Riemann-Roch Theorem and Mac Williams identities for an additive code with respect to a saturated lattice
Азнив Каспарян · 2020
Let (G, +) be a finite abelian group and (C, +)n, +) be an additive code with dual (C⊥,.)n, + .). Consider a lattice (L, ∩, +) of subgroups of (Gn, +) and its dual lattice (L⊥, ., ∩) of subgroups of the character variety (Gn, .). We say that L and L⊥are saturated if their unique elements of maximal order are Gn∈ L, respectively, G⊥∈ L⊥. The present note generalizes Randriambololona's Riemann-Roch Theorem for sections of Fq-linear codes C ⊂ Fnqwith values in subspaces of Fnqto 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 ⊂ Fnqis an Fq-linear code and (H, ∩, +) is the Hamming lattice of the coordinate subspaces of Fnq, our Mac Williams identities reduce to the classical ones.