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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 ().
It is used as one of the steps in decoding BCH codes and Reed–Solomon codes (a subclass of BCH codes). ... Handout #28 (PDF), Stanford University, pp. ...
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).
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 ...
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 ...
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A monic irreducible polynomial of degree n having coefficients in the finite field GF(q), where q = p t for some prime p and positive integer t, is called a primitive polynomial if all of its roots are primitive elements of GF(q n). [2] [3] In the polynomial representation of the finite field, this implies that x is a primitive element.
Berlekamp–Massey algorithm. The Berlekamp–Massey algorithm is an algorithm that will find the shortest linear-feedback shift register (LFSR) for a given binary output sequence.