On the Weight Distribution of Codes over Finite Rings

TL;DR

This paper investigates the weight distribution of codes over finite Frobenius rings, providing constructions for codes with small spectra and analyzing their properties using trace maps and algebraic structures.

Contribution

It introduces new methods to compute weight distributions of codes over finite Frobenius rings and constructs codes with small spectra using trace maps and algebraic techniques.

Findings

Codes over finite Frobenius rings with computable weight distributions

Construction of codes over integer modular rings with small spectra

Analysis of homogeneous weight distribution in these codes

Abstract

Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.