It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. 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 rational function fields, where q is a prime power. Our codes are constructed by evaluations of rational functions at all the rational places (including the place of "infinity") of the rational function field. Compared to the rational 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 with length q+1 over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. As a result, our codes have reasonable parameters as they are very close to the Singleton bound. Furthermore, our codes have better parameters than those obtained from MDS codes via code alphabet restriction or extension.
Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions
Southwest Jiaotong University（西南交通大学）；University of Paris VIII（巴黎第八大学）；CNRS, UMR 7539 LAGA, University of Paris XIII（法国国家科学研究中心，UMR 7539 LAGA，巴黎第十三大学）；China West Normal University（西华师范大学）
Construction of Rational Surfaces Yielding Good Codes
本文研究了某些有理曲面上的代数几何码。将最小距离的估计转化为平面曲线上的点计数问题。这个问题是通过应用上界的“à la Weil”的Aubry和Perret与霍马和金的平面曲线的界限。计算了几种有理曲面编码的参数。其中，研究了由椭圆二次曲面上的三次形式的求值所定义的码。据我们所知，这样的代码以前从未被处理过。另外两个合理的曲面进行了研究，并发现他们非常好的代码。特别地，在$\mathbf{F}_7$上发现了一个[57，12，34]码，在$\mathbf{F}_9$上发现了一个[91，18，53]码，这些码击败了目前已知的最好的码.
A $q$-analogue of the rational normal curve and linearized Reed-Solomon codes
Inria（法国国家信息与自动化研究所）；LIX, École polytechnique, Institut Polytechnique de Paris（巴黎理工学院高等学院计算机科学实验室）；Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539（巴黎第八大学，索邦巴黎北大学，法国国家科学研究中心几何、分析与应用实验室）；MPI Leipzig（莱比锡马克斯·普朗克数学研究所）
