论文标题
通过代数函数字段的非线性代码的新结构
A new construction of nonlinear codes via algebraic function fields
论文作者
论文摘要
在编码理论中，构建具有良好参数的代码是最重要，最根本的问题之一。尽管已经生产了许多好的代码，但其中大多数是在等于主要功率的字母上定义的。在本文中，我们通过代数函数字段提供了$（q+1）$ - ARY非线性代码的新的明确结构，其中$ Q $是主要功率。我们的代码是通过对代数函数字段的所有合理位置的理性函数的评估来构建的。与代数几何代码相比，主要区别在于，我们允许在极点评估有理功能。在评估了从Riemann-Roch空间联合的合理函数之后，我们通过字母$ \ Mathbb {f} _ {q} \ CUP \ cup \ {\ infty \} $获得了非线性代码系列。事实证明，我们的代码比从MDS代码或通过代码字母扩展和限制获得的良好代数几何代码获得的参数更好。
In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of $(q+1)$-ary nonlinear codes via algebraic function fields, where $q$ is a prime power. Our codes are constructed by evaluations of rational functions at all rational places of the algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. It turns out that our codes have better parameters than those obtained from MDS codes or good algebraic geometry codes via code alphabet extension and restriction.
