4.5 Algebraic Geometry Codes
53
4.5 Algebraic Geometry Codes
In this part, we recall necessary concepts and results on algebraic geometry codes
that will be utilized in our constructions. More detailed results concerning such codes
can be found in [120, 152]. We follow the notation of [152].
The class of algebraic geometry (AG) codes was introduced by Goppa in his
seminal work [52]. These codes have nice properties; among them is the fact that such
codes are asymptotically good. There exist several works dealing with investigations
concerning AG codes (see for instance [71, 106, 119, 125]). We next present this
well-known class of codes. For this subsection, we assume that the reader is familiar
with the concepts of field extension and algebraic extension of fields. For details, we
refer to [125].
Definition 4.5.1 Let F and K be fields. An algebraic function field (function field
for short) F/K of one variable over K is a field extension K ⊆ F such that F is a
finite algebraic extension of K (x) for some element x ∈ F which is transcendental
over K .
Definition 4.5.2 A valuation ring of the algebraic function field F/K is a ring O ⊆
F satisfying the following conditions:
(i) K O F;
(ii) ∀z ∈ F, z ∈ O or z
−1
∈ O.
It is well-known that a valuation ring is a local ring, i.e., it has a unique maximal
ideal (see [152, Proposition 1.1.5.]). In our context we always consider that K = F q
is the finite field with q elements.
Definition 4.5.3 Let F/K be an algebraic function field. A place P of F/K is the
maximal ideal of some valuation ring O P of F/K .
We write P F to denote the set of places, i.e., P F := {P|P is a place of F/K }. As
usual, we denote by Z the ring of integers.
Definition 4.5.4 Let F/K be a function field. A discrete valuation of F/K is a
function v : F −→ Z ∪ {∞} satisfying the following conditions:
(1) v(x) = ∞ ⇐⇒ x = 0;
(2) ∀ x, y ∈ F, v(x y) = v(x) + v(y);
(3) ∀ x, y ∈ F, v(x + y) ≥ min{v(x), v(y)};
(4) there exists z ∈ F such that v(z) = 1;
(5) ∀ a ∈ K , a = 0, v(a) = 0.
In the following, we recall the concept of divisor.
Definition 4.5.5 A divisor of F/K is a formal sum of places given by
D :=
P∈P F
n P P, where n P is an integer number and almost all n P = 0.
53
4.5 Algebraic Geometry Codes
In this part, we recall necessary concepts and results on algebraic geometry codes
that will be utilized in our constructions. More detailed results concerning such codes
can be found in [120, 152]. We follow the notation of [152].
The class of algebraic geometry (AG) codes was introduced by Goppa in his
seminal work [52]. These codes have nice properties; among them is the fact that such
codes are asymptotically good. There exist several works dealing with investigations
concerning AG codes (see for instance [71, 106, 119, 125]). We next present this
well-known class of codes. For this subsection, we assume that the reader is familiar
with the concepts of field extension and algebraic extension of fields. For details, we
refer to [125].
Definition 4.5.1 Let F and K be fields. An algebraic function field (function field
for short) F/K of one variable over K is a field extension K ⊆ F such that F is a
finite algebraic extension of K (x) for some element x ∈ F which is transcendental
over K .
Definition 4.5.2 A valuation ring of the algebraic function field F/K is a ring O ⊆
F satisfying the following conditions:
(i) K O F;
(ii) ∀z ∈ F, z ∈ O or z
−1
∈ O.
It is well-known that a valuation ring is a local ring, i.e., it has a unique maximal
ideal (see [152, Proposition 1.1.5.]). In our context we always consider that K = F q
is the finite field with q elements.
Definition 4.5.3 Let F/K be an algebraic function field. A place P of F/K is the
maximal ideal of some valuation ring O P of F/K .
We write P F to denote the set of places, i.e., P F := {P|P is a place of F/K }. As
usual, we denote by Z the ring of integers.
Definition 4.5.4 Let F/K be a function field. A discrete valuation of F/K is a
function v : F −→ Z ∪ {∞} satisfying the following conditions:
(1) v(x) = ∞ ⇐⇒ x = 0;
(2) ∀ x, y ∈ F, v(x y) = v(x) + v(y);
(3) ∀ x, y ∈ F, v(x + y) ≥ min{v(x), v(y)};
(4) there exists z ∈ F such that v(z) = 1;
(5) ∀ a ∈ K , a = 0, v(a) = 0.
In the following, we recall the concept of divisor.
Definition 4.5.5 A divisor of F/K is a formal sum of places given by
D :=
P∈P F
n P P, where n P is an integer number and almost all n P = 0.