Chapter 1 Quantum Codes from Two-Point Divisors on ... · The theory of algebraic geometry (AG) codes has its origin in the mathe-matics of algebraic curves. At the core of which

June 3, 2019 23:32 ws-rv9x6 Book Title EzermanKirov page 1

Chapter 1

Quantum Codes from Two-Point Divisors on

Hermitian and Suzuki Curves

Martianus Frederic Ezerman∗

School of Physical and Mathematical Sciences,Nanyang Technological University, 21 Nanyang Link, Singapore.

[email protected]

Radoslav Kirov

Google Inc.,1600 Amphitheatre Parkway Mountain View, CA 94043 U.S.A.

[email protected]

Sarvepalli and Klappenecker showed how to use classical one-point codeson the Hermitian curve to construct symmetric quantum codes. Forthe said curve, Homma and Kim determined the parameters of a largerfamily of codes, the two-point codes. For the Suzuki curve, the bounddue to Duursma and Kirov gave the exact minimum distance of Suzukitwo-point codes over F8 and F32. The observation that different typesof errors in binary quantum channels are not equiprobable led to themathematical study of asymmetric quantum error-correction.

This work considers quantum codes, both symmetric and asymmet-ric, constructed from two-point divisors on Hermitian and Suzuki curves.In the asymmetric case, we show strict improvements over all suitable fi-nite fields for a range of designed distances. We produce pure asymmetricquantum codes (AQCs), with large dimensions, and impure AQCs, withsmall dimensions, that have better parameters than the best-possibleAQCs from one-point codes on the corresponding families of curves.The gain is illustrated by exact numerical results.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Goppa Construction of Algebraic Geometry Codes . . . . . . . . . . . . . . . . 4

3. Two-Point Codes from Hermitian and Suzuki Curves . . . . . . . . . . . . . . 7

∗Corresponding author.

1


2 M. F. Ezerman and R. Kirov

4. From Classical to Quantum Codes . . . . . . . . . . . . . . . . . . . . . . . . . 11

5. Improved Quantum Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6. Conclusions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1. Introduction

The theory of algebraic geometry (AG) codes has its origin in the mathe-

matics of algebraic curves. At the core of which lies some beautiful mathe-

matics connecting the set of rational points on the curves to the parameters

of the resulting codes and of their duals. Goppa1 introduced a standard

construction method. A comprehensive survey of basic notions and meth-

ods in AG codes can be found in the work of Tsfasman et al.2 In the

classical set up, long AG codes constructed from curves with many rational

points have excellent properties.

Quantum codes help protect information-carrying quantum states

against decoherence and play an important part in making fault-tolerant

quantum computation possible. Firm connections between classical error-

correcting codes and quantum codes are now well-established. We can

construct quantum codes from classical codes by using the stabilizer for-

malism.3,4 The resulting quantum codes are called stabilizer codes. A

subclass of these codes can be derived by using the CSS method attributed

to Calderbank, Shor, and Steane.

The study of asymmetric quantum codes (AQCs) began with the obser-

vation that, in many qubit systems, phase-flips (or Z-errors) occur more

frequently than bit-flips (or X-errors) do.5 Steane6 first hinted the idea of

adjusting the error-correction to the particular characteristics of the quan-

tum channel and, later, Wang et al.7 established a mathematical model

of AQCs in the general qudit system that also includes the CSS construc-

tion. An earlier discussion on the constructions and bounds of AQCs can

be found in the work of Sarvepalli et al.8

Quantum codes can be further distinguished into pure and impure (or

degenerate). In the stabilizer context, purity can be deduced from the

weight distributions of the corresponding pair of classical codes. AG codes

fit nicely into the stabilizer framework, and into the CSS method more

specifically, since many of the ingredients needed to construct good quan-

tum codes can be explicitly computed once the curve and the set of rational

points have been properly chosen. The length and the dimension of an AG

code, seen as an evaluation code, are straightforward to determine. The

dual code can be easily defined. Their distances have bounds that follow


Quantum Codes from 2-Point Divisors on Hermitian and Suzuki Curves 3

immediately from the degree bounds for divisors and for rational functions.

Some constructions of symmetric quantum codes from AG codes have

been discussed previously.9–11 Maximal curves12 were used to derive a num-

ber of stabilizer codes with better parameters than the so-called quantum

twisted codes.13 Asymptotically good quantum codes from AG codes were

treated in the work of Feng et al.14 and several follow-up works. The

most cited AG-based quantum codes are those constructed from one-point

Hermitian codes since they are easy to describe, to encode and to decode.

Vector spaces of functions that correspond to two-point divisors were first

studied by Matthews.15 Using Goppa’s construction, we can derive two-

point Hermitian codes from them. A complete description of the minimum

distances of all two-point Hermitian codes was given by Homma and Kim.16

Further results17–19 have advanced our understanding of these codes.

Two-point codes often have better parameters than one-point codes,

while maintaining their ease of construction. A systematic comparison be-

tween one-point and two-point Hermitian codes showing strict improvement

for a range of designed parameters is detailed in [19, Section IV]. A similar

approach has been successfully implemented to compute the exact mini-

mum distance of codes and the minimum weight of their cosets for other

families of curves, also showing strict improvements.

In this work, we show that the parameter improvement gained from

considering the two-point codes is often carried over, but not always, to

the quantum setting. We focus on the Hermitian and Suzuki curves since

much is known about their properties. The number of rational points on the

Hermitian curve is maximal, reaching the Hasse-Weil-Serre upper bound.

On the Suzuki curve, the number of rational points is optimal, i.e., the

highest possible for the given genus. In both symmetric and asymmetric

models, the codes constructed from a pair of nested two-point Hermitian

or Suzuki codes have parameters at least as good as and often improve

significantly on the dimension and the distances if compared with their

one-point counterparts. Illustrative numerical results will be supplied to

highlight the improvements.

In line with the spirit of a CIMPA research school, the chapter consists

of two main parts. The first part contains known material to build upon. A

summary of the Goppa construction of AG codes is provided in Section 2.

Two-point codes from Hermitian and Suzuki curves are treated in Section 3.

Relevant constructions of quantum codes that can be applied to nested

AG codes form Section 4. The second part collects new contributions and

open directions. Section 5 presents numerous concrete parameters of the



improved quantum codes. It also explains two options for applying the CSS

construction that yield significant improvements. Section 6 provides a brief

summary and states some directions to explore.

2. Goppa Construction of Algebraic Geometry Codes

Let Fq denote the finite field of cardinality q = pm for a prime p and a

positive integer m. Let wt(u) denote the Hamming weight of a vector u.

Given distinct (nonempty) subsets C and D of Fnq , let wt(C \ D) denote

minwt(u) : u ∈ (C\D), u 6= 0. Let u = (u1, . . . , un) and v = (v1, . . . , vn)

be vectors in Fnq . The Euclidean inner product of u and v is 〈u,v〉E =∑ni=1 uivi. The Hermitian inner product of u and v is 〈u,v〉H =

∑ni=1 uiv

ei

if q = e2.

