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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 ().
By 1963 (or possibly earlier), J. J. Stone (and others) recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, [4] but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation ...
It is used as one of the steps in decoding BCH codes and Reed–Solomon codes (a subclass of BCH codes). George David Forney Jr. developed the algorithm. [1]
Each non-zero may be expressed as for some , where is a primitive element of (), is the power number of primitive element . Thus the powers α i {\displaystyle \alpha ^{i}} for 0 ≤ i < ( q − 1 ) {\displaystyle 0\leq i<(q-1)} cover the entire field (excluding the zero element).
If the generator polynomial is primitive, then the resulting code has Hamming distance at least 3, provided that ... This is the case for BCH codes.
Chapter 5 studies cyclic codes and Chapter 6 studies a special case of cyclic codes, the quadratic residue codes. Chapter 7 returns to BCH codes. [1] [6] After these discussions of specific codes, the next chapter concerns enumerator polynomials, including the MacWilliams identities, Pless's own power moment identities, and the Gleason ...
The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows: [13] The elements X and Y are primitive, so and are grouplike; so their product is also grouplike; so its logarithm ( ()) is primitive; and hence can be written as an infinite sum of elements of the Lie algebra generated by X and Y.
A negacyclic code is a constacyclic code with λ=-1. [8] A quasi-cyclic code has the property that for some s, any cyclic shift of a codeword by s places is again a codeword. [9] A double circulant code is a quasi-cyclic code of even length with s=2. [9] Quasi-twisted codes and multi-twisted codes are further generalizations of constacyclic ...