An Introduction to Single- and Double-Error-Correcting Codes
T. J. Cinderella · 2026
This chapter provides a foundational overview of key concepts in coding theory with a focus on error-detecting and error-correcting codes. It begins with the construction and properties of the Hamming (7,4) code, a classic example of a single-error-correcting code, and explores its significance through dual codes and weight distributions. The discussion extends to the principles of maximum likelihood decoding and syndrome decoding, which are central to efficient error correction in linear codes. The concept of the weight of a code word and its impact on code performance is analyzed in detail. The chapter also introduces maximum distance separable codes, highlighting their optimality in terms of error correction capability. To support the construction of advanced codes, the development of a finite field with 16 elements is presented, laying the groundwork for understanding Bose–Chaudhuri–Hocquenghem (BCH) codes. The chapter concludes with the construction and decoding strategies of BCH codes, emphasizing their ability to correct multiple errors.