A linear [n, k, d]q-code C is a k-dimensional subspace of Fnq with mini-

mum distance d = minwt(c) : c ∈ C \ 0. Let 〈u,v〉∗ represent either

the Euclidean or the Hermitian inner product of u and v. The dual code

C⊥∗ of C is given by C⊥∗ :=u ∈ Fnq : 〈u,v〉∗ = 0 for all v ∈ C

, while

the dual distance d⊥∗ is defined to be d(C⊥∗). A code C is self-orthogonal

(respectively, self-dual) if C ⊆ C⊥∗ (respectively, C = C⊥∗). In this work,

two codes C1 and C2 are equivalent if there exists a nonzero vector a ∈ Fnqsuch that C1 = a ? C2 where u ? v = (u1v1, . . . , unvn).

We recall Goppa’s general construction of codes from curves. Let X/Fqbe an algebraic curve (absolutely irreducible, smooth, projective) of genus g

over Fq. The Hasse-Weil-Serre bound on the number #X(Fq) of Fq-rational

points on X/Fq reads

#X(Fq) ≤

(q + 1) + 2g

√q if q is a square,

(q + 1) + gb2√qc if q is not a square.(1)

X/Fq is an Fq-maximal curve if the equality in (1) is reached.

A rational function on a curve is of the form F (Z)G(Z) , where F (Z) and

G(Z) are polynomials in some indeterminate Z. Suppose P is a point on

the curve such that G(P ) = 0 but F (P ) 6= 0, then P is a pole of f and a

zero of f−1. Point P is a pole of order ` of f , i.e., a zero of order ` of f−1 if

G(Z) = (Z −P )`g(Z) with g(P ) 6= 0. A pole is simple if ` = 1. A function

f is regular at P if G(P ) 6= 0. The valuation ring OP of P is the set of

rational functions that are regular at P . It is a local ring [20, Theorem

1.1.13] whose unique maximal ideal is mP := f ∈ OP : f(P ) = 0. Any

function tP ∈ OP such that mP = tPOP is called a local parameter at P .



A divisor D on X/Fq is a formal finite sum D =∑aPP of points on

the curve with aP ∈ Z. It is effective if all of its aP s are nonnegative. Its

support, denoted by SuppD, is P ∈ X/Fq : aP 6= 0 while its degree is

deg(D) :=∑P∈SuppD aP . The divisors on X/Fq form an Abelian group

Div(X). The divisor (f) :=∑ordP (f)P is associated with a rational

function f 6= 0, where ordP (f) := maxj : f ∈ mjP but f /∈ mj+1P . Let

(f)0 =∑

ordP (f)>0

ordP (f)P and (f)∞ =∑

ordP (f)<0

(− ordP (f))P.

Then effective divisors (f) = (f)0− (f)∞ are called principal. They form a

subgroup P (X) of Div(X). Divisors D1 and D2 such that D1−D2 ∈ P (X)

are equivalent. All divisors in an equivalence class have the same degree.

An algebraic function field is the field of rational functions on a curve.

Let Fq(X) be the function field of the curve X/Fq. Given a divisor D on

X/Fq, let L(D) := f ∈ Fq(X)\0 : (f)+D ≥ 0∪0. We say that L(D)

is the Riemann-Roch space of the divisor D and that D has a base point P

if L(D) = L(D − P ). The dimension of L(D) is denoted by dimL(D).

A function ω is a differential form, or simply a differential, if ω is reg-

ular on some open subset U of X/Fq. Such a function defines a rational

differential form on the curve. Let Ω(X) denote the set of all rational dif-

ferential forms on X/Fq. The genus of the curve is the dimension of Ω(X).

The Fq-rational automorphism group of the curve is denoted by AutFq (X).

Let (ω) denote the divisor associated with ω on X. Let t = tP be a local

parameter at P and let ω = f dt for some function f (see, e.g., [20, Propo-

sition 4.1.8 (a)], for why this can be done). We can expand f uniquely

into a power series in t as f =∑∞i=−M ait

i. The coefficient a−1 is the

residue, denoted by ResP (ω), of ω at P . The residue formula for function

fields [20, Theorem 2.2.8] states that, for any differential form ω ∈ Ω(X),

we have∑P∈X/Fq ResP (ω) = 0.

Let Ω(D) := ω ∈ Ω(X) \ 0 : (ω) ≥ D ∪ 0. Let K represent the

canonical divisor class, which is the linear equivalence class of (ω). It is a

common abuse of notation to also denote by K any divisor from the said

canonical class. For n distinct rational points P1, . . . , Pn and for disjoint

divisors D = P1 + · · · + Pn and G, the geometric Goppa codes CL(D,G)

and CΩ(D,G) are defined, respectively, as the images of the evaluation and

the residue maps

αL : L(G) −→ Fnq , sending f 7→ (f(P1), . . . , f(Pn)).

αΩ : Ω(G−D) −→ Fnq , sending ω 7→ (ResP1(ω), . . . ,ResPn(ω)).



It follows from the residue formula that CL(D,G)⊥E = CΩ(D,G). More-

over, the residue construction can also be represented as an evaluation.

Lemma 1. ( [20, Proposition 2.2.10]) Let ν be a differential with simple

poles and residue 1 at the points P1, P2, . . . , Pn of D. Then

CL(D,G)⊥E = CΩ(D,G) = CL(D,D −G+ (ν)).

Hence, both codes have length n with respective dimensions

k(CL(D,G)) = dimL(G)− dimL(G−D) and

k(CΩ(D,G)) = dimL(K −G+D)− dimL(K −G).

For the minimum distance, we have the following basic bound.

Proposition 1. (Goppa bound)

d(CL(D,G)) ≥ deg(D −G) and d(CΩ(D,G)) ≥ deg(G−K).

We call δ := deg(D −G) the designed distance of CL(D,G).

The nestedness property of AG codes is certified by the nestedness of

divisors. Let A and B be two divisors of X/Fq. If A ≤ B then CL(D,A) ⊆CL(D,B) and CΩ(D,B) ⊆ CΩ(D,A). Similarly for equivalence, if A ∼ B

then CL(D,A) is equivalent to CL(D,B) and CΩ(D,B) is equivalent to

CΩ(D,A).

Let P be disjoint from D and consider the subcodes CL(D,G − P ) ⊆CL(D,G) and CΩ(D,G + P ) ⊆ CΩ(D,G). The order bound combines

estimates for the weight of a codeword c ∈ CL(D,G) \ CL(D,G − P ) or

v ∈ CΩ(D,G) \ CΩ(D,G + P ). The basic version of the bound says that

d(CL(D,G)) is given by

mini≥0minwt(c) : c ∈ CL(D,G− iP ) \ CL(D,G− (i+ 1)P )

while d(CΩ(D,G)) is given by

mini≥0minwt(c) : c ∈ CL(D,G+ iP ) \ CΩ(D,G+ (i+ 1)P ) .

