On a generalization of the Deligne–Lusztig curve of Suzuki type and application to AG codes

Marco Timpanella

doi:10.1515/jmc-2023-0024

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On a generalization of the Deligne–Lusztig curve of Suzuki type and application to AG codes

Marco Timpanella

Published/Copyright: February 15, 2024

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From the journal Journal of Mathematical Cryptology Volume 18 Issue 1

Abstract

In this article, Algebraic-Geometric (AG) codes and quantum codes associated with a family of curves that includes the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turns out to be a Castle curve over some finite field and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered.

Keywords: AG codes; quantum codes; Deligne–Lusztig curves

MSC 2010: 94B27

1 Introduction

Let F q be a finite field with q elements, where q is a power of a prime p . An algebraic curve over F q is a projective, absolutely irreducible, non-singular, algebraic variety of dimension 1 defined over F q . Among algebraic curves over finite fields (i.e., projective, absolutely irreducible, non-singular, algebraic varieties of dimension 1 defined over a finite field F q ), a prominent role is played by the so-called Deligne–Lusztig curves associated with the Projective Special Unitary group PSU ( 3 , q ) , the Suzuki group, and the Ree group. In fact, these curves are exceptional both for being optimal with respect to the number of F ℓ -rational points for some ℓ and for having a very large automorphism group with respect to their genus.

Curves possessing a large number of rational points hold significant interest both in their own right and for their applications in Coding Theory. Goppa’s work [1] introduced a fundamental concept: linear codes (the so-called Algebraic-Geometric [AG] codes) can be derived from an algebraic curve X defined over F q by evaluating specific rational functions. These functions are chosen in such a way that their poles align with a given F q -rational divisor G , while the evaluation is performed at a distinct F q -rational divisor D whose support is disjoint from that of G . AG codes are proven to have good performances provided that X , G , and D are carefully chosen in an appropriate way. In particular, as the relative Singleton defect of an AG code from a curve X is upper bounded by the ratio g ∕ N , where g is the genus of X and N can be as large as the number of F q -rational points of X , it follows that curves with many rational points with respect to their genus are of great interest in Coding Theory. In particular, AG codes from maximal curves (namely, curves with the maximum possible number of rational points) have been widely investigated in the last years, see, e.g., [2–7] and the references therein.

In this article, we investigate a generalization of the Deligne–Lusztig curve of Suzuki type originally defined in the study by Giulietti and Korchmáros [8], where it was noted that the number of automorphisms exceeds the Hurwitz bound. Our main original contribution is the investigation of the Weierstrass semigroup of the curve at a specific point, see Propositions 5 and 9, which leads to the proof that the curve is actually a Castle curve over F q and a weak Castle curve over F q i for all i ≥ 1 . In this article, we also provide the proofs of some facts that are stated in the study by Giulietti and Korchmáros [8] without proofs. Both Castle and weak Castle curves are of particular interest in the context of applications of curves to linear codes. In fact, they combine the good properties of having a reasonable simple handling and giving codes with excellent parameters. In addition, these codes have self-orthogonality properties, which are very close to those required for obtaining quantum stabilizer codes [9–11]. As an application of the curve being Castle, we provide a construction of quantum codes associated with the curve, see Proposition 10 and the discussion at the end of Section 5.2.

2 Background on algebraic curves and AG codes

For a curve X , we adopt the usual notation and terminology [12,13]. In particular, F q ( X ) and X ( F q ) denote the field of F q -rational functions on X and the set of F q -rational points of X , respectively, and Div ( X ) denotes the set of divisors of X , where a divisor D ∈ Div ( X ) is a formal sum n 1 P 1 + ⋯ + n r P r , with P i ∈ X , n i ∈ Z , and P i ≠ P j if i ≠ j . The support Supp ( D ) of the divisor D is the set of points P i such that n i ≠ 0 , while deg ( D ) = ∑ i n i f i is the degree of D , where f i is the degree of P i . The divisor D is F q -rational if it is invariant under the F q -Frobenius action. For a function f ∈ F q ( X ) , ( f ) , ( f ) 0 , and ( f ) ∞ , are the divisor of f , its divisor of zeros, and its divisor of poles, respectively. The Weierstrass semigroup H ( P ) at P ∈ X is

H ( P ) ≔ { n ∈ N 0 ∣ ∃ f ∈ F q ( X ) , ( f ) ∞ = n P } = { ρ 0 = 0 < ρ 1 < ρ 2 < ⋯ } .

The Riemann–Roch space associated with an F q -rational divisor D is

ℒ ( D ) ≔ { f ∈ X ( F q ) : ( f ) + D ≥ 0 } ∪ { 0 } ,

and its vector space dimension over F q is ℓ ( D ) .

