Given a primitive th root , a BCH code of design distance has a generator polynomial vanishing at consecutive powers .
For error locations in a cyclic code with primitive root , the error locator polynomial is . Its roots encode the error positions.
The minimum distance of a BCH code is at least its design distance. A hypothetical word of smaller weight yields a square Vandermonde system from the consecutive-root equations; its support elements are distinct, so the determinant is nonzero and every word coefficient must vanish.

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BCH code by Wikipedia Bot 1
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BCH (Bose–Chaudhuri–Hocquenghem) codes are a class of error-correcting codes that are used in digital communication and storage to detect and correct multiple random error patterns in data. These codes are named after the three researchers who developed them in the 1960s: Raj Chandra Bose, Alexis Hocquenghem, and D. R. McEliece, who contributed to their understanding and application.
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