Riemann-Roch Theorem and Mac Williams identities for an additive code with respect to a saturated lattice

Let (G, +) be a finite abelian group and (C, +)<; (G
ⁿ
, +) be an additive code with dual (C
^⊥
,.) <; (G
ⁿ
, + .). Consider a lattice (L, ∩, +) of subgroups of (G
ⁿ
, +) and its dual lattice (L
^⊥
, ., ∩) of subgroups of the character variety (G
ⁿ
, .). We say that L and L
^⊥
are saturated if their unique elements of maximal order are G
ⁿ
∈ L, respectively, G
^⊥
∈ L
^⊥
. The present note generalizes Randriambololona's Riemann-Roch Theorem for sections of F
_q
-linear codes C ⊂ F
ⁿ
_q
with values in subspaces of F
ⁿ
_q
to L-valued sections of C and L
^⊥
-valued sections of C
^⊥
. That provides natural Mac Williams identities for C, C
^⊥
with respect to saturated L, L
^⊥
. If C ⊂ F
ⁿ
_q
is an F
_q
-linear code and (H, ∩, +) is the Hamming lattice of the coordinate subspaces of F
ⁿ
_q
, our Mac Williams identities reduce to the classical ones.