On the existence of optimum cyclic burst correcting codes over GF(q)

A cyclic b-burst correcting code over GF(q) of redundancy r and length n=(q/sup r-b+1/-1)/(q-1) is said to be optimum. It is proved that a necessary condition for the existence of such a code is the existence of a square-free polynomial in GF(q)(x) of degree b-1 which is not divisible by x such that its period and the degrees of its irreducible factors are relatively prime to q-1. Moreover, if such a polynomial exists, then there are an infinite number of optimum cyclic b-burst correcting codes over GF(q).<
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