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m Task 18 (cosmetic): eval 13 templates: del empty params (14×); hyphenate params (9×); cvt lang vals (1×); |
→Primitive narrow-sense BCH codes: primitive element links to a wrong article; GF(qⁿ) with prime q doesn't necessarily have a primitive (qⁿ-1)th root of unity, but since GF(qⁿ) is taken to be a simple extension of GF(q), it has a field primitive element, i.e. the generator element of the simple extension |
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Given a [[prime number]] {{mvar|q}} and [[prime power]] {{math|''q''<sup>''m''</sup>}} with positive integers {{mvar|m}} and {{mvar|d}} such that {{math|''d'' ≤ ''q''<sup>''m''</sup> − 1}}, a primitive narrow-sense BCH code over the [[finite field]] (or Galois field) {{math|GF(''q'')}} with code length {{math|''n'' {{=}} ''q''<sup>''m''</sup> − 1}} and [[Block code#The distance d|distance]] at least {{mvar|d}} is constructed by the following method.
Let {{mvar|α}} be a [[
For any positive integer {{mvar|i}}, let {{math|''m''<sub>''i''</sub>(''x'')}} be the [[minimal polynomial (field theory)|minimal polynomial]] with coefficients in {{math|GF(''q'')}} of {{math|α<sup>''i''</sup>}}.
The [[generator polynomial]] of the BCH code is defined as the [[least common multiple]] {{math|''g''(''x'') {{=}} lcm(''m''<sub>1</sub>(''x''),…,''m''<sub>''d'' − 1</sub>(''x''))}}.
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