Duursma's reduced polynomial

The weight distribution {W-C((w)))(w=0)(n) of a linear code C subset of F-q(n) is put in an explicit bijective correspondence with Duursma's reduced polynomial D-C(t) is an element of 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) is an element of Z[t] of the function field F = F-q(X) of a curve X of genus g over F-q is shown to provide a generating function D-F(t)/(1 - t) (1 - qt) = Sigma(infinity)(i=0) B(i)t(i) for the numbers B-i of the effective divisors of degree i >= 0 of a virtual function field of a curve of genus g - 1 over F-q.