Recursive Polynomial Sets and Their Some Algebraic Applications

This paper primarily defines the framework for a new class of polynomial sets over a finite field GF (2), providing a recursive definition and delving into pertinent algebraic properties. We also studied some applications of the obtained polynomial classes on coding theory, such as obtaining new code classes. Our focus lies on polynomial sets with degrees equal to or less than n, for which we present a methodology for encoding and decoding utilizing an irreducible polynomial p(x) = xm+xs+1, (m = 2n-1). Furthermore, as an application of this method in coding theory, we created new code classes and studied some features of these codes.