Codes Correcting Few Restricted Errors

TL;DR

This paper introduces new constructions of linear codes over Gaussian and Eisenstein integers that correct a small number of restricted errors, with potential applications in cryptography and related fields.

Contribution

It provides novel geometric and algebraic construction methods for codes correcting two or three restricted errors over complex integer rings.

Findings

New code constructions for Gaussian and Eisenstein integers

Effective correction of two or three restricted errors

Adaptation of techniques from Lee metric codes

Abstract

We consider linear codes over a field in which the error values are restricted to a subgroup of its unit group. This scenario captures Lee distance codes as well as codes over the Gaussian or Eisenstein integers. Codes correcting restricted errors gained increased attention recently in the context of code-based cryptography. In this work we provide new constructions of codes over the Gaussian or Eisenstein integers correcting two or three errors. We adapt some techniques from Roth and Siegel's work on codes for the Lee metric. We propose two construction methods, which may be seen of geometric and algebraic flavor, respectively.