BoseChaudhuriHocquenghemkoder

Bose-Chaudhuri-Hocquenghem codes, commonly called BCH codes, are a family of cyclic error-correcting codes over finite fields named after Raj Bose, A. Chaudhuri, and Alexander Hocquenghem who independently contributed to their development in the 1960s. They are designed to correct multiple random errors and can be constructed to guarantee a specified minimum distance, making them versatile for reliable data transmission and storage.

In the binary case, a BCH code has length n = 2^m − 1 for some integer m, and is built over the finite field GF(2^m) using a primitive element α. For a chosen designed distance δ, the generator polynomial g(x) is the least common multiple of the minimal polynomials of α^b, α^{b+1}, ..., α^{b+δ−2}. For the common narrow-sense BCH codes, b = 1. The code’s dimension satisfies k ≥ n − m(δ − 1), and the code can correct up to t = floor((δ − 1)/2) errors; the actual minimum distance is at least δ.

Extensions and related families include extended BCH codes (which add a parity bit to achieve even length) and primitive BCH codes (constructed with the primitive element for GF(2^m)). BCH codes have found use in various data transmission and storage systems, where they provide efficient multi-error correction and can be tailored to align with specific reliability and latency requirements. They are part of the broader landscape of algebraic error-correcting codes alongside Reed-Solomon and others.

---