By the Singleton bound, i ≤ g. The order bound gives an improvement

over all other prior bounds if, for each 0 ≤ i ≤ g, there is a better estimate.

The diagram in [21, Introduction] explains how to relate various bounds on

the minimum distance of AG codes. The best-known for general AG codes

are the extensions of the order bound. We call them the Duursma-Park

(DP) and the Duursma-Kirov (DK) bounds.

The Riemann-Roch Theorem is central in deriving properties of AG

codes.



Theorem 1. (Riemann-Roch Theorem [20, Theorem 1.5.15])

Let X/Fq be an algebraic curve and D be any divisor on it. Then

dimL(D)− dimL(K −D) = deg(D)− g + 1. (2)

Let 0 ≤ deg(G) < n and the genus g be given. Then CL(D,G) has k ≥deg(G)− g+ 1 and d ≥ n−deg(G). If 2g− 2 < deg(G), then CΩ(D,G) has

k ≥ n− deg(G) + g − 1 and d ≥ deg(G)− 2g + 2.

3. Two-Point Codes from Hermitian and Suzuki Curves

Classical two-point Goppa codes are CL(D,G) and CΩ(D,G) with divisor

G = iP + jQ for some suitable i and j. To construct them, one fixes two

distinct rational points P and Q. For both Hermitian and Suzuki curves,

it is standard to let P be the point at infinity, denoted by P∞, and Q be

the origin, denoted by P0.

There are at least three good reasons for this choice. First, it makes the

Weierstrass semigroup at P and Q to be the same. This holds because the

action of the automorphism group, of the respective curves, puts P and Q

in the same orbit, which is the set of Fq-rational points. This fact makes the

task of constructing two-point codes easier. Second, for the Suzuki curve,

the Weierstrass semigroup at points which are not Fq-rational appears to

be not explicitly known yet. Third, it is known from [22, Remarks 5.11 and

5.13] that, for the Suzuki curve, the set of Fq-rational points is the set of

Weierstrass points of the curve. In terms of their respective affine equations,

which will be explicitly given below, therefore, P∞ is the common pole of x

and y and P0 the common zero of x and y. For both curves, the canonical

divisor class is represented by K = (2g − 2)P∞ ∼ (2g − 2)P0.

The two-point codes are one coordinate shorter than the one-point

codes. In order to compare the two families, we use the fact that short-

ening a code preserves the minimum distance. Since the automorphism

group of a one-point code acts transitively on the set of coordinates, the

choice of coordinate is non-essential. A more detailed discussion can be

found in [23, Section 4]. This feature makes it easy to compare two-point

codes of length 2 less than the number of rational points with the shortened

one-point codes of equal dimension.

The Hermitian curve Hq2 is the smooth projective curve over Fq with

affine equation yq + y = xq+1. It is an Fq2 -maximal curve with q3 + 1

Fq2-rational points and genus g = q(q − 1)/2. Let P and Q be arbitrary

Fq2-rational points. Geometrically, if P = P∞, then, for any point Q 6= P∞



on Hq2 , the function f that ensures (q+1)P∞ ∼ (q+1)Q is the tangent line

of Hq2 at Q. The function that yields the general equivalence (q + 1)P ∼(q + 1)Q for P 6= P∞ can be obtained by considering σ(f), where σ is

an automorphism of Hq2 that fixes Q but sends P∞ to P . The stabilizer

of Q in AutFq2 (Hq2) is transitive on the set of all Fq2-rational points on

Hq2 \Q. Hence, the equivalent divisors (q+1)P ∼ (q+1)Q belong to the

hyperplane divisor class H. The divisor sum R of all q3 + 1 rational points

belongs to the divisor class (q2 − q + 1)H and the canonical divisor class

K = (q−2)H. One can refer to24,25 as well as the treatment in [20, Section

8.3] and [2, Section 4.4.3] for the details.

Henceforth, for Hermitian codes we fix D := R− P∞ − P0, making the

length of the constructed two-point codes q3 − 1. The Euclidean dual of a

one-point Hermitian code is also a one-point Hermitian code. We extend

this property to two-point codes.

Proposition 2. Let the Hermitian curve Hq2 be given. Then

CL(D, iP∞ − jP0)⊥E = CL(D, (q3 + q2 − q − 2− i)P∞ + (j − 1)P0).

Proof. Following the proof for one-point codes in [20, Proposition 8.3.2],

we select ν = dt/t, with t = xq2 − x. This is a good choice since ν has

simple poles and residue 1. Note that t is a prime element for all places

≤ D. Its principal divisor is (t) = R− (q3 + 1)P∞ = D+P0− q3P∞. Since

dt = d(xq2 − x) = −dx, the differential dt has the divisor (dt) = (dx) =

(q2 − q − 2)P∞. Applying Lemma 1 completes the proof.

We derive conditions for Hermitian codes to be Euclidean self-orthogonal.

Corollary 1. Given the Hermitian curve Hq2 , let i, i′, j, and j′ be integers

such that q3 + q2 − q − 2 ≤ i+ i′ and −1 ≤ j + j′. If G = iP∞ − jP0, and

G′ = i′P∞ − j′P0, then CL(D,G)⊥E ⊆ CL(D,G′) and CL(D, iP∞ − jP0)

is Euclidean self-orthogonal if 2i ≤ q3 + q2 − q − 2 and j < 0.

Since (q+1)P∞ ∼ (q+1)P0, every two-point Hermitian code is uniquely

equivalent to a code of the form CL(D, iP∞ − jP0) with 0 ≤ j ≤ q. We

use this representation as a canonical one. Note that for two-point Suzuki

codes we use (q2 + 1)P∞ ∼ (q2 + 1)P0, instead. A particularly favorable

feature of the Hermitian curves is that one can explicitly write a monomial

basis for the Riemann-Roch space of a two-point divisor of that form.

Lemma 2. ( [18, Section 4]) Given the Hermitian curve Hq2 , let a, b, and

c be integers with 0 ≤ a, b ≤ q. Let D = c(q + 1)P∞ − aP∞ − bP0. Then

L(D) has a basis given by the monomials xiyj where



(1) 0 ≤ i ≤ q, 0 ≤ j, and i+ j ≤ c,(2) a ≤ i for i+ j = c, and

(3) b ≤ i for j = 0.

The actual minimum distance of two-point Hermitian codes was deter-

mined by Homma and Kim in [26, Theorems 5.2 and 6.1] for b = 0 and

b = q, as well as in [16, Theorems 1.3 and 1.4] for 0 < b < q. Using order

bound techniques, Beelen in [17, Theorem 17] gives lower bounds for the

cases deg(G) > deg(K), i.e., for a + b > (q − 2)(q + 1). Park settles all

cases in [18, Theorems 3.3 and 3.5]. Park moreover shows that Beelen’s

lower bounds are sharp, i.e., they correspond to the actual minimum dis-

tances. Duursma and Kirov19 show that among all divisors G = iP∞+ jP0

of a given degree, the optimal minimum distance is attained for a choice of

the form G = aP∞ − 2P0.