Fix a set of pairwise distinct F q -rational points { P 1 , … , P N } , and let D = P 1 + ⋯ + P N . Take another F q -rational divisor G whose support is disjoint from the support of D . The AG code C ( D , G ) is the (linear) subspace of F q N , which is defined as the image of the evaluation map e v : ℒ ( G ) → F q N given by e v ( f ) = ( f ( P 1 ) , f ( P 2 ) , … , f ( P N ) ) . In particular, C ( D , G ) has length N . Moreover, if N > deg ( G ) , then e v is an embedding and ℓ ( G ) equals the dimension of C ( D , G ) . The minimum distance d of C ( D , G ) usually depends on the choice of D and G . A lower bound for d is d * = N − deg ( G ) , where d * is called the Goppa designed minimum distance of C ( D , G ) . Furthermore, if deg ( G ) > 2 g − 2 , then k = deg ( G ) − g + 1 by the Riemann–Roch theorem [14, Theorem 2.65].

The dual code C ⊥ ( D , G ) can be obtained in a similar way from the F q ( X ) -vector space Ω ( X ) of differential forms over X . For a differential ω ∈ Ω ( X ) , there is associated a divisor ( ω ) of X , whose degree is 2 g − 2 . For an F q -rational divisor D ,

Ω ( D ) ≔ { ω ∈ Ω ( X ) : ( ω ) ≥ D } ∪ { 0 }

is a F q -vector space of rational differential form over X . Then, the code C ⊥ ( D , G ) coincides with the (linear) subspace of F q N , which is the image of the vector space Ω ( G − D ) under the linear map res D : Ω ( G − D ) ↦ F q N given by res D ( ω ) = ( res P 1 ( ω ) , … , res P N ( ω ) ) , where res P i ( ω ) is the residue of ω at P i . In particular, C ⊥ ( D , G ) is an AG code with dimension k ⊥ = N − k and minimum distance d ⊥ ≥ deg ( G ) − 2 g + 2 .

In the case where G = α P , α ∈ N 0 , and P ∈ X ( F q ) , the AG code C ( D , G ) is referred to as one-point AG code. For a Weierstrass semigroup H ( P ) = { ρ 0 = 0 < ρ 1 < ρ 2 < ⋯ } and an integer ℓ ≥ 0 , the Feng–Rao function is

ν ℓ ≔ ∣ { ( i , j ) ∈ N 0 2 : ρ i + ρ j = ρ ℓ + 1 } ∣ .

Consider

C ℓ ( P ) = C ⊥ ( P 1 + P 2 + ⋯ + P N , ρ ℓ P ) ,

with P , P 1 , … , P N pairwise distinct points in X ( F q ) . The number

d ORD ( C ℓ ( P ) ) ≔ min { ν m : m ≥ ℓ }

is a lower bound for the minimum distance d ( C ℓ ( P ) ) of the code C ℓ ( P ) , which is called the order bound or the Feng-Rao designed minimum distance of C ℓ ( P ) ; see [14, Theorem 4.13].

3 Preliminaries

Throughout the article, q = 2 s and q 0 = 2 h with 2 h < s . In addition, q ¯ = q ∕ q 0 and n 1 ≔ q ¯ ∕ q 0 . Let C be the plane curve defined over F q by the equation

X q 0 ( X q + X ) = Y q + Y .

Furthermore, let

(1) v ( X , Y ) ≔ Y q ¯ + X q ¯ + 1 , w ( X , Y ) ≔ Y q ¯ X n 1 − 1 + v ( X , Y ) q ¯ .

Note for s odd and 2 h + 1 = s , the curve C is the Deligne–Lusztig curve of Suzuki type.

The condition 2 h < s is motivated as follows. For 2 h = s , i.e., q 0 = q ¯ = q , the curve X q 0 ( X q + X ) = Y q + Y is reducible as X q 0 ( X q + X ) + Y q + Y = Π α q 0 = α ( X q 0 + 1 + Y q 0 + Y + α ) . For 2 h > s , the curve X 2 h ( X q + X ) = Y q + Y is birationally equivalent to X ′ q ∕ 2 h ( X ′ q + X ′ ) = Y q + Y by setting X ′ = Y q ∕ 2 h + X ( q ∕ 2 h ) + 1 .

Proposition 1

(Proposition 6.7 in [8]) The curve C is absolutely irreducible. Moreover, there exists a unique place of F q ( C ) centered at Y ∞ , the infinite point of the Y-axis.

