A new class of the universal representation for the positive integers is proposed. The positive integers are divided into infinite groups, and each positive integer n is represented by a pair of integers (p,q), which means that n is the q-th number in the p-th group. It is shown that the new class includes the message length strategy as a special case, and the asymptotically optimal representation can easily be realized. Furthermore, a new asymptotically and practically efficient representation scheme is proposed, which preserves the numerical, lexicographical, and length orders.