Problems of Information Transmission, Vol. 38, No. 4, 2002, pp. 268–279. Translated from Problemy Peredachi Informatsii, No. 4, 2002, pp. 24–36.
Original Russian Text Copyright
2002 by Tsybakov, Rubinov.
INFORMATION THEORY AND CODING THEORY
Some Constructions of Conﬂict-Avoiding Codes
B. S. Tsybakov and A. R. Rubinov
Received August 14, 2001; in ﬁnal form, June 18, 2002
Abstract—Constructions of conﬂict-avoiding codes are presented. These codes can be used as
protocol sequences for successful packet transmission over a collision channel without feedback.
We give a relation between codes that avoid conﬂicts of diﬀerent numbers of colliding users.
Upper bounds on the maximum code size and three particular code constructions are presented.
We consider binary conﬂict-avoiding codes. A length-N codeword of such a code is assigned to a
user as his protocol sequence. Each user uses its protocol sequence for transmission over a collision
channel without feedback. The maximum number of channel users can at most be equal to the
number of codewords in the code. The channel is slotted. The users have slot synchronization.
No other synchronization is assumed. A user can transmit his packets in his session with a length of
N slots. It is assumed that any slot can belong to at most k sessions. During his session, each user
transmits packets according to his protocol sequence. If a user begins his session at time T (T is
the beginning of slot T ) and his protocol sequence has symbol 1 (symbol 0) at the ﬁrst position,
then the user transmits (does not transmit, respectively) a packet in slot T . Similarly, if there is
symbol 1 (symbol 0) at the ith position, then the user transmits (does not transmit, respectively)
apacketinslotT + i − 1, 1 ≤ i ≤ N . If exactly one user sends a packet in a slot, the packet
has successful transmission. If more than one user transmit packets in a slot, there is a conﬂict in
the slot and none of the packets has successful transmission. The protocol sequences must be such
that they guarantee successful transmission of at least r packets for each user within his session.
The set of such protocol sequences, or words, is called a conﬂict-avoiding code with the following
parameters: N, the word length; k, the maximum number of sessions intersecting in a slot; and r,
the minimum number of successfully transmitted packets in a session.
Important results on a collision channel without feedback and on conﬂict-avoiding codes are
presented in [1–7]. The problems considered in these papers and our present paper are illustrated
in Fig. 1 for the case of k =2. Fork = 2, precise formulations of the problems considered in this
paper are given below in Sections 2 and 3.
Fig. 1a shows the problem considered in [1–6]. In this problem, one session, called the “signal
codeword,” can collide in the channel with two other sessions, called “noise codewords,” which
go one after another without a gap. Collision of the signal codeword with each of the two noise
codewords is partial; otherwise, the signal codeword collides fully with only one of these noise
codewords. These two noise codewords are called the same in the sense that they are sessions of
the same user. This means that the signal codeword can collide with any cyclic shift of another
codeword. The problem was motivated by the traditional-for-information-theory consideration of
Fig. 1b shows the problem considered in . In this problem, each codeword must have length
− 1 and at least N
− 1 zeros at the end. This means that a signal codeword of length N
extended with the all-zero sequence of length N
− 1 can collide fully with any cyclic shift of
2002 MAIK “Nauka/Interperiodica”