Proposition 3. ( [19, Theorem 4.3]) Given the Hermitian curve Hq2 , let

B be a divisor that satisfies B 6= 0, deg(B) ≥ 0, and B = cH − aP∞− qP0,

for 0 ≤ a ≤ q. Let G := K + B. Then the two-point Hermitian code

CL(D,G) has dimension deg(G)− g + 1 and dual distance

d⊥E =

deg(B) + max0, q − c if a = q,

deg(B) + max0, q − c+ max0, a− c, otherwise.

The code CL(D,G′) where G′ = (deg(B)+2g−2)P∞ is a one-point Hermi-

tian code of the same dimension as CL(D,G). If a 6= q, then d(CL(D,G))

is max0, q − c higher than d(CL(D,G′)).

Using Proposition 2, we can restate the result for the minimum distance.

The range for r can be extended, but outside the given range there is no

improvement over the one-point codes.

Corollary 2. Let 0 ≤ r ≤ q2 + q. Let 1 ≤ c and 0 ≤ a ≤ q be the

unique numbers such that r + q = c(q + 1) − a. Then the Hermitian code

CL(D, (q3 − r + 1)P∞ − 2P0) is a [q3 − 1, k(r), d(r)]q2 code, where

k(r) = q3 − q(q − 1)

2− r and

d(r) =

r + max0, q − c if a = q,

r + max0, q − c+ max0, a− c, otherwise.

We now review basic facts and results on the Suzuki function field and

the corresponding codes.27,28 Results on a special class of invariant codes



are available.29 Duursma et al. give a thorough discussion on the distance

bounds and the determination of the exact distances of two-point codes

from the Suzuki curves.21 We give just enough details for self containment.

Let q0 = 2s ≥ 2 and q = 2q20 . The affine equation defining the Suzuki

curve Sq over Fq is yq + y = xq0(xq +x). The curve has genus g = q0(q−1)

and q2 +1 rational points, making it optimal. Any curve over Fq with genus

g = q0(q − 1) and q2 + 1 rational points is isomorphic to Sq [22, Theorem

5.1]. The Fq-automorphism group of the curve is none other than the Suzuki

group of order q2(q2 + 1)(q − 1). A result analogous to Proposition 2 for

the Suzuki two-point codes can be derived in a similar manner.

Proposition 4. Let R be the divisor sum of all q2 + 1 rational points on

the Suzuki curve Sq. If D = R− P∞ − P0, then

CL(D, iP∞ − jP0)⊥E = CL(D, (q2 + 2q0(q − 1)− 2− i)P∞ + (j − 1)P0).

Proof. Let ν = dt/t with t = xq + x. This leads to

(dt) = (q2 + 2q0(q − 1)− 2− r)P∞.

We then follow the argument in the proof of Proposition 2 to reach the

conclusion.

Corollary 3. Given the Suzuki curve Sq, let i, i′, j, and j′ be integers such

that q2 + 2q0(q − 1) − 2 ≤ i + i′ and −1 ≤ j + j′. If G′ = i′P∞ + j′P0,

then CL(D,G)⊥E ⊆ CL(D,G′) and CL(D, iP∞ + jP0) is Euclidean self-

orthogonal if 2i ≤ q2 + 2q0(q − 1)− 2 and j < 0.

For two-point codes from S8 and S32, the DK bound coincides with the

actual distance. The bound is established in the language of the divisor

semigroups using the divisor class C = D−G for CL(D,G) and C = G−Kfor CΩ(D,G). We call C the designed minimum support of the code. Let

Γ := A : L(A) 6= 0 be the semigroup of effective divisor classes. The

subset of Γ with no base point at P∞ is ΓP∞ := A : L(A) 6= L(A− P∞).The semigroup Γ(C) is a subset of Γ where L(A− C) 6= 0 while ΓP∞(C) :=

A : L(A) 6= L(A−P∞) and L(A−C) 6= L(A−C −P∞) is the semigroup

defined by C and P∞. Its complement in ΓP∞ is denoted by ∆P∞(C). For

a finite set S of points in X/Fq, let ΓS :=⋂P∈S ΓP with Γ∅ := Γ.

The DK bound for two-point Suzuki codes is computed by using S =

P∞, P0, based on the following result.

Theorem 2. ( [30, Theorem 4.1]) Let Pi ∈ S for 1 ≤ i ≤ `. Let a sequence

of divisors A1 ≤ A2 ≤ . . . ≤ A` ⊂ ∆S(C) with Ai ∈ ∆Pi(C) satisfies



Table 1. Important Properties of the Curves.

Curve X Affine Equation g #X K

H9 x4 = y3 + y 3 28 4P∞ ∼ 4P0

H16 x5 = y4 + y 6 65 10P∞ ∼ 10P0

H25 x6 = y5 + y 10 126 18P∞ ∼ 18P0

H49 x8 = y7 + y 21 344 40P∞ ∼ 40P0

H64 x9 = y8 + y 28 513 54P∞ ∼ 54P0

S8 y8 + y = x2(x8 + x) 14 65 26P∞ ∼ 26P0

S32 y32 + y = x4(x32 + x) 124 1025 246P∞ ∼ 246P0

Ai − Pi ≥ Ai−1 for 2 ≤ i ≤ `. Then deg(A) ≥ ` for every A ∈ ΓS(C) with

support disjoint from A` −A1.

To determine the lower bounds from the theorem, one needs to actually

build the sequences of divisors in ∆S(C). Two algorithmic tools from graph

theory, namely the weight maximizing and the flow maximizing algorithms,

are used to implement the computation efficiently.

Table 1 lists important properties of the curves H9, H16, H25, H49, H64,

S8, and S32. The numbers 26 and 246 in K for S8, and S32 come from a

known stronger linear equivalence when P∞ and P0 are Fq-rational, namely

(q + 2q0 + 1)P∞ ∼ (q + 2q0 + 1)P0 [31, Example 9.80].

Figure 1 provides a graphical summary of the gain, in dimension and

minimum distance, of two-point codes over their one-point counterparts.

The comparisons on the distance and dimension were made by comparing

the best possible one-point and best possible two-point codes. In general the

respective (one-point and two-point) best codes may come from different

choices of divisors.

4. From Classical to Quantum Codes

Let η = e2π√−1p ∈ C, the field of complex numbers. Let Vn = (Cq)⊗n be the

nth tensor product of Cq with|c〉 = |c1c2 . . . cn〉 : c = (c1, . . . , cn) ∈ Fnq

as

an orthonormal basis where |c1c2 . . . cn〉 abbreviates |c1〉 ⊗ |c2〉 ⊗ · · · ⊗ |cn〉.For two quantum states |ϕ〉 and |ψ〉 in Vn with |ϕ〉 =

∑c∈Fnq

α(c)|c〉 and

|ψ〉 =∑

c∈Fnqβ(c)|c〉, where α(c), β(c) ∈ C, the inner product of |ϕ〉 and |ψ〉

is 〈ϕ|ψ〉 =∑

c∈Fnqα(c)β(c) ∈ C, where α(c) is the complex conjugate of α(c).

