介绍三种最熟知的Goppa代码 (Introducing Three Best Known Goppa Codes)

The current best known $[239, 21], \, [240, 21], \, \text{and} \, [241, 21]$ binary linear codes have minimum distance 98, 98, and 99 respectively. In this article, we introduce three binary Goppa codes with Goppa polynomials $(x^{17} + 1)^6, (x^{16} + x)^6,\text{ and } (x^{15} + 1)^6$. The Goppa codes are $[239, 21, 103], \, [240, 21, 104], \, \text{and} \, [241, 21, 104]$ binary linear codes respectively. These codes have greater minimum distance than the current best known codes with the respective length and dimension. In addition, with the techniques of puncturing, shortening, and extending, we find more derived codes with a better minimum distance than the current best known codes with the respective length and dimension.

翻译：目前最著名的 $[239, 21], \, [240, 21],\,\,\\ text{和} \, [241, 21]$二进制线性代码有最小距离 98, 98, 和 99。在本条中, 我们引入了三种二进制哥帕代码, 与Goppa 多边代码$( x ⁇ 17} + 1), 16, (x ⁇ 16} + x), 6,\ text { 和 } (x ⁇ 15} + 1), 。 Goppa 代码是 $ 239, 21, 103], \, [240, 21, 104], \, \,\ text{和} $ 。这些代码的最小距离比目前已知的最佳代码的长度和维度要大。此外, 我们发现比当前已知的最佳代码的长度和维度要远得多, 我们发现比当前最已知的代码的最小距离。