48 IEEE TRANSACTIONS ONINFORMATION THEORY, VOL. 41, NO.2, MARCH 1995 New Constructions of Optimal Cyclically Permutable Constant Weight Codes oo, Membe, IEEE, Zhn Zhn, e Membe, EEE, . , Mebe, IEEE, . o s-Three new construcions for familis of cyclic con stant weight cods are prsented. All are asymptoically optimum in the sense that in each case, as the length of the sequencs within the family approachs ininity, tbe raio of family size the maximum possible under the Johnson upper bound, approacbes unity. s-, cyclic constant weight cdes, cyclically permutahle codes, optical orthogonal codes, protocol sequences, correlation, crosscorrelation. I. INTRODUCTION A N (n, w,') binary cyclically permutable constant weight code (we call it here a CPCW code) C, where 1 :: , :: w :: n, is a family of 1 }-sequences of length and Hamming weight w satisfying the following two conditions: - L x(k ) x(k " ) , (1) =O for all sequences x C and all integers T (mod n) and n- l Lx(k)y(klnT):: , (2) =O for all pairs of distinct sequences x(,), C and all integers T, where denotes addition modulo n. Codes with these propcrties have been called optical orthog onal codes in papers [1]-[3J in connection with applications for optical channels and cyclically permutable constant wcight codes (see [4] and references there) in connection to con structing protocol sequences for the multiuser collision channel without fecdback. We use the latter terminology because it its better the general results given here, which are not restricted to applications for optical communication channels. Manuscript receivedMay 21,1993; revised Januar 4,1994. This research was supported in part by the National Science Foundation under GrantsRI!- 9014056,NCR-80505, and NCR-9016077, by the Ofice of Naval Research under Grant NOOOI4-90-J-1301, and by the Computational Mathematics Group of the EPSCoR of Puerto Rico Grant. This paper as presented at the IEEE InteationalSmposium on InformationTheor,SanAntonio, TX, Jan. 17-22, 1993. O. Moreno is with the Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 0931 USA V. A. Zinovie iswith the Departmentof Mathematics, University ofPuerto Rico, Rio Piedras, PR 0931 USA, and with the Institute for Problems of Information Transmission, Russi an Academy of Sciences, GSP-4, Moscow, 101447, Russia. Z. Zhang and P. V. Kuar are with Communication Sciences Institute, University of Sohe Califonia, Los Angeles, CA 90089-2565 USA. IEEE Log Number 9406688. For a given set of values of n, w, A, let >(n,w, ) denote the largest possible cardinality of an (n, w, ,) CPCW code. Upper bounds for this function and several optimal constructions can be found in [1]-[41. An easy upper bound derived from the Johnson bound A(n,2(w - ,),w) (see [5]) statcs that >(n , w , ,):: l A(n,2(W n -,),w) J (n-l)(n-2)",(n -,) , (3) - w(w - ) . . · (w A) Constant weight codes, which are optimal relative to this Johnson upper bound, are also of interest from the combina torial point of view (see [6] and references there). In this paper, three constructions ( B, and C) for families of CPCW codes are presented. In every case, the families ,are asymptotically optimum in the sense that, as the length of the sequence family goes to infinity, the ratio of the size of the code to that of the maximum permissible as determined by the bound in (3) above, approaches unity. All three constructions make use of the following two ideas, Let n be an integer that can be expressed as the product n = f!1!2 of two relatively prime integers nl and n2. Then, from an application of the Chinese remainder theorem, it follows that the construction of sets of {O, 1} sequences with periodic correlation bounded above by , is completely equivalent to the task of constructing a collection of arrays whose doubly periodic correlation is bounded above by A_ Secondly, the codewords within each family are required to have constant weight. The sequences in each of the three families A B, and C when represented in matrix form appear the graph of a function mapping Zn2 > Z"' This guarantees that they all have constant weight (approximately) 1,2. The functions in A and B are polynomials, whereas construction C uses rational functions, Precise parameters of the three families constructed are tabulatcd below. Reference l4J appeared after we had written the initial version of this paper. The two papers share some material in common such as the idea behind the construction as well as some featurcs of consuction A. On the other hand, Section III of this paper was written after paper [4], and was inspired by it. II. CONSTRUCTIONS The three constructions that follow depend upon the fol lowing observation: 0018-9448/95$04.00 ) 1995 IEEE