in a series of many papers, including [3, 4, 8, 10, 9, 22, 23, 5, 7, 13, 24, 32]. Nowadays the theory of quantum error-correcting codes is a very active area of research (see [29, 30, 15, 16, 17, 25, 18] for some recent publications).

A classical linear error-correcting code is called self-orthogonal if it is contained in its dual code. The CSS (Calderbank-Shor-Steane) construction [9, 39] showed that classical self-orthogonal codes with certain properties are useful in the construction of quantum error-correcting codes. As a result, people looking for good quantum error-correcting codes started trying to find classical self-orthogonal codes with the required properties.

In the 1970s and early 1980s, using concepts and tools coming from algebraic geometry, Goppa constructed error correcting linear codes from smooth and geometrically irreducible projective curves defined over a finite field (see [20, 21, 40, 27]). They are called Goppa or algebraic geometry (AG) codes and have played an important role in the theory of error-correcting codes. They were used to improve the Gilbert-Varshamov bound [41] which was a surprising result at that time. In fact, every linear code can be realized as an algebraic geometry code [36]. In the area of quantum information processing, what is important is that AG codes provide a natural method for finding classical self-orthogonal codes. Thus, some researchers focussed on finding suitable AG codes, codes that would give us the required classical self-orthogonal codes, which in turn would give good quantum error-correcting codes.

Many of the properties of AG codes that give rise to good quantum error-correcting codes were captured in the definition of Castle codes by Munuera, Sepúlveda, and Torres [34]. In [35], Munuera, Tenório and Torres use the good properties of algebraic geometry codes coming from Castle and weak Castle curves to provide new sequences of self-orthogonal codes. Essentially, they use Lemma 2 and Proposition 2 of [35] to come up with interesting families of curves giving rise to those sequences.

The main purpose of this paper is to extend the construction of Castle curves. This allows us a larger family of curves from which to obtain self-orthogonal AG codes, and good quantum codes. We provide families of curves giving sequences of one-point self-orthogonal AG codes which are not covered by [35].

This paper is laid out as follows. In Section 2 we briefly summarize the construction of AG codes and establish some notation that will be used in the paper. In Section 3 we state and prove the main theoretical results (Theorem 3.1 and corollaries) that generalize the construction of Castle curves and will allow us to present the afore-mentioned sequences of self-orthogonal codes. The next sections are devoted to applying these results and obtaining explicit families of curves giving rise to those sequences. Finally, in Section 7, we will use them to obtain quantum codes with good parameters.

In the experimental examples we use the computational algebra system MAGMA [6].

2. AG codes and their duals

Throughout this and next section, we fix an arbitrary finite field . Let be a nonsingular, projective and geometrically irreducible curve of genus over (we will say simply ‘curve’ for abbreviation). We write for an algebraic closure of and denotes the set of -valued points of for any field extension .

A divisor of is a formal sum , where is a positive integer, and for all , and moreover if . We will say that the divisor is -rational if , where , and is the Frobenius -automorphism. Equivalently, can be regarded as a linear combination of places of with integer coefficients [40, Def. 1.1.8], where denotes the function field of . The support of , denoted by , is the set of points , and the degree of is defined as , where denotes the cardinality of the orbit of under the action of (or, equivalently, the degree of the extension , where is the residue field of ). Notice that a point is -rational (i.e. ) if and only if .

A divisor as above is effective if for all ; we write then . Also, given two divisors and , the notation means that the divisor is effective. We also consider the following finite-dimensional

-vector space associated with


where denotes the divisor associated to .

For a fixed set of -rational points on , set , and let be another -rational divisor of whose support is disjoint from . Consider the -vector space

where is the -vector space of rational differential forms over , and denotes the divisor associated to any .

Definition 2.1.

The AG code associated to the triple is the linear code of length over given by the image of the linear map

defined by .

It can be seen that its dual code, , coincides with the image of the map defined by , where stands for the residue of at for all . Furthermore, if is a differential form in with simple poles at and such that for all , then it holds that

(see, for instance, [12, Lemma 1.38]). Notice that a differential with these conditions does always exist.

Definition 2.2.