We say |ϕ〉 and |ψ〉 are orthogonal if 〈ϕ|ψ〉 = 0.



(a) Comparison for H9, n = 26 (b) Comparison for H25, n = 124

(c) Comparison for H16, n = 63 (d) Comparison for S8, n = 63

(e) Comparison for H49, n = 342 (f) Comparison for H64, n = 511

Fig. 1. Comparisons on the dimension and minimum distance between one-point and

two-point codes.

Standard mathematical error model is available for both the symmetric

case3,4 and the asymmetric case.7 To define a quantum code Q, we need to

consider the set of error operators that Q can handle. Let α, β ∈ Fq. The



unitary operators X(α) and Z(β) on Cq are

X(α)|ϕ〉 = |ϕ+ α〉 and Z(β)|ϕ〉 = ηTr(〈β,ϕ〉E)|ϕ〉. (3)

Based on (3), for a = (α1, . . . , αn) ∈ Fnq , we can write X(a) = X(α1) ⊗. . . ⊗ X(αn) and Z(a) = Z(α1) ⊗ . . . ⊗ Z(αn) for the tensor product of

n error operators. The set En := X(a)Z(b) : a,b ∈ Fnq is a nice error

basis on Vn. The error group Gn of order pq2n is generated by the matrices

in En and is given by Gn := ηcX(a)Z(b) : a,b ∈ Fnq , c ∈ Fp. Let

E = ηcX(a)Z(b) ∈ Gn. The quantum weight wtQ(E) of E is the number

of coordinates with (αi, βi) 6= (0, 0). The number of X-errors wtX(E) and

the number of Z-errors wtZ(E) in E are given, respectively, by wt(a) and

wt(b).

Definition 1. A q-ary quantum code of length n is a subspace Q of Vn with

dimension K ≥ 1. A quantum code Q of dimension K ≥ 2 is said to detect

d−1 qudits of errors for d ≥ 1 if, for every orthogonal pair |ϕ〉 and |ψ〉 in Qand every E ∈ Gn with wtQ(E) ≤ d− 1, |ϕ〉 and E|ψ〉 are orthogonal. In

this case, we call Q a symmetric quantum code with parameters ((n,K, d))qor [[n, k, d]]q, where k = logq K. Such a quantum code is called pure if |ϕ〉and E|ψ〉 are orthogonal for any (not necessarily orthogonal) |ϕ〉 and |ψ〉in Q and any E ∈ Gn with 1 ≤ wtQ(E) ≤ d− 1. A quantum code Q with

K = 1 is assumed to be pure.

Let dx and dz be positive integers. A quantum code Q in Vn with

dimension K ≥ 2 is called an asymmetric quantum code with parameters

((n,K, dz, dx))q or [[n, k, dz, dx]]q, where k = logq K, if Q detects dx − 1

qudits of X-errors and, at the same time, dz − 1 qudits of Z-errors. That

is, if 〈ϕ|ψ〉 = 0 for |ϕ〉, |ψ〉 ∈ Q, then |ϕ〉 and E|ψ〉 are orthogonal for

any E ∈ Gn such that wtX(E) ≤ dx − 1 and wtZ(E) ≤ dz − 1. Such an

asymmetric quantum code Q is pure if |ϕ〉 and E|ψ〉 are orthogonal for

any |ϕ〉, |ψ〉 ∈ Q and any E ∈ Gn such that 1 ≤ wtX(E) ≤ dx − 1 or

1 ≤ wtZ(E) ≤ dz − 1. Any code Q with K = 1 is assumed to be pure.

Remark 1. An asymmetric quantum code with parameters ((n,K, d, d))qis symmetric with parameters ((n,K, d))q, but the converse is not true since,

for E ∈ Gn with wtX(E) ≤ d − 1 and wtZ(E) ≤ d − 1, wtQ(E) may be

bigger than d− 1.

It is well-known that quantum codes can be constructed from classical

codes. We will use the following three constructions tailored to classi-

cal codes from Hermitian and Suzuki curves. The first two, presented in



Lemma 3 and Lemma 4, yield symmetric codes while the last, given in

Lemma 5, produces asymmetric ones.

Lemma 3. (CSS Construction) [4, Lemma 20]

Let Ci be an [n, ki, di]q-code for i = 1, 2 with C⊥E1 ⊆ C2. Then there is a

symmetric quantum code Q with parameters

[[n, k1 + k2 − n,minwt(C2 \ C⊥E1 ),wt(C1 \ C⊥E

2 )]]q

which is pure whenever minwt(C2 \ C⊥E1 ),wt(C1 \ C⊥E

2 ) = mindi. If

C ⊆ C⊥E where C is an [n, k, d]q-code, then Q has parameters

[[n, n− 2k,wt(C⊥E \ C)]]q.

It is pure whenever d⊥E = wt(C⊥E \ C).

If, instead of the Euclidean, we use the Hermitian inner product, then

there is a construction of a q-ary stabilizer quantum code from a Hermitian

self-orthogonal code C ⊆ Fnq2 . Notice that the classical codes are over Fq2while the resulting quantum codes are q-ary. In particular, we can not apply

this construction to Suzuki codes since they are over Fq with q = 22s+1,

which is never a square.

Lemma 4. (Stabilizer Construction) [4, Corollary 19]

Let C be an [n, k, d]q2-code such that C ⊆ C⊥H . Then there exists a quan-

tum code Q with parameters [[n, n−2k,wt(C⊥H \C)]]q which is pure when-

ever wt(C⊥H \ C) = d⊥H .

The CSS construction extends to the AQCs. For any q we can use

the Euclidean inner product. Either the Euclidean or the Hermitian inner

product can be used if the underlying field is a square since, over Fq2 , C⊥E

and C⊥H share the same MacWilliams transform, making d⊥E = d⊥H .

Lemma 5. (CSS-like Construction for AQC) [32, Theorem 3.5]

Let Ci be an [n, ki, di]q-code for i = 1, 2. Let C⊥∗1 ⊆ C2. Let

dz := maxwt(C2 \ C⊥∗1 ),wt(C1 \ C⊥∗2 ) and

dx := minwt(C2 \ C⊥∗1 ),wt(C1 \ C⊥∗2 ).

Then there exists an AQC Q with parameters [[n, k1 + k2 − n, dz, dx]]q. It

is pure whenever dz, dx = d1, d2. If we have C ⊆ C⊥∗ where C is an

[n, k, d]q-code, then Q is an [[n, n−2k, d′, d′]]q-code where d′ = wt(C⊥∗ \C).

The code Q is pure whenever d′ = d⊥∗ .



5. Improved Quantum Codes

One can apply Lemma 3 on Euclidean self-orthogonal two-point Hermi-

tian codes from Corollary 1 and two-point Suzuki codes from Corollary 3.

