4.5 Algebraic Geometry Codes
55
A divisor W is called canonical if W = (w) for some w ∈ F . For every nonzero
differential w, its canonical divisor is denoted by (w) :=
P∈P F
v P (w)P, where
v P (w) := v P ((w)). All canonical divisors are equivalent and have degree 2g − 2.
Given a divisor G, we define
F (G) := {w ∈ F | w = 0 or(w) ≥ G}.
The dimension of F (G) as F q -vector space is denoted by i(G).
A fundamental result concerning AG codes is the well-known Riemann–Roch
theorem:
Theorem 4.5.1 (Riemann–Roch) (Thm. 1.5.15 of [152]) Let W be a canonical divisor of F/F q . Then for each divisor A ∈ D F , the dimension of L(A) is given by
l(A) = deg(A) + 1 − g + l(W − A).
The definition of an algebraic geometry (AG) code is given in the sequence.
Definition 4.5.11 Let P 1 , . . . , P n be pairwise distinct places of F/F q of degree
1, and let D = P 1 + . . . + P n be a divisor. Let G be a divisor of F/F q such that
supp G ∩ supp D = ∅. The algebraic geometry code C L (D, G) associated with D
and G is defined by
C L (D, G) := {(x(P 1 ), . . . , x(P n ))|x ∈ L(G)} ⊆ F
n
q .
If G = m Q, where Q is a rational place (place of degree 1) not belonging in the
support of D, then the code C L (D, G) is said to be a one-point AG code.
The following result establishes the parameters of an AG code.
Proposition 4.5.1 ([152, Thm. 2.2.2./Cor.2.2.3]) Let F/F q be an algebraic function
field of genus g. Then C L (D, G) is an [n, k, d] q code with k = l(G) − l(G − D) and
d ≥ n − deg(G). In addition, if 2g − 2 < deg(G) < n, then one has k = deg(G) +
1 − g. Moreover, if {x 1 , . . . , x k } is a basis of L(G), then the matrix
H δ,b =
⎡
⎢
⎣
x 1 (P 1 ) x 1 (P 2 ) . . . x 1 (P n )
. . .
. . .
. . .
x k (P 1 ) x k (P 2 ) . . . x k (P n )
⎤
⎥
⎦
is a generator matrix for C L (D, G).
The Weierstrass semigroup of a divisor Q plays an important role in our constructions.
Definition 4.5.12 Assume that Q is a divisor of degree 1 of F/F q . Let us consider
L(∞Q) =
r ≥0
L(r Q) be the space of rational functions having poles only at Q. Then
the Weierstrass semigroup of Q is defined as
55
A divisor W is called canonical if W = (w) for some w ∈ F . For every nonzero
differential w, its canonical divisor is denoted by (w) :=
P∈P F
v P (w)P, where
v P (w) := v P ((w)). All canonical divisors are equivalent and have degree 2g − 2.
Given a divisor G, we define
F (G) := {w ∈ F | w = 0 or(w) ≥ G}.
The dimension of F (G) as F q -vector space is denoted by i(G).
A fundamental result concerning AG codes is the well-known Riemann–Roch
theorem:
Theorem 4.5.1 (Riemann–Roch) (Thm. 1.5.15 of [152]) Let W be a canonical divisor of F/F q . Then for each divisor A ∈ D F , the dimension of L(A) is given by
l(A) = deg(A) + 1 − g + l(W − A).
The definition of an algebraic geometry (AG) code is given in the sequence.
Definition 4.5.11 Let P 1 , . . . , P n be pairwise distinct places of F/F q of degree
1, and let D = P 1 + . . . + P n be a divisor. Let G be a divisor of F/F q such that
supp G ∩ supp D = ∅. The algebraic geometry code C L (D, G) associated with D
and G is defined by
C L (D, G) := {(x(P 1 ), . . . , x(P n ))|x ∈ L(G)} ⊆ F
n
q .
If G = m Q, where Q is a rational place (place of degree 1) not belonging in the
support of D, then the code C L (D, G) is said to be a one-point AG code.
The following result establishes the parameters of an AG code.
Proposition 4.5.1 ([152, Thm. 2.2.2./Cor.2.2.3]) Let F/F q be an algebraic function
field of genus g. Then C L (D, G) is an [n, k, d] q code with k = l(G) − l(G − D) and
d ≥ n − deg(G). In addition, if 2g − 2 < deg(G) < n, then one has k = deg(G) +
1 − g. Moreover, if {x 1 , . . . , x k } is a basis of L(G), then the matrix
H δ,b =
⎡
⎢
⎣
x 1 (P 1 ) x 1 (P 2 ) . . . x 1 (P n )
. . .
. . .
. . .
x k (P 1 ) x k (P 2 ) . . . x k (P n )
⎤
⎥
⎦
is a generator matrix for C L (D, G).
The Weierstrass semigroup of a divisor Q plays an important role in our constructions.
Definition 4.5.12 Assume that Q is a divisor of degree 1 of F/F q . Let us consider
L(∞Q) =
r ≥0
L(r Q) be the space of rational functions having poles only at Q. Then
the Weierstrass semigroup of Q is defined as