28 research outputs found
The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over F3
Cyclic codes have efficient encoding and decoding algorithms. The decoding
error probability and the undetected error probability are usually bounded by
or given from the weight distributions of the codes. Most researches are about
the determination of the weight distributions of cyclic codes with few
nonzeros, by using quadratic form and exponential sum but limited to low
moments. In this paper, we focus on the application of higher moments of the
exponential sum to determine the weight distributions of a class of ternary
cyclic codes with three nonzeros, combining with not only quadratic form but
also MacWilliams' identities. Another application of this paper is to emphasize
the computer algebra system Magma for the investigation of the higher moments.
In the end, the result is verified by one example using Matlab.Comment: 10 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
Two classes of reducible cyclic codes with large minimum symbol-pair distances
The high-density data storage technology aims to design high-capacity storage
at a relatively low cost. In order to achieve this goal, symbol-pair codes were
proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output
pairs of overlapping symbols. Such a channel is called symbol-pair read
channel, which introduce new concept called symbol-pair weight and minimum
symbol-pair distance. In this paper, we consider the parameters of two classes
of reducible cyclic codes under the symbol-pair metric. Based on the theory of
cyclotomic numbers and Gaussian period over finite fields, we show the possible
symbol-pair weights of these codes. Their minimum symbol-pair distances are
twice the minimum Hamming distances under some conditions. Moreover, we obtain
some three symbol-pair weight codes and determine their symbol-pair weight
distribution. A class of MDS symbol-pair codes is also established. Among other
results, we determine the values of some generalized cyclotomic numbers