Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of $q$-ary projective two-weight linear codes is presented, where $q$ is a power of 2. The parameters of both the codes and their duals are excellent. As applications, the codes are used to derive strongly regular graphs with new parameters and secret sharing schemes with interesting access structures.
It is well-known that few-weight linear codes have better applications in secret sharing schemes \cite{JY2006,CC2005}.In particular, projective two-weight codes are very precious as they are closely related to finite projective spaces, strongly regular graphs and combinatorial designs \cite{RC1986,CD2018,P1972}. Here, we present the following two applications.
In this paper, for an odd prime $p$, several classes of two-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from the defining sets, and then their complete weight distributions are determined by employing character sums. These codes can be suitable for applications in secret sharing schemes. Furthermore, two new classes of projective two-weight codes are obtained, and then two new classes of strongly regular graphs are given.
Boolean functions can be used to construct binary linear codes in many ways, and vice versa. The objective of this short article is to point out a connection between the weight distributions of all projective binary linear codes and the Walsh spectra of all Boolean functions. New research problems are also proposed.
Maximum number of zeroes of polynomials on weighted projective spaces over a finite field
Univ Rennes, CNRS, IRMAR - UMR 6625（雷恩大学）；Department of Mathematics Virginia Tech（弗吉尼亚理工大学数学系）；IMUVA-Mathematics Research Institute, Universidad de Valladolid（巴利亚多利德大学 IMUVA 数学研究所）
In this paper we find the second generalized Hamming weight of some evaluation codes arising from a projective torus, and it allows to compute the second generalized Hamming weight of the codes parameterized by the edges of any complete bipartite graph. Also, at the beginning, we obtain some results about the generalized Hamming weights of some evaluation codes arising from a complete intersection when the minimum distance is known and they are non--degenerate codes. Finally we give an example where we use these results to determine the complete weight hierarchy of some codes.