Deriving the parameters of the resulting quantum codes is almost trivial.

Lemma 4 states a construction that yields (symmetric) q-ary quantum

codes instead of q2-ary. To use the construction we need the following re-

sult about the dual codes with respect to the Hermitian inner product. The

one-point version of the proposition was proved by Sarvepalli and Klappe-

necker.11 A similar proposition appears in [9, Proposition 12.23], but the

statements differ due to technical details.

Proposition 5. Let i ≤ q2−2 and 1 ≤ j. Then, under the Hermitian inner

product, a two-point Hermitian code CL(D, iP∞ − jP0) is self-orthogonal.

Proof. Since CL(D, iP∞ − jP0) ⊆ CL(D, iP∞ − P0), it suffices to prove

the theorem with j = 1. By Lemma 2 we know that a basis for the two-

point vector space can be obtained by monomial evaluation. Codewords

which are Hermitian dual to xayb(P∞) are Euclidian dual to words of the

form xqayqb(P∞) which live in CL(D, qiP∞). Adding −P0 to the divisor

removes only the constants and any non-constant monomial to the qth power

is also non-constant. Thus the Hermitian dual of CL(D, iP∞−P0) contains

CL(D, qiP∞−P0)⊥E . Under the degree assumption we can use Corollary 1

to show that CL(D, qG) is Euclidean self-orthogonal. Hence the original

code CL(D,G) is Hermitian self-orthogonal.

Unfortunately, this requirement is too restrictive on the range of G. Due

to the small degree of G, the dual code is outside the range of improvements

given in Proposition 3. Thus, for this particular construction, two-point

codes do not improve on one-point codes. As we have noted, the Hermitian

inner product is not well-defined for the Suzuki codes.

Knowing that there are ranges in which two-point Hermitian and Suzuki

codes exceed their respective one-point counterparts leads us to two basic

options in implementing Lemma 5. One can select C1 and C2 in the im-

proved range. The resulting quantum codes are pure, with high dimension

but with relatively low dx and dz. Alternatively, one may opt for C1 and

C⊥?2 in the improved range. The derived quantum codes have small di-

mension, high dz, low dx, and are often impure. The high ratio of dz over

dx suggests better alignment with observations from actual physical qubit

(q = 2) systems,5 although care remains to be exercised when extrapolating



from q = 2 to higher values.

We begin by applying Lemma 5 with C1 and C⊥E2 in the range of im-

provement. This construction produces pure long quantum codes with ex-

cellent parameters. The next proposition provides the parameters of the

resulting quantum codes in the case of the Hermitian curves.

Proposition 6. Given the curve Hq2 , let 0 ≤ r1 ≤ r2 ≤ q2 + q. For q ≥ 4,

there exists a pure AQC with parameters

[[q3 − 1, q3 − q(q − 1)− (r1 + r2) + 1, d(r2), d(r1)]]q2

where d(r1) and d(r2) are computed according to Corollary 2.

Proof. We apply Lemma 5 on

C1 = CL(D, (q3 − r1 + 1− (q + 1))P∞ + (q − 1)P0) and

C2 = CL(D, (q3 − r2 + 1)P∞ − 2P0).

The nestedness C⊥E1 ⊆ C2 is guaranteed by Corollary 1, given the range

ri ≤ q2 +q. The minimum distance can be computed from Corollary 2 since

C1 is equivalent to a code having a divisor of the form G = iP∞− 2P0. By

the degree bounds, d(C⊥Ei

)= q3 − 1 − r − (q − 2)(q + 1). If q ≥ 4, this

value is larger than d(ri) for the given range. Thus the derived quantum

code is pure.

Let the designed distance δ be fixed. Table 2 gives the best dimensions

obtainable from the classical one-point and two-point codes. They are based

on Proposition 3, along with the design parameter r used in Corollary 2,

for the Hermitian curves. For the Suzuki S8 curve the numerical values are

computed directly from the DK bound. There are 13 two-point codes for

a given degree. For each 0 ≤ δ ≤ 27 there are 2gm = 364 two-point codes.

The respective data and the source code are available online.33,34

By Proposition 6, the inner and the outer codes can be independently

selected to be optimal when constructing an AQC, as long as they are within

the specified range. This effectively doubles the gain when switching to two-

point codes, one from the better inner codes and the other from the better

outer codes.

Example 1. The best 16-ary AQC with dz = 9 and dx = 5 we can con-

struct has parameters [[63, 39, 9, 5]]16 if only one-point Hermitian codes are

used. Using two-point Hermitian codes, the value of k increases to 42.



Table 2. Dimensions of 1- and 2-point codes on the specified curves

H16 with n = 63 S8 with n = 63

No. δ Best Dimension r No. δ Best Dimension

1-point 2-point 1-point 2-point

1 5 53 55 3 1 5 50 532 7 52 53 5 2 7 47 49

3 9 49 50 8 3 9 41 44

4 11 47 48 10 4 11 39 425 13 37 40

6 15 36 37

7 17 33 348 19 31 32

9 21 29 3010 23 27 2811 25 25 26

12 27 23 24

H64 with n = 511

No. δ Best Dimension r No. δ Best Dimension r1-point 2-point 1-point 2-point

1 9 475 481 3 13 33 451 454 30

2 11 474 481 3 14 35 449 454 30

3 13 474 481 3 15 37 447 450 344 15 474 475 9 16 39 447 448 36

5 17 467 472 12 17 41 443 445 39

6 19 465 472 12 18 43 441 443 417 21 465 472 12 19 45 439 441 43

8 23 465 466 18 20 47 438 439 459 25 459 463 21 21 49 435 436 48

10 27 457 463 21 22 51 433 434 50

11 29 456 459 25 23 53 431 432 5212 31 456 457 27 24 55 429 430 54

We now consider the second option. Recent results on the minimum dis-

tance bounds produce better bounds for the cosets.21 A particular feature

of the coset bounds on the Hermitian curve is that they are non-monotonic.

This lack of monotonicity can be exploited to produce excellent impure

AQCs based on Lemma 5. In terms of the parameters of the quantum

codes, our analysis is from their coset bound properties, not from the di-

mension and distance parameters of the classical ingredient codes. This

more refined approach is novel.

Example 2. For the Hermitian curve H64 we simulated all possible pairs



of two-point divisors (up to equivalence) G1 ≤ G2 of degrees 0 ≤ deg(G1) ≤deg(G2) ≤ q(q−1) and calculated the parameters of the impure asymmetric

quantum code constructed based on the nested pair C1 = CΩ(D,G2) ⊆CΩ(D,G1) = C2.

The DP and DK bounds are sufficient to calculate dz = wt(C2 \C1). A

brief high level explanation has been given at the end of Section 3 above.

The complete details can be found in the work of Duursma et al.21 Note

that dx = wt(C⊥E1 \C⊥E

2 ) = n− deg(G1) + 2g+ 2 since it falls in the range

where it can be completely determined by the Riemann-Roch Theorem.