The code is said to be self-orthogonal if .

There is a particular class of curves among those satisfying the definition of AG codes. These are called Castle and weak Castle (pointed) curves, see [35, 34]. A pointed curve is a pair , where is a curve and is a rational point on .

Castle and weak Castle curves are defined taking into consideration the following notion. Let be a curve and an -rational point on , and consider the valuation (attached to the local ring) at . The set

is an additive subsemigroup of which is called the Weierstraß semigroup at the rational point of . We say that a pointed curve is Castle if

  1. is symmetric, i.e., if and only if for all .

  2. If , then .

If we substitute condition (2) by

  1. There exist a morphism with as well as elements such that and for all ,

then the pointed curve is said to be weak Castle. Notice that the terminology makes sense, since Castle curves are always weak Castle curves [34].

3. Main results

We start this section with some definitions and conventions. An affine plane curve over will be a curve defined by an equation , where , being affine coordinates. Considering projective coordinates such that and , we will denote by the projectivization of , and by the associated normalization morphism; in this way is a nonsingular model of .

For every , (resp., ) will denote the affine line over defined by the equation (resp., the projective line over , called line at infinity, with equation ).

Definition 3.1.

An affine plane curve over has only one place at infinity if it is geometrically irreducible, there exists an -rational point such that , has only one branch at and this branch is defined over . We impose the additional condition that is not a line.

Notice that, in the situation of Definition 3.1, there exists a unique point such that and, moreover, is -rational. Since and are isomorphic, we will identify the points of both curves.

If and are affine or projective plane curves (with respective equations and ) and is a common point of them, then we write (and also ) the intersection multiplicity of and at [27, Def. 2.22].

Definition 3.2.

Given two affine plane curves and over , we will say that and are transversal if for all . Also, we will say that and are -transversal if they are transversal and, in addition, all the points in are -rational.

Fixing a curve , for every subset of , we will define by

and we will be studying the polynomial (where is finite)

and its derivative . For any rational function , we will denote by (resp. by ) its divisor of zeroes (resp.of poles). We will consider the divisor of zeros of the rational function , and if

where the are points in the affine chart and is the point at infinity of the curve, then we define a divisor by . It is easy to show that the divisor is -rational. We call the divisor of affine zeroes of the rational function defined by the derivative .

Theorem 3.1.

Let be a smooth affine plane curve over with only one place at infinity. Let be the genus of and let

Let . Let be the divisor of affine zeroes of the rational function of defined by the derivative , as defined above.

Then the following hold:

  • If is the divisor , and is another -rational divisor such that , then

  • If, in addition, then .

  • For all let . In view of the choice of , the image of at the local ring at is a uniformizing parameter. Consider the following differential form of :

    Clearly, for any , we have

    Therefore has poles at the points of , which are of order 1 and have residue 1. Since and are -transversal for every root of , the associated divisor to is

    and the result now follows from [27, Th. 2.72].

  • It follows immediately from (a).

The following corollary (that is straightforward from Theorem 3.1) concerns AG codes defined from divisors of type and yields a range of values of for which the associated code is self-orthogonal.

Corollary 3.2.

Assume the notation and hypotheses of Theorem 3.1 and suppose that with . Then if .

In the specific case of curves defined by a separable equation , the degree of the divisor mentioned in the statement of Theorem 3.1 can be explicitly computed from the equation of and the degree of the polynomial :

Corollary 3.3.

Assume the notation and hypotheses of Corollary 3.2 and suppose that has an equation of the type , where are polynomials with coefficients in . Then .

Furthermore, if


Let be the distinct roots of the polynomial and consider the decomposition , . For each , let be the different roots of and consider the decomposition

Notice that the points in the support of are those in the set .

The coefficient in of one of the points is , where is the valuation defined by the curve at ; then


The last part of the statement follows from Corollary 3.2. ∎

Remark 3.4.

In practice, the main difficulty in applying Corollary 3.3 is that the polynomial and its derivative can be hard to compute. As an example, the curve is maximal over , having affine rational points. The polynomial can be computed using MAGMA and has degree 405; its derivative has degree 324. Applying Corollary 3.3 gives self-orthogonal curves for in the range . We are unable to compute by hand.

