The BCH code with has generator polynomial It has minimal Hamming distance at least 7 and corrects up to three errors. Since the generator polynomial is of degree 10, this code has 5 data bits and 10 checksum bits. (This particular generator polynomial has a real-world application, in the format patterns of the QR code.) The BCH code with and higher has generator polynomial This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. In fact, this code has only two codewords: 000000000000000 and 111111111111111. General BCH codes General BCH codes differ from primitive narrow-sense BCH codes in two respects. First, the requirement that be a primitive element of can be relaxed. By relaxing this requirement, the code length changes from to the order of the element Second, the consecutive roots of the generator polynomial may run from instead of Definition. Fix a finite field where is a prime power. Choose positive integers such that and is the multiplicative order of modulo As before, let be a primitive th root of unity in and let be the minimal polynomial over of for all The generator polynomial of the BCH code is defined as the least common multiple Note: if as in the simplified definition, then is 1, and the order of modulo is Therefore, the simplified definition is indeed a special case of the general one. Special cases A BCH code with is called a narrow-sense BCH code. A BCH code with is called primitive. The generator polynomial of a BCH code has coefficients from In general, a cyclic code over with as the generator polynomial is called a BCH code over The BCH code over and generator polynomial with successive powers of as roots is one type of Reed–Solomon code where the decoder (syndromes) alphabet is the same as the channel (data and generator polynomial) alphabet, all elements of . The other type of Reed Solomon code is an original view Reed Solomon code which is not a BCH code. Properties The generator polynomial of a BCH code has degree at most . Moreover, if and , the generator polynomial has degree at most . Each minimal polynomial has degree at most . Therefore, the least common multiple of of them has degree at most . Moreover, if then for all . Therefore, is the least common multiple of