Ideal access structures based on a class of minimal linear codes

Abstract

Coding theory is one of ways to construct ideal access structures. However, in general, determining the ideal access structures of the secret sharing schemes based on linear codes is very hard. According to the concept of minimal liner codes we proposed, this paper concentrates on irreducible cyclic codes and constructs the ideal access structures of the schemes based on the duals of minimal irreducible cyclic codes. In order to study the conditions whether several types of irreducible cyclic codes are minimal, we investigate the weight enumerators of certain irreducible cyclic codes by means of cyclotomic classes and Gaussian periods. On the basis of our aforementioned studies, we obtain ideal access structures and show the corresponding examples through programming.

Keywords:

• ideal access structure
• irreducible cyclic code
• minimal linear code
• period polynomial
• weight enumerator

2010 AMS Subject Classifications:

• 11T71
• 94B15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant No. 61373150], the Key Technologies R&D Program of Shaanxi Province [Grant No. 2013k0611] and the National Natural Science Foundation of China [Grant No. 11426148].