Linear and group perfect codes over skew fields and quasi skew fields

TL;DR

This paper introduces a general construction and classification of linear perfect codes over infinite skew fields and quasi skew fields, extending previous work to uncountable cases and infinite code lengths.

Contribution

It provides a new construction method and complete classification of linear perfect codes over infinite skew fields, removing previous countability restrictions.

Findings

Codes over infinite skew fields have infinite length.

Complete classification of such codes over associative skew fields.

Extended previous results to uncountable skew fields and code lengths.

Abstract

In this paper, we propose a general construction of linear perfect codes over infinite skew fields and quasi skew fields with right (left) unity. A complete classification of such codes over associative skew fields is given. Since the cardinality of the considered skew fields is infinite, the constructed codes have an infinite length. In the previous work, we considered codes over infinite countable fields, the length of which was also countable. We now remove this restriction and consider that the cardinality of the skew field and the length of the codes can be arbitrary (not necessarily countable).