格拉斯曼轨道结构G$2, m}美元和格拉斯曼代码解码器$C(2, m美元) (Orbit Structure of Grassmannian $G_{2, m}$ and a decoder for Grassmann code $C(2, m)$)

In this manuscript, we consider decoding Grassmann codes, linear codes associated to Grassmannian of planes in an affine space. We look at the orbit structure of Grassmannian arising from the natural action of multiplicative group of certain finite field extension. We project the corresponding Grassmann code onto these orbits to obtain a few subcodes of certain Reed-Solomon code. We prove that some of these projected codes contains an information set of the parent Grassmann code. By improving the efficiency of Peterson's decoding algorithm for the projected subcodes, we prove that one can correct up to $\lfloor d-1/2\rfloor$ errors for Grassmann code, where $d$ is the minimum distance of Grassmann code.

翻译：在此手稿中, 我们考虑解码格拉斯曼代码, 与格拉斯曼空域中的飞机的格拉斯曼代码相关的线性代码。我们查看格拉斯曼的轨道结构, 由某些有限外延的倍数组的自然动作产生。我们将这些轨道上相应的格拉斯曼代码投射到这些轨道上, 以获得某些 Reed- Solomon 代码的几分子代码。我们证明其中的一些预测代码包含一个母格拉斯曼代码的信息集。通过提高彼得森为预测的子代码解码算法的效率, 我们证明可以纠正格拉斯曼代码中高达$\ lplage d-1/2\r 底值的错误, 在那里, $d是格拉斯曼代码的最小距离。