In the next sections we will give some infinite families of curves where we are able to compute by hand.

Next we present a special case of Corollary 3.3, where the range of values of for which the codes are self-orthogonal depends only on the genus of and :

Corollary 3.5.

Assume the notation and hypotheses of Corollary 3.2 and suppose that has an equation of the type , where are polynomials with coefficients in . Then if


First we will prove that . Notice that coincides with the cardinality of ; hence it is enough to show that for every . For this purpose, notice that for all because and are transversal. Then

where the last two equalities are deduced from the fact that and are -transversal.

Finally, the result follows from

where the first equality is consequence of Corollary 3.3. ∎

Remark 3.6.

There are examples where this bound is tight, in the sense that when , and for the smallest with One example is over , the number of rational points is . The derivative so it is not constant. The genus is and so becomes . We confirm with MAGMA that for we have that but not for .

To finish this section, we prove that the AG codes coming from Corollary 3.3 arise from weak Castle curves.

Proposition 3.7.

If is a curve satisfying the hypotheses of Corollary 3.3 then the pointed curve is weak Castle.


Assume the notation of Theorem 3.1 and suppose, without loss of generality, that .

Consider an arbitrary element and the divisor of the rational function . Since and are -transversal one has that and

where and, for every , equals if belongs to , and 0 otherwise.

Notice that, independently of , the point belongs to if and only if ; moreover, in this case, equals (the multiplicity of at ) because the line is not tangent to at (notice that is not a line). This shows that the value does not depend on and that . Therefore

In particular, .

Now, consider the morphism associated with the rational function defined by . From the previous paragraphs, it holds that and, for all , and . Hence, taking into account [35, Prop. 3 (2)], the pointed curve is weak Castle. ∎

Remark 3.8.

We would like to comment on how our results differ from the results in [35] and [19]. All the families of curves in [35] satisfy the hypotheses of Lemma 2 in that paper. Under the assumptions (and notation) of Theorem 3.1, the pointed curve satisfies the hypotheses of [35, Lemma 2] if and only if the polynomial is a nonzero constant (if and only if the divisor in Theorem 3.1 is the zero divisor). In this paper we will present some families with non-constant derivative, which are the first of this kind as far as we are aware.

To emphasize this point, we partition the curves satisfying the hypotheses of Theorem 3.1 into two types:
Type I: those where is a nonzero constant.
Type II: those where is not constant.

The curves in [35] are of Type I, however many of the codes introduced in our paper come from curves of Type II. Therefore, we are presenting a new type of code. By Proposition 3.7 both types of curves are weak Castle. Most of the Type II curves in this paper are not Castle, as we will see.

The curves in [19] after Theorem 1 are of Type I. The curves in [19] after Theorem 2 are not necessarily of Type I, however the codes are not one-point AG codes. All codes in our paper are one-point AG codes, and hence our results and examples are different from [19]. Also, all the sets in [19] are multiplicative subgroups after removing 0.

Families of self-orthogonal AG codes

The aim of this subsection is to provide a lemma which will allow us to obtain several families of curves satisfying the hypotheses of Corollary 3.3 and, therefore, to obtain families of self-orthogonal AG codes.

Lemma 3.9.

Let be a finite field of characteristic and let be an affine plane curve over with equation

where and are polynomials with coefficients in such that is a nonzero constant and . Then

  • is smooth.

  • If or with such that and moreover , then has only one place at infinity.

  • The genus of is .


Statement (a) is obvious, since the partial derivative with respect to of the defining equation of is a nonzero constant. We split the proof of (b) in two cases:

Case 1: . In this case, is the unique intersection point of and the line at infinity. Set , , , , and , where denotes the resultant (with respect to ) of and .

It is easily checked that . Therefore, and . Since and is a multiple of , Proposition 3.5 of [11] (see also the original source [1] by Abhyankar) implies that has only one place at infinity.

