844 research outputs found
How many weights can a cyclic code have ?
Upper and lower bounds on the largest number of weights in a cyclic code of
given length, dimension and alphabet are given. An application to irreducible
cyclic codes is considered. Sharper upper bounds are given for the special
cyclic codes (called here strongly cyclic), {whose nonzero codewords have
period equal to the length of the code}. Asymptotics are derived on the
function {that is defined as} the largest number of nonzero
weights a cyclic code of dimension over \F_q can have, and an algorithm
to compute it is sketched. The nonzero weights in some infinite families of
Reed-Muller codes, either binary or -ary, as well as in the -ary Hamming
code are determined, two difficult results of independent interest.Comment: submitted on 21 June, 201
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
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