论文标题
几类的最佳$ p $ - ary循环代码，距离很小四个
Several classes of optimal $p$-ary cyclic codes with minimal distance four
论文作者
论文摘要
循环代码是线性代码的子类，由于其有效的编码和解码算法，在数据存储系统，通信系统和消费电子产品中具有广泛的应用。令$ p \ ge 5 $为奇数，$ m $为正整数。令$ \ MATHCAL {C} _ {（1，E，S）} $表示$ P $ -ARY环状代码，带有三个nonzeros $α$，$α^e $和$α^s $，其中$α$是$ {\ MathB f} _ {p^m} _ {p^m} $}的生成器$ s = \ frac {p^m-1} {2} $和$ 2 \ le e \ le p^m-2 $。在本文中，我们介绍了四类最佳$ p $ -ary环状代码$ \ mathcal {c} _ {（1，e，e，s）} $，带有参数$ [p^m-1，p^m-1，p^m-2m-2,4] $，通过分析在有限田地上的某些多功能物的求解。关于具有参数的最佳Quinary Cyclic代码$ [5^M-1,5^M-2M-2,4] $的一些先前结果是我们结构的特殊情况。此外，通过分析$ {\ mathbb f} _ {5^m} $的某些多项式的不可约因素，我们提出了两类最佳的Quariny Cyclic Codes $ \ MATHCAL {C} _ {（1，E，e，s）} $。
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 $p\ge 5$ be an odd prime and $m$ be a positive integer. Let $\mathcal{C}_{(1,e,s)}$ denote the $p$-ary cyclic code with three nonzeros $α$, $α^e$, and $α^s$, where $α$ is a generator of ${\mathbb F}_{p^m}^*$, $s=\frac{p^m-1}{2}$, and $2\le e\le p^m-2$. In this paper, we present four classes of optimal $p$-ary cyclic codes $\mathcal{C}_{(1,e,s)}$ with parameters $[p^m-1,p^m-2m-2,4]$ by analyzing the solutions of certain polynomials over finite fields. Some previous results about optimal quinary cyclic codes with parameters $[5^m-1,5^m-2m-2,4]$ are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over ${\mathbb F}_{5^m}$, we present two classes of optimal quinary cyclic codes $\mathcal{C}_{(1,e,s)}$.
