论文标题
循环代码和扩展的原始循环代码及其应用的构造
Constructions of cyclic codes and extended primitive cyclic codes with their applications
论文作者
论文摘要
具有几个权重的线性代码具有许多不错的应用程序，包括组合设计，分布式存储系统，秘密共享方案等。在本文中，我们根据有限领域的特殊多项式构建了两个线性代码的家族。线性代码的第一个家族是仿射不变的扩展原始循环代码。线性代码的第二家族是可还原的循环代码。确定这些代码及其双重的参数。作为第一个应用程序，我们证明了这两个线性代码的家族含有$ t $ -Designs，其中$ t = 2,3 $。作为第二个应用程序，也确定了代码的最低位置，并得出了最佳的局部可回收代码。
Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold $t$-designs, where $t=2,3$. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived.
