5.4 Algebraic Geometry Codes
109
(2) the Hermitian construction applied to Hermitian self-orthogonal codes [4, 25,
71, 73, 80, 91, 100];
(3) the Steane enlargement of CSS construction applied to Euclidean self-orthogonal
codes [89, 100, 147, 148].
In particular, the CSS construction was also utilized in chains of nested linear codes
to construct quantum codes whose parameters are asymptotically good [12, 26, 27,
81, 117]. All these latter asymptotically good quantum codes were constructed by
applying the CSS construction to families of AG codes. In fact, the class of AG
codes is a good source in order to obtain asymptotically good codes (see for example
[47, 151]). In Refs. [12, 26, 27, 117], the authors constructed asymptotically good
binary quantum codes and, in Ref. [81], the authors presented families of nonbinary
asymptotically good quantum codes by means of one-point AG codes.
The aim here is to construct classical t-point (t ≥ 1) AG codes (which are a
generalization of one-point AG codes) as well as AG codes whose divisor G is not a
rational place, after applying the CSS construction to these codes, in order to generate
nonbinary quantum codes with good parameters. Additionally, we also construct
sequences of classical t-point AG codes to obtain sequences of asymptotically good
quantum codes by means of the CSS construction.
The constructions performed here are natural generalizations of the works dealing
with constructions of quantum codes derived from one-point AG codes (see for
example [12, 26, 27, 117]).
5.4.1 Preliminaries
Recall that a q-ary quantum code Q of length n is a K -dimensional subspace of
the q
n -dimensional Hilbert space (C
q
)
⊗n , where ⊗n denotes the tensor product of
vector spaces. If K = q
k we write [[n, k, d]] q to denote a q-ary quantum code of
length n and minimum distance d. Let [[n, k, d]] q be a quantum code. The quantum
Singleton bound (QSB) asserts that k + 2d ≤ n + 2. If the equality holds then the
code is MDS.
5.4.2 Our Codes
This subsection is divided into three parts. The first part deals with constructions of
quantum t-point algebraic geometry codes. In the second, we construct AG codes
where the divisor G is a sum of non-rational places and, in the third part, we construct
sequences of asymptotically good quantum codes derived from AG codes.
109
(2) the Hermitian construction applied to Hermitian self-orthogonal codes [4, 25,
71, 73, 80, 91, 100];
(3) the Steane enlargement of CSS construction applied to Euclidean self-orthogonal
codes [89, 100, 147, 148].
In particular, the CSS construction was also utilized in chains of nested linear codes
to construct quantum codes whose parameters are asymptotically good [12, 26, 27,
81, 117]. All these latter asymptotically good quantum codes were constructed by
applying the CSS construction to families of AG codes. In fact, the class of AG
codes is a good source in order to obtain asymptotically good codes (see for example
[47, 151]). In Refs. [12, 26, 27, 117], the authors constructed asymptotically good
binary quantum codes and, in Ref. [81], the authors presented families of nonbinary
asymptotically good quantum codes by means of one-point AG codes.
The aim here is to construct classical t-point (t ≥ 1) AG codes (which are a
generalization of one-point AG codes) as well as AG codes whose divisor G is not a
rational place, after applying the CSS construction to these codes, in order to generate
nonbinary quantum codes with good parameters. Additionally, we also construct
sequences of classical t-point AG codes to obtain sequences of asymptotically good
quantum codes by means of the CSS construction.
The constructions performed here are natural generalizations of the works dealing
with constructions of quantum codes derived from one-point AG codes (see for
example [12, 26, 27, 117]).
5.4.1 Preliminaries
Recall that a q-ary quantum code Q of length n is a K -dimensional subspace of
the q
n -dimensional Hilbert space (C
q
)
⊗n , where ⊗n denotes the tensor product of
vector spaces. If K = q
k we write [[n, k, d]] q to denote a q-ary quantum code of
length n and minimum distance d. Let [[n, k, d]] q be a quantum code. The quantum
Singleton bound (QSB) asserts that k + 2d ≤ n + 2. If the equality holds then the
code is MDS.
5.4.2 Our Codes
This subsection is divided into three parts. The first part deals with constructions of
quantum t-point algebraic geometry codes. In the second, we construct AG codes
where the divisor G is a sum of non-rational places and, in the third part, we construct
sequences of asymptotically good quantum codes derived from AG codes.