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On certain projective geometry codes (Corresp.)

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Let V be an (n, k, d) binary projective geometry code with n = (q^{m}-1)/(q - 1), q = 2^{s} , and d \geq [(q^{m-r}-1)/(q - 1)] + 1 . This code is r -step majority-logic decodable. With reference to the GF (q^{m}) = {0, 1, \alpha , \alpha ^{2} , \cdots , \alpha ^{n(q-1)-1} } , the generator polynomial g(X) , of V , has \alpha ^{\nu} as a root if and only if \nu has the form \nu = i(q - 1) and \max _{0 \leq l < s} W_{q}(2^{l} \nu) \leq (m - r - 1)(q - 1) , where W_{q}(x) indicates the weight of the radix- q representation of the number x . Let S be the set of nonzero numbers \nu , such that \alpha ^{\nu} is a root of g(X) . Let C_{1}, C_{2}, \cdots , C_{\nu} be the cyclotomic cosets such that S is the union of these cosets. It is clear that the process of finding g(X) becomes simpler if we can find a representative from each C_{i} , since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases of m-r = 2 , with s = 2, 3 , and m-r=3 , with s=2 .

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Information Theory, IEEE Transactions on  (Volume:30 ,  Issue: 2 )