MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings

TL;DR

This paper characterizes bi-invariant weights on finite principal ideal rings that satisfy the Extension Property, expanding understanding beyond finite fields and Frobenius rings to a broader class of rings.

Contribution

It provides a complete characterization of bi-invariant weights with the Extension Property on finite principal ideal rings, including non-commutative cases.

Findings

Characterization of bi-invariant weights on finite principal ideal rings.

Extension Property holds for these weights under specified conditions.

Includes non-commutative rings in the analysis.

Abstract

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.