【摘要】 纠错码可以用来构造秘密共享方案,每个线性码均对应一个秘密共享方案.然而,一般情况下,基于线性码所得到的秘密共享方案的存取结构难以确定.循环码是线性码的一类非常重要的子码,由于其具有高效的编码和解码算法,循环码在数据存储系统、通信系统和消费者电子系统中有广泛应用.令p是一个奇素数, m是一个正整数.设α是■的一个生成元, 0≤e1, e2, e3≤p^m-2,■ 令■表示具有三个非零根■的p元循环码.首先,本文通过分析有限域上某些多项式的根的个数,给出两类参数为[5^m-1, 5^m-2m-2, 4]的最优五元循环码■.其次,给出了循环码■和■之间的一个联系.该联系表明可以利用最优循环码C_（0,1,e）来构造最优循环码C_（1,e,s）.最后,本文给出了一些基于五元循环码的秘密共享方案,结果表明所构造的秘密共享方案具有良好的存取结构.更多还原

【Abstract】 Error-correcting codes can be used to construct secret sharing schemes. Every linear code can be used to construct a secret sharing scheme. However, it is usually difficult to determine the access structure of the secret sharing scheme based on a linear code. Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let p be an odd prime and m be a positive integer. Let α be a generator of■ denote the p-ary cyclic code with three nonzeros ■This paper presents two classes of optimal quinary cyclic codes■ with parameters [5^m-1, 5^m-2m-2, 4] by analyzing the number of roots of certain polynomials. Then a connection between p-ary cyclic codes■ is given. This connection can be used to obtain optimal p-ary cyclic codes C_（1,e,s） from optimal p-ary cyclic codes C_（0,1,e）. Finally, some secret sharing schemes based on quinary cyclic codes are proposed. Those secret sharing schemes are shown to have good access structures.更多还原