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4 Linear Block Codes
We denote by D F the free group of divisors of F/K . The support and the degree of a
divisor D are defined, respectively, by supp D := {P ∈ P F |n P = 0} and deg(D) :=
P∈P F
n P deg(P), where deg(P) is the degree of the place P.
For each x ∈ F/K , the principal divisor (x) of x is defined by
(x) :=
P∈P F
v P (x)P,
where v P is the discrete valuation corresponding to the place P.
For every x ∈ O P , we define x(P) ∈ O P /P as the residue class of x modulo P;
if x ∈ F − O P , we put x(P) := ∞.
Definition 4.5.6 Let A be a divisor of F/K . The Riemann–Roch space associated
to A is defined as
L(A) := {x ∈ F| (x) ≥ −A} ∪ {0}.
The integer l(A) := dim L(A) is called the dimension of the divisor A.
We here define the genus of a function field, which is the most important invariant
of a function field.
Definition 4.5.7 Let F/K be a function field. The genus g of F/K is defined as
g := max{deg(A) − l(A) + 1 | A ∈ D F }.
Definition 4.5.8 An adele of F/K is a mapping α : P F −→ F, defined by α(P) =
α P , such that α P ∈ O P for almost all P ∈ P F .
An adele can be considered as an element of the direct product
P∈P F
F; hence,
we utilize the notation α = (α P ) P∈P F or α = (α P ).
The set A F := {α|α is an adele of F/K } is said to be the adele space of F/K .
Definition 4.5.9 Let A ∈ P F . Then we define A F (A) := {α ∈ A F |v P (α) ≥ −v P
(A) ∀ P ∈ P F }, where v P (A) = n P and v P (α) := v P (α P ).
It is easy to see that A F (A) is a K -subspace of A F . We next present the concept
of Weil differential.
Definition 4.5.10 Let F/K be a function field. A Weil differential of F/K is a K -
linear map ω : A F −→ K which vanishes on A F (A) + F for some divisor A ∈ D F .
Let F be the differential space of F/K , i.e.,
F := {w | w is a Weil differential ofF/K }.