서강대학교

초록/요약

In this thesis, we consider various orthogonals defined by four dualities on additive codes and apply a special class of them to secret sharing schemes. For additive codes, we examine four dualities and define various orthogonals over the fields of even order. We relate the MacWilliams relations and the duals of F_{2^{2s}} codes for these dualities. We also study self-dual codes with respect to four dualities and prove that any subgroup of order 2^s of an additive group is a self-dual code with respect to some dualities. The building-up constructions are used to additive self-dual codes to obtain additive self-orthogonal codes. Additive codes, especially those over F_4, are applied to Secret Sharing Schemes (SSSs) which were first introduced by Shamir. Contemplating that additive codes form a natural generalization of linear codes, a construction of SSSs based on additive codes over F_4 can be considered as a generalization of the previous SSSs. We construct SSSs based on additive codes over F_4, showing that the scheme requires at least two steps of calculations to reveal a secret. We also define minimal access structures of SSSs from additive codes over F_4 and describe SSSs using some interesting additive codes over F_4 which contain generalized 2-designs.