Abstract
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.
Highlights
The classes of cyclic codes play a very significant role in the theory of error-correcting codes
We present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums
We mainly investigate the weight distributions of reducible cyclic codes which had been determined by exponential sums
Summary
The classes of cyclic codes play a very significant role in the theory of error-correcting codes. We survey some results on the weight distributions of cyclic codes over finite fields that have been recently determined by exponential sums. The weight distributions of cyclic codes had been determined in a few cases by using mathematical tools, such as Gauss periods, Gauss sums, quadratic forms, and the numbers of the solutions of equations over finite fields. We mainly investigate the weight distributions of reducible cyclic codes which had been determined by exponential sums. Weight distributions of cyclic codes over finite fields of cyclic codes whose duals have arbitrary zeros. We hope that this paper will show that weight distributions of cyclic codes which are determined by exponential sums in general
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