56
4 Linear Block Codes
(Q) := {−v Q ( f )| f ∈ L(∞Q)} = {0 = ρ 1 < ρ 2 < · · · },
where v Q denotes the valuation at Q.
Another interesting type of AG code is the code C (D, G), defined in the
sequence.
Definition 4.5.13 Let F/F q be a function field of genus g. Let G and D be divisors
as in Definition 4.5.11. The code C (D, G) ⊆ F
n
q is defined by
C (D, G) := {(resp P 1 (w), . . . , resp P n (w))|w ∈ F (G − D)},
where resp P i (w) is the residue of w at P i .
Theorem 4.5.2 computes the parameters of the code C (D, G).
Theorem 4.5.2 ([152, Thm. 2.2.7.]) Assume the notation of Definition 4.5.11. The
code C (D, G) is an [n, k
∗
, d
∗
] q code, where k
∗
= i(G − D) − i(G) and d
∗
≥
deg(G) − (2g − 2). Additionally, if 2g − 2 < deg(G) < n one has k
∗
= n + g −
1 − deg(G).
C L (D, G) and C (D, G) have an important correlation.
Theorem 4.5.3 ([152, Thm. 2.2.8.]) The codes C L (D, G) and C (D, G) are
(Euclidean) dual of each other, i.e., C (D, G) = C L (D, G)
⊥ .
The following result characterizes when an AG code is self-orthogonal.
Proposition 4.5.2 ([152, Cor. 2.2.11.]) Suppose there exists a Weil differential η such
that 2G − D ≤ (η) and η P i (1) = 1 for i = 1, 2, . . . , n. Then C L (D, G) is Euclidean
self-orthogonal, that is, C L (D, G) ⊆ C
⊥
L (D, G).
Remark 4.5.1 Note that the Hermitian self-orthogonal condition to AG code can
be easily obtained from the Euclidean condition due to the fact that C L (D, G) ⊆
C
⊥ H
L (D, G) if and only if C
q
L (D, G) ⊆ C
⊥
L (D, G).