27,155 research outputs found
The Weight Enumerator of Three Families of Cyclic Codes
Cyclic codes are a subclass of linear codes and have wide applications in
consumer electronics, data storage systems, and communication systems due to
their efficient encoding and decoding algorithms. Cyclic codes with many zeros
and their dual codes have been a subject of study for many years. However,
their weight distributions are known only for a very small number of cases. In
general the calculation of the weight distribution of cyclic codes is heavily
based on the evaluation of some exponential sums over finite fields. Very
recently, Li, Hu, Feng and Ge studied a class of -ary cyclic codes of length
, where is a prime and is odd. They determined the weight
distribution of this class of cyclic codes by establishing a connection between
the involved exponential sums with the spectrum of Hermitian forms graphs. In
this paper, this class of -ary cyclic codes is generalized and the weight
distribution of the generalized cyclic codes is settled for both even and
odd alone with the idea of Li, Hu, Feng, and Ge. The weight distributions
of two related families of cyclic codes are also determined.Comment: 13 Pages, 3 Table
Recent progress on weight distributions of cyclic codes over finite fields
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions
A Class of Three-Weight 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, a class of
three-weight cyclic codes over \gf(p) whose duals have two zeros is
presented, where is an odd prime. The weight distribution of this class of
cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a
subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page
Five Families of Three-Weight Ternary Cyclic Codes and Their Duals
As a subclass of linear codes, cyclic codes have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, five families of
three-weight ternary cyclic codes whose duals have two zeros are presented. The
weight distributions of the five families of cyclic codes are settled. The
duals of two families of the cyclic codes are optimal
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