Codes Correcting Few Restricted Errors

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.