Covering and Secret Sharing with Linear Codes

Secret sharing has been a subject of study for over twenty years, and has had a number of real-world applications. There are several approaches to the construction of secret sharing schemes. One of them is based on coding theory. In principle, every linear code can be used to construct secret sharing schemes. But determining the access structure is very hard as this requires the complete characterisation of the minimal codewords of the underlying linear code, which is a difficult problem. In this paper we present a sufficient condition under which we are able to determine all the minimal codewords of certain linear codes. The condition is derived using exponential sums. We then construct some linear codes whose covering structure can be determined, and use them to construct secret sharing schemes with interesting access structures.