82,998 research outputs found
Binary linear codes with few weights from two-to-one functions
In this paper, we apply two-to-one functions over in two
generic constructions of binary linear codes. We consider two-to-one functions
in two forms: (1) generalized quadratic functions; and (2)
with and . Based on the study of the Walsh transforms of those functions
or their related-ones, we present many classes of linear codes with few nonzero
weights, including one weight, three weights, four weights and five weights.
The weight distributions of the proposed codes with one weight and with three
weights are determined. In addition, we discuss the minimum distance of the
dual of the constructed codes and show that some of them achieve the sphere
packing bound. { Moreover, several examples show that some of our codes are
optimal and some have the best known parameters.
On the weight distributions of several classes of cyclic codes from APN monomials
Let be an odd integer and be an odd prime. % with ,
where is an odd integer.
In this paper, many classes of three-weight cyclic codes over
are presented via an examination of the condition for the
cyclic codes and , which have
parity-check polynomials and respectively, to
have the same weight distribution, where is the minimal polynomial of
over for a primitive element of
. %For , the duals of five classes of the proposed
cyclic codes are optimal in the sense that they meet certain bounds on linear
codes. Furthermore, for and positive integers such
that there exist integers with and satisfying , the value
distributions of the two exponential sums T(a,b)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where , are
settled. As an application, the value distribution of is utilized to
investigate the weight distribution of the cyclic codes
with parity-check polynomial . In the case of and
even satisfying the above condition, the duals of the cyclic codes
have the optimal minimum distance
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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