Coding protection for magnetic tapes: A generalization of the Patel - Hong code

1985 where Pmin = ypk F PInax = map, k occur together on every line. Lemma 1 applies and hence the system (7) reduces to one equation, 62(KPrnin-l)-6(0KP~~-Pain)-eP~a.x r O. (12) The coefficient of a2 and the free term both are negative. Hence, (12) admits positive solutions if and only if the discriminant is positive and the coefficient of-S is negative. We get e2K2 2 Pmax-2~(2PIn,-KP~, pain) + p&n 2 0 (134 The coefficient of o2 in (13a) is positive, and the left side of (13a) is negative for 6 = p,J(Kp,,). Hence, the solution of (13) is the smaller of the two roots, so and D,* is given by It is easy to check that 0: > 0 if pmin > 0, i.e., if there are no nonzero transitions. For K = 2, D,* equals Gray's formula [2] for the binary symmetric source, namely,