On the Existence of Optimum Cyclic Burst Correcting Codes Over GF(q)

A cyclic b-burst correcting code over GF(a) of redundancy r and length n #x003D; (qr-b+1 #x2014; 1)/(q - 1) is said to be optimum. We will prove that a necessary condition lor the existence of such code is the existence of a square-free polynomial in GF(q)[x] of degree b #x2014; 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). © 1988 IEEE