This paper presents an in-depth exploration of Hamming Quasi-Cyclic (HQC) codes, an algorithmic framework that has garnered significant attention within the domain of post-quantum cryptography. Acknowledged as a third-round finalist in the NIST competition, this cryptographic approach demonstrates promising capabilities in addressing the imminent challenges posed by quantum computing to contemporary cryptographic protocols. The paper begins by elucidating the foundational principles underlying HQC codes, tracing their evolution from traditional cyclic codes to the more specialized domain of quasi-cyclic structures. It subsequently delves into the distinctive characteristics and cryptographic properties exhibited by HQC Hamming Quasi-Cyclic codes, emphasizing their suitability for cryptographic applications in the post quantum era. A detailed examination of the construction methodologies employed in HQC code generation follows, encompassing algebraic, combinatorial, and iterative techniques. These methodologies are scrutinized not only for their efficacy in producing robust and secure codes but also for their compatibility with the stringent requirements of post quantum cryptographic frameworks. In conclusion, this survey paper serves as a comprehensive resource for researchers and practitioners seeking to comprehend the cryptographic significance of HQC Hamming Quasi-Cyclic codes. By elucidating their role as third-round finalists in the NIST competition and their potential impact on post-quantum cryptographic paradigms, this paper aims to catalyze further advancements in the field of quantum-resistant cryptography.