Proof

Let X be any component of C , with equation H ( X , Y ) = 0 for some irreducible factor H ( X , Y ) of X q 0 ( X q + X ) − Y q − Y . Then, F q ( X ) = F q ( x , y ) , with

(2) x q 0 ( x q + x ) = y q + y .

From

( y q + y ) x q 0 ( n 1 − 1 ) = x q + n 1 q 0 + x n 1 q 0 + 1 ,

it follows that

( y q ¯ x n 1 − 1 + x n 1 + q ¯ ) q 0 = x n 1 q 0 + 1 + y x q 0 ( n 1 − 1 ) ,

hence

(3) ( x n 1 + q ¯ ) q 0 + x n 1 q 0 + 1 = ( y q ¯ x n 1 − 1 ) q 0 + y x q 0 ( n 1 − 1 ) .

Now define v ≔ v ( x , y ) and w ≔ w ( x , y ) as in equation (1). Then, by straightforward, computation,

(4) v q + v = x q ¯ ( x q + x ) .

By equation (3),

( y q ¯ x n 1 − 1 ) q 0 + y x q 0 ( n 1 − 1 ) = v q ¯ q 0 + v ,

hence

(5) w q 0 = y x q 0 ( n 1 − 1 ) + v .

Now, let P be any place of F q ( x , y ) centered at Y ∞ . Let s ≔ v P ( x ) . By equation (2),

(6) − q ≤ s < 0 .

Note that showing s = − q is enough to prove both the statements in the proposition. By equations (2) and (4), it follows that

(7) − q 0 ≤ s < 0 s ( q 0 + q ) = q v P ( y ) , s ( q ¯ + q ) = q v P ( v ) .

In particular, q ¯ divides s . By equation (5),

q 0 v P ( w ) ≥ min { v P ( y ) + q 0 ( n 1 − 1 ) v P ( x ) , v P ( v ) } ,

i.e.,

q 0 v P ( w ) ≥ min s q 0 + q q + q 0 ( n 1 − 1 ) s , s q ¯ + q q .

Since s < 0 and n 1 > 1 , s q 0 + q q + q 0 ( n 1 − 1 ) s < s q ¯ + q q holds, hence

q 0 v P ( w ) = s q 0 + q q + q 0 ( n 1 − 1 ) s .

This implies that q 0 divides s q ¯ , which together with equation (7) yields q ∣ s . Finally, s = − q follows from equation (6), and the proposition is proved.□

According to the proof of Proposition 1, from now on, x and y denote the algebraic functions in F q ( C ) such that F q ( C ) = F q ( x , y ) with x q 0 ( x q + x ) = y q + y . Moreover, let P ∞ be the only place of F q ( C ) centered at Y ∞ . Finally, we set v = v ( x , y ) and w = w ( x , y ) . The following statement follows from the proof of Proposition 1.

Proposition 2

In F q ( C ) ,

v P ∞ ( x ) = − q , v P ∞ ( y ) = − q 0 − q ;
v P ∞ ( v ) = − q ¯ − q ;
v P ∞ ( w ) = − ( q ( n 1 − 1 ) + q ¯ + 1 ) .

Corollary 1

The rational function v x n 1 − 2 ∕ w ∈ F q ( C ) is a local parameter at P ∞ .

Proposition 3

(Proposition 6.8 in [8]) The genus of C is g C = 1 2 q ¯ ( q − 1 ) .

Proof

We apply Hilbert’s different formula to the extension F q ( C ) ∕ F q ( x ) . It is easy to see that F q ( C ) ∕ F q ( x ) is a Galois extension. Its Galois group Γ ( F q ( C ) ∕ F q ( x ) ) consists of the automorphisms δ a , where

(8) δ a ≔ x ↦ x , y ↦ y + a ,

with a ranging over F q . Then, the Hurwitz’s genus formula gives

(9) 2 g C − 2 = − 2 q + d ( P ∞ ∣ Q ) ,

where Q being the infinite place of F q ( x ) . For a ∈ F q , we compute v P ∞ ( δ a ( t ) − t ) , where t = v x n 1 − 2 ∕ w . By straightforward computation,

δ a ( t ) − t = v x n 1 − 2 + a q ¯ x n 1 − 2 w + a q ¯ x n 1 − 1 + a q ¯ 2 − v x n 1 − 2 w = w ( v x n 1 − 2 + a q ¯ x n 1 − 2 ) − ( w + a q ¯ x n 1 − 1 + a q ¯ 2 ) v x n 1 − 2 ( w + a q ¯ x n 1 − 1 + a q ¯ 2 ) w = a q ¯ ( w x n 1 − 2 − v x 2 n 1 − 3 − a q ¯ v x n 1 − 2 ) ( w + a q ¯ x n 1 − 1 + a q ¯ 2 ) w .