This assertion, in fact, follows from the degree bounds for divisors and

rational functions, i.e., the number of zeros of a polynomial is at most

its degree. To find the exact improvement, the parameters of the best

AQCs derivable from one-point codes, i.e.codes with divisors G1 = iP∞ and

G2 = jP∞, were stored separately and then compared with the parameters

of the AQCs constructed from nested two-point codes.

Based on the computational data, Table 3 presents all two-point Her-

mitian codes which strictly improve on Hermitian one-point codes. The re-

sulting quantum codes are of parameters [[511, k, dz, dx]]64. Similar compu-

tations can be performed on other curves by utilizing the online tools. The

required values for S8 and S32 can also be found, respectively, in [21, Table

6] and [30, Table 3].

In cases where we obtain good impure codes, relation to quantum hybrid

codes35 is interesting to study. To obtain good quantum hybrid codes one

can start by developing methods to construct good non-trivial impure codes.

In the qudit cases covered by Hermitian curves, with well-chosen classical

ingredients, our approach above appear to be an excellent fit.

6. Conclusions and Open Problems

This work uses Hermitian and Suzuki two-point codes as the ingredient

classical codes to construct improved quantum codes. The main steps in the

construction of the classical AG codes as well as in deriving the parameters

of the quantum codes have all been previously well-established.

Our contribution lies in combining the relevant steps from both frame-

works into a unified approach. In the process we identify the range of

improvement when we move into the quantum setup. The stabilizer con-

struction does not, in general, yield significant improvement going from

one-point to two-point codes. The same occurs when we apply the CSS

construction of symmetric quantum codes. It is in the CSS-like construc-



Table 3. Best AQCs from two-point codes on H64 with n = 511. Im-

provement is measured by adding the gain to the closest one-point code

in terms of k, dx, and dz .

Best 2-point Closest 1-point Total G1 G2

(k, dz , dx) (k, dz , dx) Gain

(1, 470, 11) (1, 470, 10) 1 35P∞ + 5P0 35P∞ + 6P0

(1, 471, 10) (1, 470, 10) 1 34P∞ + 5P0 34P∞ + 6P0

(2, 469, 11) (2, 469, 10) 1 35P∞ + 5P0 35P∞ + 7P0

(2, 470, 10) (1, 470, 10) 1 34P∞ + 5P0 34P∞ + 7P0

(2, 486, 5) (2, 486, 4) 1 17P∞ + 6P0 17P∞ + 8P0

(2, 487, 4) (2, 486, 4) 1 16P∞ + 6P0 17P∞ + 7P0

(3, 460, 14) (3, 460, 12) 2 44P∞ + 4P0 44P∞ + 7P0

(3, 461, 13) (3, 460, 12) 2 43P∞ + 4P0 43P∞ + 7P0

(3, 463, 12) (3, 460, 12) 3 41P∞ + 4P0 43P∞ + 5P0

(3, 477, 7) (3, 477, 5) 2 26P∞ + 5P0 26P∞ + 8P0

(3, 479, 6) (3, 477, 5) 3 24P∞ + 5P0 26P∞ + 6P0

(3, 486, 4) (2, 486, 4) 1 16P∞ + 6P0 17P∞ + 8P0

(4, 462, 12) (3, 460, 12) 3 41P∞ + 4P0 43P∞ + 6P0

(4, 468, 9) (4, 468, 6) 3 35P∞ + 4P0 35P∞ + 8P0

(4, 471, 8) (4, 468, 6) 5 32P∞ + 4P0 35P∞ + 5P0

(4, 478, 6) (3, 477, 5) 3 24P∞ + 5P0 26P∞ + 7P0

(5, 461, 12) (3, 460, 12) 3 41P∞ + 4P0 43P∞ + 7P0

(5, 463, 10) (5, 459, 7) 7 40P∞ + 3P0 44P∞ + 4P0

(5, 470, 8) (4, 468, 6) 5 32P∞ + 4P0 35P∞ + 6P0

(5, 477, 6) (5, 476, 5) 2 24P∞ + 5P0 26P∞ + 8P0

(6, 462, 10) (5, 459, 7) 7 40P∞ + 3P0 44P∞ + 5P0

(6, 469, 8) (6, 467, 6) 4 32P∞ + 4P0 35P∞ + 7P0

(7, 461, 10) (7, 458, 7) 6 40P∞ + 3P0 44P∞ + 6P0

(7, 468, 8) (6, 467, 6) 4 32P∞ + 4P0 35P∞ + 8P0

(8, 460, 10) (7, 458, 7) 6 40P∞ + 3P0 44P∞ + 7P0

(9, 459, 10) (7, 458, 7) 6 40P∞ + 3P0 44P∞ + 8P0

tion of asymmetric quantum codes that marked improvement takes place.

The ease of implementation of the order bounds21 help us tremendously

here. We can derive the parameters of the resulting quantum codes from

the coset bound properties, yielding exact values of the distances, unlike

the usual route of providing only their lower bounds.

Combining recent studies on quantum codes and AG codes reveals sev-

eral insights. The search for improved quantum codes provides a strong

motivation to deepen our understanding of t-point AG codes. We have

demonstrated that strict improvements are obtained in specific ranges of

values in the case of Hermitian and Suzuki curves for t = 2. It is very

likely that the same takes place for many other families of curves such as

Castle curves,36 curves over Kummer Extensions,37 Giulietti-Korchmaros



curves,38,39 Deligne-Lusztig curves,40 as well as Garcia-Guneri-Stichtenoth

curves.41 The geometry of these curves are relatively well-studied although,

in most cases, a significant amount of work remains to be done in deter-

mining the exact minimum distance.

Another important direction is to consider if the strict gain extends to

t ≥ 3-point codes. Results on the designed distance and the Weierstrass gap

on Hermitian codes from higher degree places have been derived42 and can

be built upon. Others may opt to go into generalized algebraic geometry

codes where many directions remain open.

References

1. V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR. 259(6),1289–1290 (1981).

2. M. Tsfasman, S. Vladut, and D. Nogin, Algebraic geometric codes: Basicnotions. vol. 139, Mathematical Surveys and Monographs, American Mathe-matical Society, Providence, RI (2007).

3. A. Ashikhmin and E. Knill, Nonbinary quantum stabilizer codes, IEEETrans. Inf. Theory. 47(7), 3065–3072 (Nov, 2001).

4. A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, Nonbinarystabilizer codes over finite fields, IEEE Trans. Inf. Theory. 52(11), 4892–4914 (Nov, 2006).

5. L. Ioffe and M. Mezard, Asymmetric quantum error-correcting codes, Phys.Rev. A. 75, 032345 (Mar, 2007).

6. A. Steane, Multiple-particle interference and quantum error correction, Pro-ceedings of the Royal Society of London A: Mathematical, Physical and En-gineering Sciences. 452(1954), 2551–2577 (1996).

