1 research outputs found
Three new classes of optimal frequency-hopping sequence sets
The study of frequency-hopping sequences (FHSs) has been focused on the
establishment of theoretical bounds for the parameters of FHSs as well as on
the construction of optimal FHSs with respect to the bounds. Peng and Fan
(2004) derived two lower bounds on the maximum nontrivial Hamming correlation
of an FHS set, which is an important indicator in measuring the performance of
an FHS set employed in practice.
In this paper, we obtain two main results. We study the construction of new
optimal frequency-hopping sequence sets by using cyclic codes over finite
fields. Let be a cyclic code of length over a finite field
such that contains the one-dimensional subcode Two codewords of are said to be equivalent if
one can be obtained from the other through applying the cyclic shift a certain
number of times. We present a necessary and sufficient condition under which
the equivalence class of any codeword in
has size . This result addresses an open question raised by Ding {\it et
al.} in \cite{Ding09}. As a consequence, three new classes of optimal FHS sets
with respect to the Singleton bound are obtained, some of which are also
optimal with respect to the Peng-Fan bound at the same time. We also show that
the two Peng-Fan bounds are, in fact, identical.Comment: to appear in Designs, Codes and Cryptograph