Taking into account Proposition 2, it follows that for δ a ≠ i d ,

v P ∞ ( δ a ( t ) − t ) = q ¯ + 2 .

Hilbert’s different formula yields d ( P ∞ ∣ Q ) = ( q − 1 ) ( q ¯ + 2 ) , hence, by equation (9), the statement is proved.□

Let f be the morphism f ≔ C → P 4 ( F ¯ q ) with coordinate functions

f ≔ ( f 0 : f 1 : f 2 : f 3 : f 4 )

such that f 0 ≔ 1 , f 1 ≔ x , f 2 ≔ v x n 1 − 2 , f 3 ≔ y , and f 4 ≔ w . They are uniquely determined by f up to a proportionality factor in F q ( C ) . For each point P ∈ C , we have f ( P ) = ( ( t − e P f 0 ) ( P ) , … , ( t − e P f 4 ) ( P ) ) , where e P = − min { v P ( f 0 ) , … , v P ( f 4 ) } for a local parameter t of C at P . It turns out that f ( C ) is a curve not contained in any hyperplane of P 4 ( F ¯ q ) . For a point P ∈ f ( C ) , the intersection multiplicity of f ( C ) with a hyperplane H of equation a 0 X 0 + … + a 4 X 4 = 0 is v P ( a 0 f 0 + … + a 4 f 4 ) + e P , and the intersection divisor f − 1 ( H ) cut out on f ( X ) by H is defined to be f − 1 ( H ) = ( a 0 f 0 + … + a 4 f 4 ) + E with E = ∑ e p P . By Proposition 2, we have v P ∞ ( f 1 ) = − q , v P ∞ ( f 2 ) = − ( q ( n 1 − 1 ) + q ¯ ) , v P ∞ ( f 3 ) = − q 0 − q , and v P ∞ ( f 4 ) = − ( q ( n 1 − 1 ) + q ¯ + 1 ) . Then, e P ∞ = q ( n 1 − 1 ) + q ¯ + 1 , and the representative ( f 0 ∕ f 4 : f 1 ∕ f 4 : f 2 ∕ f 4 : f 3 ∕ f 4 : 1 ) of f is defined on P ∞ . Hence, f ( P ∞ ) = ( 0 : 0 : 0 : 0 : 1 ) . For a point P ∈ C , an integer j is called a Hermitian P -invariant if there exists a hyperplane intersecting f ( C ) at f ( P ) with multiplicity j . There are exactly five pairwise distinct Hermitian P -invariants. Such integers arranged in increasing order define the order sequence of C at P .

Proposition 4

f ( C ) is a non-singular model defined over F q of C .

Proof

We showed that f is a closed embedding. By the above discussion, f is bijective and f ( C ) has no singular point.□

3.1 Some automorphisms of F q ( C )

For b , c , d ∈ F q with d ≠ 0 , we define the following automorphisms of F q ( C ) :

α b , c ≔ x ↦ x + b , y ↦ y + b q 0 x + c ;

(10) β d ≔ x ↦ d x , y ↦ d q 0 + 1 y .

Note that α b , c 2 = δ b q 0 + 1 , with δ b q 0 + 1 as in equation (8). Let A , ℬ , and D be the following subgroups of Aut ( C ) :

A ≔ { α b , c ∣ b , c ∈ F q } , ℬ ≔ { β d ∣ d ∈ F q , d ≠ 0 } , D ≔ { δ a ∣ a ∈ F q } .

Let Γ be the automorphism group of F q ( C ) generated by A and ℬ . The number of elements in Γ is at least q 2 ( q − 1 ) , i.e., # Aut ( F q ( X ) ) > 84 ( g C − 1 ) apart from the case q ≤ 16 . The sets { P ∞ } and { C ( F q ) } \ { P ∞ } are two shorts orbits of Aut ( F q ( X ) ) . The former is a non-tame orbit, while the latter is tame.

4 Weierstrass semigroup

The aim of this section is to prove the following result.

Proposition 5

The Weierstrass semigroup at P ∞ is H ( P ∞ ) = ⟨ q , q + q 0 , q + q ¯ , q ( n − 1 ) + q ¯ + 1 ⟩ .

Let A be the numerical semigroup generated by { q , q + q 0 , q + q ¯ , q ( n − 1 ) + q ¯ + 1 } . To prove A = H ( P ∞ ) , we will make use of the following definitions and results from the theory of numerical semigroups.

Definition 1

Let S ⊂ N be a numerical semigroup.

