On certain projective geometry codes (Corresp.)
J. Huang, S. Shiva, Gérald E. Séguin · IEEE Transactions on Information Theory · 1984
LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd \geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = \{0, 1, \alpha , \alpha^{2} , \cdots , \alpha^{n(q-1)-1} \}, the generator polynomialg(X), ofV, has\alpha^{ u}as a root if and only if uhas the form u = i(q - 1)and\max_{0 \leq l < s} W_{q}(2^{l} u) \leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbers u, such that\alpha^{ u}is a root ofg(X). LetC_{1}, C_{2}, \cdots, C_{ u}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{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 ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.