Duursma's reduced polynomial

TL;DR

This paper explores the properties of Duursma's reduced polynomial, establishing its connection to code weight distributions, self-duality conditions, and bounds on field size, with implications for algebraic geometry over finite fields.

Contribution

It links Duursma's polynomial to code properties, proves conditions for the Riemann Hypothesis Analogue, and relates it to divisor counts on algebraic curves.

Findings

Weight distribution corresponds bijectively to Duursma's polynomial

Riemann Hypothesis Analogue implies code self-duality and bounds on q

Polynomial encodes divisor counts on algebraic curves

Abstract

The weight distribution of a linear code C is put in an explicit bijective correspondence with Duursma's reduced polynomial of C. We prove that the Riemann Hypothesis Analogue for a linear code C requires the formal self-duality of C and imposes an upper bound on the cardinality q of the basic field, depending on the dimension and the minimum distance of C. Duursma's reduced polynomial of the function field of a curve X of genus g over the field with q elements is shown to provide a generating function for the numbers of the effective divisors of non-negative degree degree of a virtual function field of a curve of genus g-1 over the same finite field.