The genus g ( S ) of S is the cardinality of the set N \ S (which, by definition, is finite);
The conductor c ( S ) of S is c ( S ) = 1 + max { x ∈ N \ S } . In addition, S is symmetric if c ( S ) = 2 g ( S ) ;
The multiplicity m ( S ) of S is m ( S ) = min { x ∈ S } ;
For a nonzero element s ∈ S , the Apéry set of s is
A p ( S , s ) ≔ { x ∈ S ∣ x − s ∉ S } .

Note that A p ( S , m ( S ) ) provides a complete set of minimal representatives for the congruence classes of Z modulo m ( S ) . As a consequence, the semigroup can also be described as S = { t m ( S ) + x : t ≥ 0 and x ∈ A p ( S , m ( S ) ) } . A strong connection between the Apéry sets, the genus, and the conductor of a numerical semigroup is given by the following well-known result [15, Proposition 2.12].

Proposition 6

Let S be a numerical semigroup and s be a nonzero element of S. Then, ∣ A p ( S , s ) ∣ = s ,

(11) g ( S ) = 1 s ∑ x ∈ A p ( S , s ) x − s − 1 2 ,

and

(12) c ( S ) = 1 + max { x ∈ A p ( S , s ) } − s .

Observe that if S ¯ ⊂ S is a complete set of representatives for the congruence classes of Z modulo m ( S ) (not necessarily minimal), then

(13) g ( S ) ≤ 1 m ( S ) ∑ x ∈ S ¯ x − m ( S ) − 1 2 ,

and the equality holds if and only if S ¯ = A p ( S , m ( S ) ) .

By Proposition 2, { q , q + q 0 , q + q ¯ , q ( n − 1 ) + q ¯ + 1 } ⊆ H ( P ∞ ) , hence A is contained in H ( P ∞ ) . In particular, g ( A ) ≥ g ( H ( P ∞ ) ) = 1 2 q ¯ ( q − 1 ) . To prove the other inequality, we explicitly compute the Apéry set A p ( A , q ) . Note that q is the multiplicity of A .

Proposition 7

The set

A ¯ ≔ { t 1 ( q + q 0 ) + t 2 ( q + q ¯ ) + t 3 ( q ( n − 1 ) + q ¯ + 1 ) : 0 ≤ t 1 ≤ n − 1 , 0 ≤ t 2 ≤ q 0 − 1 , 0 ≤ t 3 ≤ q 0 − 1 }

is a complete set of representatives for the congruence classes of Z modulo q .

Proof

Clearly, the size of A ¯ is at most n q 0 2 = q . To prove the claim, we show that if a ¯ and a ¯ ′ are two distinct elements of A ¯ , then a ¯ ≢ a ¯ ′ ( mod q ) . Indeed, let

a ¯ = t 1 ( q + q 0 ) + t 2 ( q + q ¯ ) + t 3 ( q ( n − 1 ) + q ¯ + 1 ) , a ¯ ′ = t 1 ′ ( q + q 0 ) + t 2 ′ ( q + q ¯ ) + t 3 ′ ( q ( n − 1 ) + q ¯ + 1 ) ,

and assume a ¯ ≡ a ¯ ′ ( mod q ) . As q 0 divides q , we have a ¯ ≡ a ¯ ′ ( mod q 0 ) , hence t 3 ≡ t 3 ′ ( mod q 0 ) . Since t 3 , t 3 ′ ∈ { 0 , … , q 0 − 1 } , we obtain t 3 = t 3 ′ . The same argument, replacing q 0 with q ¯ , yields a ¯ ≡ a ¯ ′ ( mod q ¯ ) , hence t 1 q 0 ≡ t 1 ′ q 0 ( mod q ¯ ) . Then, q ¯ = n q 0 yields t 1 ≡ t 1 ′ ( mod n ) , which, combined with t 1 , t 1 ′ ∈ { 0 , … , n − 1 } , gives t 1 = t 1 ′ . Finally, t 2 ( q + q ¯ ) ≡ t 2 ′ ( q + q ¯ ) ( mod q ) yields t 2 ≡ t 2 ′ ( mod q 0 ) and so t 2 = t 2 ′ . Therefore, a ¯ = a ¯ ′ , which completes the proof.□

We are now in position to prove Proposition 5.

Proposition 8

H ( P ∞ ) = A .

Proof

As we already observed, A ⊆ H ( P ∞ ) and hence g ( A ) ≥ g ( H ( P ∞ ) ) . On the other hand, Proposition 7 together with equation (13) yields

g ( A ) ≤ 1 q ∑ x ∈ A ¯ x − q − 1 2 .

