On Bounds for Ring-Based Coding Theory

TL;DR

This paper introduces a new weight called the overweight for ring-based coding theory, generalizes classical bounds like Plotkin and Gilbert-Varshamov, and proves a Johnson bound for homogeneous weights on Frobenius rings.

Contribution

It proposes the overweight weight and extends classical coding bounds to this new metric over rings and Frobenius rings.

Findings

Established a Plotkin bound for the overweight weight.

Derived a sphere-packing bound using the overweight weight.

Proved a Johnson bound for the homogeneous weight on Frobenius rings.

Abstract

Coding Theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. Such codes over rings had important applications and many interesting mathematical problems are related to this line of research. It has been well established, that with the generalization of the algebraic structure to rings there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a new weight, called the overweight, which can be seen as a generalization of the Lee weight on the integers modulo $4$. For this new weight we provide a number of well-known bounds, like a Plotkin bound, a sphere-packing bound, and a Gilbert-Varshamov bound. A further highlight is the proof of a Johnson bound for the homogeneous weight on a general finite Frobenius ring.