Algebraic Construction of Non-Primitive BCH Codes on GF(2^m) with Minimum Distance Checking for Degrees up to 10

In coding theory, Bose-Chaudhuri-Hocquenghem (BCH) codes possess an excellent error correction capability, which allows them to become crucial for use in communication and storage systems. However, non-primitive BCH codes that hide interesting properties have received less attention than their primitive counterpart. In this work, we present an algebraic method to construct non-primitive BCH codes of length and dimension fixed, with a varying minimum distance, on a Galois finite field $G F\left(2^{m}\right)$ for extensions of degree m between 4 and 10. In this method, we will describe the construction of generating polynomials using non-primitive algebraic roots and verify the parameters of the codes obtained with an Algebraic Calculator. The resultant list of these non-primitive BCH codes give optimal properties in terms of coding rate and minimum distance, providing valuable flexibility for the construction of Low-density parity check (LDPC) codes.