By straightforward computation, we have

∑ x ∈ A ¯ x = ∑ t 1 = 0 n − 1 ∑ t 2 = 0 q 0 − 1 ∑ t 3 = 0 q 0 − 1 ( t 1 ( q + q 0 ) + t 2 ( q + q ¯ ) + t 3 ( q ( n − 1 ) + q ¯ + 1 ) ) = ∑ t 1 = 0 n − 1 ∑ t 2 = 0 q 0 − 1 t 1 q 0 ( q + q 0 ) + t 2 q 0 ( q + q ¯ ) + q 0 ( q 0 − 1 ) 2 ( q ( n − 1 ) + q ¯ + 1 ) = ∑ t 1 = 0 n − 1 t 1 q 0 2 ( q + q 0 ) + q 0 2 ( q 0 − 1 ) 2 ( q + q ¯ ) + q 0 2 ( q 0 − 1 ) 2 ( q ( n − 1 ) + q ¯ + 1 ) = n ( n − 1 ) q 0 2 2 ( q + q 0 ) + n q 0 2 ( q 0 − 1 ) 2 ( q + q ¯ ) + n q 0 2 ( q 0 − 1 ) 2 ( q ( n − 1 ) + q ¯ + 1 ) = n q 0 2 2 ( q ¯ q − q ¯ + q − 1 ) = q 2 ( q ¯ q − q ¯ + q − 1 ) ,

whence

g ( A ) ≤ 1 2 ( q ¯ q − q ¯ + q − 1 ) − q − 1 2 = q ¯ ( q − 1 ) 2 = g C = g ( H ( P ∞ ) ) .

Therefore, g ( A ) = g ( H ( P ∞ ) ) and A = H ( P ∞ ) .□

Remark 1

By the proof of Proposition 8, g ( A ) = 1 q ∑ x ∈ A ¯ x − q − 1 2 . Therefore, A ¯ is exactly the Apéry set A p ( A , q ) = A p ( H ( P ∞ ) , q ) .

Proposition 9

The Weierstrass semigroup at P ∞ is symmetric.

Proof

By Propositions 6 and 8, together with Remark 1, the conductor of H ( P ∞ ) is

c ( H ( P ∞ ) ) = 1 + max { x ∈ A ¯ } − q ,

i.e.,

c ( H ( P ∞ ) ) = 1 + ( n − 1 ) ( q + q 0 ) + ( q 0 − 1 ) ( q + q ¯ ) + ( q 0 − 1 ) ( q ( n − 1 ) + q ¯ + 1 ) − q = n q 0 q − q ¯ = q ¯ q − q ¯ = 2 g C = 2 g ( H ( P ∞ ) ) ,

whence the claim follows.□

5 AG codes and AG quantum codes

5.1 Number of rational points

By the non-singularity of any affine point in C and by Proposition 1, it follows that the number of F q -rational points of C is N 1 ( C ) = q 2 + 1 . This means that

N 1 ( C ) g C = 2 q 0 + 2 q + 4 q ¯ ( q − 1 ) > 2 q 0 .

By Proposition 5, the smallest positive nongap at the F q -rational point P ∞ is q . Hence, the curve is F q -optimal with respect to the Lewittes bound [16].

Let N i ( C ) be the number of F q i -rational points of C . By computer results, we checked that

q = 16 , q 0 = 2 , g C = 60 : N 3 ( C ) = N 2 ( C ) = N 1 ( C ) = 1 + 256 , N 4 ( C ) = 65537 = q 4 + 1 ;
q = 32 , q 0 = 2 , g C = 248 : N 2 ( C ) = N 1 ( C ) = 1 + 1024 , N 3 ( C ) = 1 + 1024 + 3 × 1024 × 31 = 96257 .

This numerical example shows that there are instances in which the curve has “many” rational points over F q 3 , since N 3 ( C ) exceeds ( 1 ∕ 2 ) ( q 3 + 1 + 2 g C q 3 ) in the second case.

5.2 Quantum codes and Castle property

Let H = ( C q ) ⊗ n = C q ⊗ ⋯ ⊗ C q be a q n -dimensional Hilbert space. Then, the q -ary quantum code C of length n and dimension k are the q k -dimensional Hilbert subspace of H . Such quantum codes are denoted by [ [ n , k , d ] ] q , where d is the minimum distance. As in the ordinary case, C can correct up to ⌊ d − 1 2 ⌋ errors. Moreover, the quantum version of the Singleton bound states that for a [ [ n , k , d ] ] q -quantum code, 2 d + k ≤ 2 + n holds. Again, by analogy with the ordinary case, the quantum Singleton defect and the relative quantum Singleton defect are defined to be δ Q ≔ n − k − 2 d + 2 and Δ Q ≔ δ Q n , respectively.

The CSS construction [17,18], named after Calderbank, Shor, and Steane, showed that quantum codes can be derived from classical linear codes verifying certain self-orthogonality properties.

