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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]
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).
A certain family of BCH codes have a particularly useful property, which is that treated as linear operators, their dual operators turns their input into an -wise independent source. That is, the set of vectors from the input vector space are mapped to an ℓ {\displaystyle \ell } -wise independent source.
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 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.
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The 95 isprint codes 32 to 126 are known as the ASCII printable characters. Some older and today uncommon formats include BOO, BTOA , and USR encoding. Most of these encodings generate text containing only a subset of all ASCII printable characters: for example, the base64 encoding generates text that only contains upper case and lower case ...