Abstract:
The study of cyclotomic co-sets over Galois fields (GF) is a fundamental aspect of algebraic number theory. Cyclotomic co-sets are sets of integers that share common properties when raised to certain powers in a given modulus. Coding theory is concerned with data transmission across noisy channels and recovering the original information. Error-correcting codes, such as BCH codes and Reed-Solomon codes, rely heavily on the properties of cyclotomic co-sets over Galois fields. The study of cyclotomic cosets in GF (37) leads to the development of new codes and, improving the efficiency and error-correction capability of existing ones. This research is aimed at investigating the number of cyclotomic co-sets over a Galois field, (GF (37)), and understanding their properties and behavior. The research employs rigorous mathematical analysis and computational methods to enumerate and analyze the cyclotomic cosets over GF (37), establishing that cyclotomic cosets over GF (37), Ca, from a cyclic group of the form } where the gcd of and 37 is 1. This research contributes to the understanding of the properties of GF (37), with potential applications in various areas of mathematics and computer science. Additionally, it contributes to the construction of BCH codes, a class of error-correcting codes widely used in digital communications to ensure data integrity. The unique property of GF (37) can also contribute to defining the parity-check and generator matrices in the cryptographic system algorithms that can help in enhancing the security of public key cryptosystems, leading to improved reliability of digital communications and the security of digital data, as well as enriching the field of research in cyclic codes over prime fields.
