Binary Linear Codes With Few Weights From Two-to-One Functions

Abstract

In this paper, we apply two-to-one functions over b F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2ⁿ</sub> in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) (x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2t</sup> +x) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> with gcd(t, n)=gcd(e, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -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.