Suppose that and are two distinct odd prime numbers with . In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over was presented and their minimal polynomials and linear complexities were derived, where with a prime number and . The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over with length and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.

中文翻译：

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通过二阶通用二素广义环原子序列的循环码

假设和是两个不同的奇质数，其中。在本文中，证明了通用二素数广义切开术具有二阶以上的统一表示。基于此广义广义切分法，提出了一种定义为的二进制序列，并推导了它们的最小多项式和线性复杂度，其中用素数和。结果表明，这些序列的线性复杂度很高，对质数没有任何特殊要求。此外，我们利用这些序列获得了一定长度的一些循环码，并开发了许多循环码最小距离的下限。需要强调的是，本文中的某些循环码是最佳的。