2 research outputs found
On an open problem about a class of optimal ternary cyclic codes
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems and communication systems as they have
efficient encoding and decoding algorithms. In this paper, we settle an open
problem about a class of optimal ternary cyclic codes which was proposed by
Ding and Helleseth. Let be a cyclic code of length over
GF(3) with two nonzeros and , where is a generator
of and e is a given integer. It is shown that is
optimal with parameters if one of the following conditions
is met. 1) , , and . 2)
, , and
A class of optimal ternary cyclic codes and their duals
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. Let for an integer
and be a generator of \gf(3^m)^*. In this paper, a class
of cyclic codes \C_{(u,v)} over \gf(3) with two nonzeros and
is studied, where , and is the
ternary Welch-type exponent. Based on a result on the non-existence of
solutions to certain equation over \gf(3^m), the cyclic code \C_{(u,v)} is
shown to have minimal distance four, which is the best minimal distance for any
linear code over \gf(3) with length and dimension
according to the Sphere Packing bound. The duals of this class of cyclic codes
are also studied