Binary linear codes with few weights from two-to-one functions

In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$ with $\gcd(t, n)=1$ and $\gcd\left(e, 2^n-1\right)=1$. 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.

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

Recursive formulas generating power moments of multi-dimensional Kloosterman sums and m-multiple power moments of Kloosterman sums

In this paper, we construct two binary linear codes associated with multi-dimensional and m-multiple power Kloosterman sums (for any fixed m) over the finite field Fq. Here q is a power of two. The former codes are dual to a subcode of the binary hyper-Kloosterman code. Then we obtain two recursive formulas for the power moments of multi-dimensional Kloosterman sums and for the m-multiple power moments of Kloosterman sums in terms of the frequencies of weights in the respective codes. This is done via Pless power moment identity and yields, in the case of power moments of multi-dimensional Kloosterman sums, much simpler recursive formulas than those associated with finite special linear groups obtained previously