BCHcodes

BCHCodes

Bose–Chaudhuri–Hocquenghem (BCH) codes are a family of cyclic error-correcting codes defined over finite fields. They were independently discovered by Bose and Chaudhuri, and by Hocquenghem in 1959, and are named after them. BCH codes are designed to correct multiple symbol errors and can be constructed to guarantee a specified minimum distance, providing predictable error-correction capabilities.

In the binary case, a common construction uses a field GF(2^m) with a primitive element α and code length n = 2^m − 1. For a chosen t, the code is designed so that the consecutive powers α^b, α^{b+1}, ..., α^{b+2t−1} are roots of the code, yielding a designed distance δ ≥ 2t + 1. The generator polynomial g(x) is the least common multiple of the minimal polynomials of these powers and divides x^n − 1. The resulting BCH code C consists of all polynomials divisible by g(x) and has parameters [n, k, d], with k ≥ n − m t, where d ≥ δ.

Decoding typically involves syndrome calculation followed by algorithms such as Berlekamp–Massey or Euclidean decoding to find the error-locator polynomial, and a Chien search to locate error positions. BCH codes can correct up to t errors in ideal conditions, with practical decoders extending the basic approach to handle real-world imperfections.

Variants include narrow-sense BCH (b = 1), extended BCH (adding a parity bit to form an extended length), and shortened BCH codes to achieve different lengths and dimensions. Applications span data storage and communications, including certain optical discs and memory systems, where multi-error correction is required and predictable performance with reasonable complexity is desirable.