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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). George David Forney Jr. developed the algorithm. [1]
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 ...
In the code examples below, C(x) is a potential instance of ... In the case of binary GF(2) BCH code, the discrepancy d will be zero on all odd steps, so a check can ...
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 ...
The examples above discuss the representation problem for the numbers 3 and 65 by the form + and for the number 1 by the form . We see that 65 is represented by x 2 + y 2 {\displaystyle x^{2}+y^{2}} in sixteen different ways, while 1 is represented by x 2 − 2 y 2 {\displaystyle x^{2}-2y^{2}} in infinitely many ways and 3 is not represented by ...
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.
Raj Chandra Bose (or Basu) (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is partly named after him.