论文标题
缩短有限字段的线性代码
Shortened Linear Codes over Finite Fields
论文作者
论文摘要
穿刺和缩短技术是构建旧线性代码的两种重要方法。在过去的70年中，已经取得了许多进展，并进行了许多关于刺穿线性代码的工作。通过刺穿技术获得了许多具有有趣参数的线性代码系列。但是，对缩短技术的研究很少，只有少数关于缩短线性代码的参考。本文的第一个目的是证明缩短线性代码的一些一般理论。第二个目标是研究一些缩短的锤码代码，单纯形码，一些芦苇 - 毛刺代码和卵形代码的代码。本文介绍了11个具有有趣参数的最佳缩短代码的家庭。作为副产品，本文中提出的一些缩短代码也构建了五个$ 2 $ designs的无限家庭。
The puncturing and shortening technique are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes with interesting parameters are presented in this paper. As a byproduct, five infinite families of $2$-designs are also constructed from some of the shortened codes presented in this paper.
