Problems of Information Transmission, Vol. 39, No. 3, 2003, pp. 239–254. Translated from Problemy Peredachi Informatsii, No. 3, 2003, pp. 11–27.
Original Russian Text Copyright
2003 by Balakirsky, Vinck.
INFORMATION THEORY AND CODING THEORY
On the Performance of Permutation Codes
for Multi-User Communication
V. B. Balakirsky and H. Vinck
Institute for Experimental Mathematics, Essen, Germany
Received March 5, 2002; in ﬁnal form, February 19, 2003
Abstract—Permutation coding for multi-user communication schemes that originate from the
Fast Frequency Hopping/Multiple Frequency Shift Keying modulation is investigated. Each
sender is either passive or sends some signal formed as the concatenation of M elementary
signals having M diﬀerent speciﬁed frequencies. There is also a jammer, who can introduce
disturbances. A single disturbance is either sending the signal that contains all M frequencies
at a certain time instant or sending some elementary signal at all time instants. Each receiver
receives a vector of M sets, where a set at each time instant contains a ﬁxed frequency if and
only if the corresponding elementary signal was sent by either some sender or the jammer.
The task of the receiver is to uniquely decode the message of his sender. We present regular
constructions of permutation codes for this scheme given the following parameters: the number
of frequencies, number of pairs (sender, receiver), number of messages per sender, and maximum
number of disturbances of the jammer.
In , Chang and Wolf introduced the T -user M-frequency noiseless multiple access channel
as a model of interest for diﬀerent communication schemes (see, for example, [2, 3]). Each of
the T senders uses the same input alphabet of M integers and the so-called “idle symbol.” The
output of the channel at each time instant is a set consisting of all integers that occur at the channel
input at this instant. This model of a communication channel can be viewed as a basis model in the
case of a system with many users which transmit signals containing M diﬀerent frequencies (the
so-called Fast Frequency Hopping/Multiple Frequency Shift Keying [FFH/MFSK] modulation).
Here, each of the T receivers has M frequency detectors to determine the set of signals transmitted
through the channel at each time instant.
We consider the further restricted situation where each independent sender is allowed to transmit
either M idle symbols (“the zero message”) or a codeword of length M obtained as a permutation of
components of the vector (0,...,M− 1), i.e., the transmitted codeword has length M and contains
each integer only once (see, for example, [4–6]). The task of the tth receiver is to determine the
message of the tth sender.
The permutation constraint on the set of all possible codewords makes sense from a commu-
nication theory point of view, since it implies that a constant energy is sent by each sender to
the channel and a ﬁxed frequency band is used within the slot of M time instants. On the other
hand, this constraint allows us to design speciﬁc codes for many values of M, T ,andL,whereL
is the number of possible nonzero messages per sender. The characteristics of these codes can be
The work was done in the Institute for Experimental Mathematics, Essen, Germany, with a partial ﬁnancial
support of the DFG.
2003 MAIK “Nauka/Interperiodica”