Lemma 1

(CSS construction) Let C 1 and C 2 be linear codes with parameters [ n , k 1 , d 1 ] q and [ n , k 2 , d 2 ] q , respectively, and assume that C 1 ⊂ C 2 . Then, there exists a [ [ n , k 2 − k 1 , d ] ] q -quantum code with

d = min { w ( c ) ∣ c ∈ ( C 2 \ C 1 ) ∪ ( C 1 ⊥ \ C 2 ⊥ ) } .

Among all the classical codes used to produce quantum codes, AG codes have received considerable attention.

As an application of Lemma 1 to AG codes, La Guardia and Pereira proposed [19] the following general t-point construction.

Lemma 2

[19, Theorem 3.1] (General t -point construction) Let F ∕ F q be an algebraic function field of genus g and with n + t distinct points F q -rational for some n , t > 0 . For every i = 1 , … , t , let a i and b i be positive integers such that a i ≤ b i and

2 g − 2 < ∑ i = 1 t a i < ∑ i = 1 t b i < n .

Then, there exists a [ [ n , k , d ] ] q -quantum code with k = ∑ i = 1 t b i − ∑ i = 1 t a i and

d ≥ min n − ∑ i = 1 t b i , ∑ i = 1 t a i − ( 2 g − 2 ) .

By applying Lemma 2 to the curve C , the following result is obtained.

Proposition 10

Let a , b ∈ N such that

q ¯ ( q − 1 ) − 2 < a < b < q 2 .

Then, there exists a [ [ q 2 , b − a , d ] ] q quantum code, where

d ≥ min { q 2 − b , a − q ¯ ( q − 1 ) + 2 } .

Many of the properties of AG codes that give rise to good quantum codes were captured in the definition of Castle curves and weak Castle curves [10,11].

Definition 2

Let X be a curve defined over F q and Q be an F q -rational place of X . Then, the pair ( X , Q ) is called Castle if the following conditions are satisfied:

The Weierstrass semigroup H ( Q ) is symmetric.
∣ X ( F q ) ∣ = q m ( H ( Q ) ) + 1 .

All the Deligne–Lusztig curves are Castle.

Definition 3

Let X be a curve defined over F q and Q be an F q -rational place of X . Then, the pair ( X , Q ) is called weak Castle if the following conditions are satisfied:

The Weierstrass semigroup H ( Q ) is symmetric;
For some integer ℓ , there exists a morphism f : X → P 1 = F ¯ q ∪ { ∞ } such that ( f ) ∞ = ℓ Q , and there exists a set U = { α 1 , … , α h } ⊆ F q , such that for every i = 1 , … , h , f − 1 ( α i ) ⊆ X ( F q ) and ∣ f − 1 ( α i ) ∣ = ℓ .

Every Castle curve is weak Castle, since the rational function f ∈ ℒ ( Q ) with ( f ) ∞ = m ( H ( Q ) ) Q and U = F q satisfy condition WC2; see [10, Proposition 2.5]. If ( X , Q ) is weak Castle, define

(14) D = ∑ i = 1 h ∑ j = 1 ℓ P j i ,

where f − 1 ( α i ) = { P 1 i , … , P ℓ i } for every i = 1 , … , h .

The one-point AG codes C ( D , r Q ) are called Castle or weak Castle codes. Thanks to the weak Castle condition, these codes can be treated in an unified way. As it was proved, Castle and weak Castle curves provide families of codes with excellent parameters that satisfy certain self-orthogonality properties, making them good candidates to obtain performing quantum stabilizer codes.

Proposition 11

[11, Proposition 1, Proposition 2, and Corollary 2] Let ( X , Q ) be a weak Castle curve of genus g and C ( D , r Q ) be a weak Castle code from X . Define r ⊥ = n + 2 g − 2 − r , where n is the length of C ( D , r Q ) . Then, the following properties hold:

