On the maximum true burst correcting capability of primitive Fire codes

Fire codes are cyclic codes generated by the product of two polynomials: a binomial that characterizes the code´s guaranteed burst correcting capability and an irreducible polynomial that characterizes the code length. However, the true burst correcting capability of a Fire code may exceed its guaranteed burst correcting capability. The true burst correcting capability of primitive Fire codes, in which the irreducible polynomial is primitive, is studied. In particular, the true burst correcting capability maximized over all primitive Fire codes with a given guaranteed burst correcting capability and a given greatest common divisor of the periods of the two factors of its generator polynomial is considered. It is shown that this maximum is attained by an infinite number of such codes.