The Existence of Linear Code for Any Access Structure

Abstract

In cryptographic field, it is open problem whether there is a linear code for given access structure $\Gamma$, and how to construct if it exists. In this paper, we mainly consider whether there is a linear code for given access structure $\Gamma$. Our contributions are that we give two necessary and sufficient conditions for given access structure$\Gamma$ which exists a linear code in finite field $\mathbb{F}_2$ and $\mathbb{F}_q$($q\neq 2$), respectively.