Let f ∈ ℒ ( Q ) be a rational function such that ( f ) ∞ = m ( H ( Q ) ) Q . If ( d f ) = ( 2 g − 2 ) Q , where ( d f ) is the divisor associated with the differential form d f , then C ( D , r Q ) ⊥ = C ( D , r ⊥ Q ) .
The divisors D and rQ are equivalent. In addition, for every r < n , C ( D , r Q ) attains the designed minimum distance d * if and only if C ( D , ( n − r ) Q ) attains the designed minimum distance as well.
( 2 g − 2 ) Q and ( n + 2 g − 2 ) Q − D are canonical divisors, and there exists x ∈ ( F q * ) n such that C ( D , r Q ) ⊥ = x ⋅ C ( D , r ⊥ Q ) .
For every i = 1 , … , r , let r i ≔ min { r : ℓ ( r Q ) − ℓ ( ( r − n ) Q ) ≥ i } and C i ≔ C ( D , r i Q ) . Then, C i has dimension i, and
C 0 = ( 0 ) ⊂ C 1 ⊂ ⋯ ⊂ C n = F q n
is a formally self-dual sequence of codes.
If 2 i ≤ n , then there exist quantum codes with parameters [ [ n , n − 2 i , ≥ d ( C n − i ) ] ] q , where d ( C n − i ) ≥ n − r n − i + γ a + 1 , with a = ℓ ( ( r n − i − n ) Q ) and
γ a + 1 = min { deg ( A ) : A is a r a t i o n a l d i v i s o r o n X with ℓ ( A ) ≥ a + 1 } .

The following statement is a consequence of Section 5.1 and Proposition 9.

Proposition 12

The pair ( C , P ∞ ) is Castle over F q .

Based on numerical computations, we conjecture that ( C , P ∞ ) is never Castle over F q i if i > 1 . However, we can prove that ( C , P ∞ ) is weak Castle over F q i for every i ≥ 1 .

Proposition 13

The pair ( C , P ∞ ) is weak Castle over F q i , i ≥ 1 .

Proof

To prove the claim, it is enough to show that there exists a function defined over F q i whose pole divisor is ℓ P ∞ , ℓ > 0 , and such that its zeros are ℓ distinct F q i -rational points of C . A possible choice is to consider x ∈ F q ( C ) , because for any element a ∈ F q ⊂ F q i , the equation

Y q + Y = a q 0 ( a q + a ) = 0

has q distinct solutions in F q i , and by Proposition 2, its pole divisor is q P ∞ .□

Now we construct quantum codes from C exploiting the Castle property of ( C , P ∞ ) . We point out that the construction that follows is standard and has been applied in other articles; consider, for instance, [20] and the references therein. However, we were unable to compare our codes with the existing ones, and determine if they improve any record.

Let D be as in equation (14), namely

D = ∑ P ∈ C ( F q ) \ { P ∞ } P .

Then, C ( D , r P ∞ ) , r > 0 , are Castle codes of length n = q 2 . Moreover, with the notations of Proposition 11, since all the zeros of x are simple and its unique pole P ∞ is totally ramified, we have div ( d x ) = ( 2 g C − 2 ) P ∞ . Therefore, by (i) of Proposition 11, C ( D , r P ∞ ) ⊥ = C ( D , r ⊥ P ∞ ) . Now, let

H ( P ∞ ) = { ρ 0 = 0 < ρ 1 < ρ 2 < ⋯ } .

For ρ a , ρ a + b ∈ H ( P ∞ ) , with a , b ≥ 1 , consider the codes

C a + b ≔ C ⊥ ( D , ρ a + b P ∞ ) and C a ≔ C ⊥ ( D , ρ a P ∞ ) ,

whose dimensions are k 1 = q 2 − h a + b and k 2 = q 2 − h a , where h i is the number of non-gaps at P ∞ that do not exceed i . Note that C a + b ⊆ C a and k 2 − k 1 = b . Then, the CSS construction yields a [ [ q 2 , b , d ] ] q -quantum code such that d ≥ min { d ORD ( C a ) , d 1 } , where d 1 is the minimum distance of the code C ( D , ρ a + b P ∞ ) . Since C ( D , ρ a + b P ∞ ) = C ⊥ ( D , ρ a + b ⊥ P ∞ ) , the lower bound on d reads

(15) d ≥ min { d ORD ( C a ) , d ORD ( C ⊥ ( D , ρ a + b ⊥ P ∞ ) ) } .

Note that the order bound can be computed only in terms of the Weierstrass semigroup H ( P ∞ ) , which we determined explicitly in Section 4.

Acknowledgements

This work has been accepted for presentation at CIFRIS23, the Congress of the Italian Association of Cryptography “De Componendis Cifris.” The research of M. Timpanella was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (Gruppo Nazionale per le Strutture Algebriche, Geometriche e loro Applicazioni – Istituto Nazionale di Alta Aatematica). The author is funded by the project “Metodi matematici per la firma digitale ed il cloud computing” (Programma Operativo Nazionale “Ricerca e Innovazione” 2014–2020, University of Perugia). The author would like to thank Massimo Giulietti for his helpful suggestions.

Conflict of interest: The authors state no conflict of interest.

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Received: 2023-09-06

Revised: 2023-10-03

Accepted: 2023-10-07

Published Online: 2024-02-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/jmc-2023-0024

Keywords for this article

AG codes; quantum codes; Deligne–Lusztig curves

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