User-Detectable Sequences for the Collision Channel Without Feedback Yijin Zhang, Kenneth W. Shum, Wing Shing Wong Department of Information Engineering The Chinese University of Hong Kong Shatin, Hong Kong Abstract—Protocol sequences are used for distributed multiple accessing in the collision channel without feedback. In this paper we consider user-detectable sequences with the property that each active user can be detected by looking at the channel activity only, within some bounded delay. It is important in some applications such as ad hoc networks. Some lower and upper bounds of its minimum period are investigated in this paper. In addition, we display some interconnections with some other sequence designs. Index Terms—Collision channel without feedback, protocol se- quences, user-irrepressible sequences, CRT construction, optical orthogonal code. I. I NTRODUCTION Massey and Mathys [1], [2] introduced the model of the collision channel without feedback for multiple access communication. Consider a time-slotted system, consisting of users and one sink. All users may be active at the same time. Each user is assigned a binary deterministic sequence with length , called a protocol sequence. For =1, 2,..., , the protocol sequence associated with user is specified by a row vector := [ (0) (1) ... ( − 1)]. When a user changes from inactive to active, protocol sequence assigned are read slot by slot periodically. It transmits a packet within the boundaries of a time slot if and only if the value of the protocol sequence at that time slot equals one. If two or more users send simultaneously, we say that there is a collision and we assume that no information can be recovered. If one and only one user transmits, the packet can be received error-free. When a user changes from active to inactive, it is assumed that after the end of the sequence, the user must keep silent for at least one period before becoming active again. There is some complication due to delay offsets. As there is no feedback from the receiver and no cooperation among the users, each user has a delay offset which is a random integer but remains fixed throughout the communication session. In other words, all protocol sequences sent by active users are not frame synchronized. Suppose user starts a protocol sequence at the time index 0 . It will send a packet if ( − 0 ) equals 1. For practical considerations, one would like to remove the assumption that the slot boundaries are synchronized, i.e., 0 This work was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region under Project 417909. here may be a non-integral number. It is, in fact, possible to do so and to allow the users to be totaly asynchronous. Our result can also be extended to this more general scenario. In multiple access transmission without packet header, three tasks [3], [4] should be solved by the receiver through observ- ing the channel activity (whether a time slot is idle, containing a collision or a successful packet), viz.: (i) to detect each active user (detection), (ii) to determine the sender of each successful received packet (decoding), and (iii) to find their delay offsets (synchronization). In this paper we investigate only the detection problem, as task (ii) and (iii) may be not necessary for some applications. We want to find protocol sequences that allow any active user be detected by the receiver via some algorithm within some bounded delay if and only if it has become active. Such protocol sequence set is said to be user-detectable (UD). The notion of user-detectability is also addressed in another context for the OR channel, under the name uniquely deci- pherable code [5], [6] with the assumption all active users start its codeword at the same time, which can be viewed as a special case of the concept discussed here. In this paper, in order to explore the minimal delay in the worst case, we are interesting in min ( ), the smallest length such that a UD sequence set exists for users. The paper is organized as follows. After setting up some notations and definitions in Section II, we establish a lower bound on min ( ) in section III. Then an upper bound and related constructions are presented in section IV. Section V gives a proof to show the existence of UD sequence set different from that in section IV. Finally, we close in Section VI with some concluding remarks. II. NOTATIONS AND DEFINITIONS We will use sequence “period” and sequence “length” inter- changeably. Given a binary sequence (), =0, 1,..., − 1, of length , we define its Hamming weight as () := −1 ∑ =0 (). Let the cyclic shift of a sequence by relative shift be {zyj007, wswong}@ie.cuhk.edu.hk wkshum@inc.cuhk.edu.hk PREPRESS PROOF FILE CAUSAL PRODUCTIONS 1