Let Cp（x） denote the cyclic code generated by p（X）

and Cpn（x） the principal ideal generated by p（X） in the algebra of polynomials modulo Xn - 1. The following theorems may make clear the idea behind the design of Fire codes.Theorem Ⅰ. If n is the length of Cp（x）q（x）

and p（X） is relatively prime to q（X）

then Cp（X）q（x） is equal to the intersection Cpn（x）Cqn（x）.Theorem Ⅱ. Consider three cyclic codes with the intersection of the first two equal to the third. The syndromes of the two distinct bursts with respect to the first or second code are not the same if and only if the syndromes with respect to the third are not the same.