Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

Abstract

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew PBW extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.