7. L. Wang, K. Feng, S. Ling, and C. Xing, Asymmetric quantum codes: Char-acterization and constructions, IEEE Trans. Inf. Theory. 56(6), 2938–2945(June, 2010).

8. P. K. Sarvepalli, A. Klappenecker, and M. Rotteler, Asymmetric quantumcodes: constructions, bounds and performance, Proceedings of the Royal So-ciety of London A: Mathematical, Physical and Engineering Sciences. 465(2105), 1645–1672 (2009).

9. J.-L. Kim and G. L. Matthews, Quantum error-correcting codes from alge-braic curves, vol. Advances in Algebraic Geometry Codes, Coding Theoryand Cryptology, Chapter 12, pp. 419–444. World Scientific (2008).

10. J.-L. Kim and J. Walker, Nonbinary quantum error-correcting codes fromalgebraic curves, Discrete Mathematics. 308(14), 3115 – 3124 (2008).

11. P. K. Sarvepalli and A. Klappenecker, Nonbinary quantum codes from Her-mitian curves, In eds. M. P. C. Fossorier, H. Imai, S. Lin, and A. Poli,Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 16th In-ternational Symposium, AAECC-16, Las Vegas, NV, USA, February 20-24,



2006. Proceedings, pp. 136–143. Springer Berlin Heidelberg, Berlin, Heidel-berg (2006).

12. L. Jin, Quantum stabilizer codes from maximal curves, IEEE Trans. Inf.Theory. 60(1), 313–316 (Jan, 2014).

13. J. Bierbrauer. Some good quantum twisted codes. Online avail-able at https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/

QTBCHIndex.html (2010). Accessed on 2019-05-23.14. K. Feng, S. Ling, and C. Xing, Asymptotic bounds on quantum codes from

algebraic geometry codes, IEEE Trans. Inf. Theory. 52(3), 986–991 (March,2006).

15. G. L. Matthews, Weierstrass pairs and minimum distance of Goppa codes,Des. Codes Cryptogr. 22(2), 107–121 (2001).

16. M. Homma and S. J. Kim, The complete determination of the minimumdistance of two-point codes on a Hermitian curve, Des. Codes Cryptogr. 40(1), 5–24 (2006).

17. P. Beelen, The order bound for general algebraic geometric codes, FiniteFields Appl. 13(3), 665–680 (2007).

18. S. Park, Minimum distance of Hermitian two-point codes, Designs, Codesand Cryptography (January, 2010).

19. I. M. Duursma and R. Kirov, Improved two-point codes on Hermitian curves,IEEE Trans. Inf. Theory. 57(7), 4469–4476 (July, 2011).

20. H. Stichtenoth, Algebraic function fields and codes, 2nd ed. vol. 254, GraduateTexts in Mathematics, Springer-Verlag, Berlin (2009).

21. I. Duursma, R. Kirov, and S. Park, Distance bounds for algebraic geometriccodes, Journal of Pure and Applied Algebra. 215(8), 1863 – 1878 (2011).

22. R. Fuhrmann and F. Torres, On Weierstrass points and optimal curves, Ren-diconti del Circolo Matematico di Palermo Suppl.. 51(1), 25–46 (1998).

23. M. Giulietti and G. Korchmaros, On automorphism groups of certain Goppacodes, Des. Codes Cryptogr.. 47(1), 177–190 (Jun, 2008).

24. H. J. Tiersma, Remarks on codes from Hermitian curves, IEEE Trans. Inf.Theory. 33(4), 605–609 (1987).

25. H. Stichtenoth, A note on Hermitian codes over GF(q2), IEEE Trans. Inf.Theory. 34(5, part 2), 1345–1348 (1988).

26. M. Homma and S. J. Kim, Toward the determination of the minimum dis-tance of two-point codes on a Hermitian curve, Des. Codes Cryptogr. 37(1),111–132 (2005).

27. G. L. Matthews, Codes from the Suzuki function field, IEEE Trans. Inf.Theory. 50(12), 3298–3302 (2004).

28. F. Torres, Algebraic curves with many points over finite fields, vol. Advancesin Algebraic Geometry Codes, Coding Theory and Cryptology, Chapter 6, pp.221–256. World Scientific, Singapore (2008).

29. A. Eid, H. Hasson, A. Ksir, and J. Peachey, Suzuki-invariant codes from theSuzuki curve, Des. Codes Cryptogr.. 81(3), 413–425 (Dec, 2016).

30. I. Duursma and R. Kirov. An extension of the order bound for AG codes. Ineds. M. Bras-Amoros and T. Høholdt, Applied Algebra, Algebraic Algorithmsand Error-Correcting Codes, pp. 11–22, Springer, Berlin (2009).



31. J. W. P. Hirschfeld, G. Korchmaros, and F. Torres, Algebraic Curves over aFinite Field, Princeton Ser. in Applied Mathematics, Princeton Univ. Press,New Jersey (2013).

32. M. F. Ezerman, S. Jitman, S. Ling, and D. V. Pasechnik, CSS-like con-structions of asymmetric quantum codes, IEEE Trans. Inf. Theory. 59(10),6732–6754 (oct, 2013).

33. R. Kirov. Parameters of AG codes. URL http://agtables-1100.appspot.

com/ (Mar. 2011).34. R. Kirov. Scripts for computing bounds on error-correcting codes from curves.

URL https://github.com/rkirov/code_bounds (Sept. 2018).35. M. Grassl, S. Lu, and B. Zeng, Codes for simultaneous transmission of quan-

tum and classical information. In 2017 IEEE International Symposium onInformation Theory (ISIT), pp. 1718–1722 (June, 2017).

36. C. Munuera, A. Sepulveda, and F. Torres, Castle curves and codes, Advancesin Mathematics of Communications. 3(4), 399–408 (Nov, 2009).

37. A. S. Castellanos, A. M. Masuda, and L. Quoos, One- and two-point codesover Kummer extensions, IEEE Trans. Inf. Theory. 62(9), 4867–4872 (Sept,2016).

38. M. Giulietti and G. Korchmaros, A new family of maximal curves over afinite field, Mathematische Annalen. 343(1), 229–245 (Aug, 2008).

39. A. S. Castellanos and G. C. Tizziotti, Two-point AG codes on the GK max-imal curves, IEEE Trans. Inf. Theory. 62(2), 681–686 (Feb, 2016).

40. P. Deligne and G. Lusztig, Representations of reductive groups over finitefields, The Annals of Mathematics. 103(1), 103–161 (Jan, 1976).

41. A. Garcia, C. Guneri, and H. Stichtenoth, A generalization of the Giuli-etti–Korchmaros maximal curve, Advances in Geometry. 10(3) (Jan, 2010).

42. G. Korchmros and G. Nagy, Hermitian codes from higher degree places,Journal of Pure and Applied Algebra. 217(12), 2371 – 2381 (2013).

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Chapter 1 Quantum Codes from Two-Point Divisors on ... · The theory of algebraic geometry (AG) codes has its origin in the mathe-matics of algebraic curves. At the core of which