Duursma's reduced polynomial
Азнив Каспарян, Ivan Marinov · Advances in Mathematics of Communications · 2017
The weight distribution $\{ \mathcal{W}_C^{(w)} \} _{w=0} ^n$ of a linear code $C \subset {\mathbb F}_q^n$ is put in an explicit bijective correspondence with Duursma's reduced polynomial $D_C(t) ∈ {\mathbb Q}[t]$ of $C$ . We prove that the Riemann Hypothesis Analogue for a linear code $C$ requires the formal self-duality of $C$ . Duursma's reduced polynomial $D_F(t) ∈ {\mathbb Z}[t]$ of the function field $F = {\mathbb F}_q(X)$ of a curve $X$ of genus $g$ over ${\mathbb F}_q$ is shown to provide a generating function $\frac{D_F(t)}{(1-t)(1-qt)} = \sum\limits _{i=0} ^{∞} \mathcal{B}_i t^{i}$ for the numbers $\mathcal{B}_i$ of the effective divisors of degree $i ≥0$ of a virtual function field of a curve of genus $g-1$ over ${\mathbb F}_q$ .