4 research outputs found
Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)
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, monomials and
trinomials over finite fields with even characteristic are employed to
construct a number of families of binary cyclic codes. Lower bounds on the
minimum weight of some families of the cyclic codes are developed. The minimum
weights of other families of the codes constructed in this paper are
determined. The dimensions of the codes are flexible. Some of the codes
presented in this paper are optimal or almost optimal in the sense that they
meet some bounds on linear codes. Open problems regarding binary cyclic codes
from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687,
arXiv:1206.437
Cyclic Codes from Dickson Polynomials
Due to their efficient encoding and decoding algorithms cyclic codes, a
subclass of linear codes, have applications in consumer electronics, data
storage systems, and communication systems. In this paper, Dickson polynomials
of the first and second kind over finite fields are employed to construct a
number of classes of cyclic codes. Lower bounds on the minimum weight of some
classes of the cyclic codes are developed. The minimum weights of some other
classes of the codes constructed in this paper are determined. The dimensions
of the codes obtained in this paper are flexible. Most of the codes presented
in this paper are optimal or almost optimal in the sense that they meet some
bound on linear codes. Over ninety cyclic codes of this paper should be used to
update the current database of tables of best linear codes known. Among them
sixty are optimal in the sense that they meet some bound on linear codes and
the rest are cyclic codes having the same parameters as the best linear code in
the current database maintained at http://www.codetables.de/