79,321 research outputs found
Binary Linear Codes With Few Weights From Two-to-One Functions
In this paper, we apply two-to-one functions over b F 2n in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x 2t +x) e with gcd(t, n)=gcd(e, 2 n -1)=1. Based on the study of the Walsh transforms of those functions or their variants, 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, examples show that some codes in this paper have best-known parameters.acceptedVersio
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.
Weight hierarchies of a family of linear codes associated with degenerate quadratic forms
We restrict a degenerate quadratic form over a finite field of odd
characteristic to subspaces. Thus, a quotient space related to is
introduced. Then we get a non-degenerate quadratic form induced by over the
quotient space. Some related results on the subspaces and quotient space are
obtained. Based on this, we solve the weight hierarchies of a family of linear
codes related to Comment: 12 page
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