Case 2: with and . In this case, is the unique intersection point of and the line at infinity. Setting one has that the equation of (in projective coordinates and ) is

where for all and . Taking coordinates and in the affine chart defined by (to which belongs), the equation of the restriction of to has the form

where is an homogeneous polynomial of degree such that and is the origin. Hence, has a unique tangent at (defined by ). Performing finitely many successive quadratic transformations we can obtain a resolution of singularities of at (so that, by composition of them, we get the normalization morphism ); see e.g. [2, Lecture 18]. The quadratic transformation (with center ) defined by and gives rise to the following equation of the proper transform of :

Hence, meets the exceptional line at a point that is -rational. Since , it is not difficult to see that all the proper transforms involved in the process meet each exceptional line at a unique -rational point, and that the last proper transform has multiplicity one at every point. Since the points of are in one-to-one correspondence with the branches of [26, Th. 5.29], it follows that has only one branch at (which is -rational).

It only remains to prove that is geometrically irreducible. Indeed, reasoning by contradiction, assume that and are two different components of . Then both curves and must meet at the point , which contradicts the conclusion of the preceding paragraph.

Statement (c) follows from [35, Prop. 3]. ∎

Next, in Sections 4, 5, and 6, we will present the families of curves where our results are applicable. From now on, will be a power of a prime number and stands for the number of -rational points of an affine curve . We will make use of the notion of trace of an element over : the trace is the sum of the conjugates of with respect to , i.e.

4. Curves

Let and denote positive integers (not both equal to 1) such that , and let be the affine curve (defined over ) with equation

The following Proposition refers to the statement of Theorem 3.1.

Proposition 4.1.

Let and let . Then is smooth over and has only one place at infinity. The set in the statement of Theorem 3.1 is equal to the set of all -coordinates of the -rational points of .

Moreover, , where , and the number of -rational points of is .


By Lemma 3.9, is a smooth affine curve having one place at infinity with genus . If is the -coordinate of an -rational point of the curve then the equation has distinct solutions for in . Hence all the points in the intersection are -rational. Moreover, if is one of these points, then

because is a simple factor of . Therefore the set

coincides with .

It remains to prove . To this aim notice that, on the one hand, . On the other hand, for every , we have and and, therefore, is a root of . Then every element of is a root of .

Conversely, let be a root of . Then and, therefore, . Hence because the equation has solutions in (by surjectivity of trace).

Finally, for every , it holds that if and only if either or . Hence, since is the image of by the trace of over , we have

Proposition 4.1 means that we can apply Corollary 3.3 to the curve , and we deduce the following result:

Corollary 4.2.

Let , let be the set of -rational points of , and let be a divisor of . Then, for any nonnegative integer , the AG code (defined from ) given by is self-orthogonal if

where if divides and otherwise.

Remark 4.3.

In [35, Example 2] the authors consider curves with and show that, when , the pointed curves satisfy the hypotheses of Lemma 2 of [35]. Hence, in these cases, this lemma implies that the code (defined as in Corollary 4.2) is self-orthogonal if

However, our Corollary 4.2 gives a larger family of curves which do not necessarily satisfy the hypotheses of Lemma 2 of [35] (see Remark 3.8).

Lastly in this section, we show that the pointed curve is almost never a Castle curve. Proposition 2 of [35] can only be applied to when the curve is Castle.

Note that we never have because is relatively prime to .

Proposition 4.4.

(1) If the pointed curve is never a Castle curve.

(2) If the pointed curve is a Castle curve if and only if


Let be the smallest nonzero element of the Weierstraß semigroup at . We know that the number of (affine) points is so the curve is Castle if and only if .

Notice that is a multiple of , since .

Proof of (1) : Suppose . In this case the smallest element of the Weierstraß semigroup is i.e. . But we always choose to be relatively prime to , so we cannot have for . Therefore the curve is never Castle in this case.

If the curve is Castle iff . Then , but also . If then , which is impossible. If then there is a divisor of which is also a divisor of , which is impossible.

Proof of (2) : Suppose In this case the smallest element of the Weierstraß semigroup is i.e. . The curve is Castle if and only if . However if and only if , by the definition of . ∎

5. Curves

Let be a positive integer and consider a polynomial such that and . Consider the unique polynomial with degree at most such that for all . We will assume that

  1. is separable, and

  2. all roots of belong to