Point-line incidence on Grassmannians and majority logic decoding of Grassmann codes

Grassmannian空间中的计数编码

Enumerative Encoding in the Grassmannian Space

格拉斯曼空间中的码最近在网络编码中得到了应用。$\F_q^n$的$k$维子空间的表示在解决格拉斯曼的编码问题中，特别是在格拉斯曼的子空间的编码中，通常具有重要的作用。给出了格拉斯曼子空间的不同表示。我们使用两个这些表示的格拉斯曼枚举编码。一种是基于子空间的费勒斯图表示的枚举编码，另一种是基于子空间的标识向量和缩减行梯形表示的枚举编码。结合前两种方法的第三种方法比其他两种枚举编码更有效。

Cliques in projective space and construction of Cyclic Grassmannian Codes

The construction of Grassmannian codes in some projective space is of highly mathematical nature and requires strong computational power for the resulting searches. In this paper was constructed, using GAP System for Computational Discrete Algebra and Wolfram Mathematica, cliques in the projective space Pq(n) and then we use these to produce cyclic Grassmannian codes.

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 corr…

Covering Grassmannian Codes: Bounds and Constructions

Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n.$ Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding solutions for generalized combination networks. An $α$-$(n,k,δ)_q^c$ covering Grassmannian code $\mathcal{C}$ is a subset of $\mathcal{G}_q(n,k)$ such that every set of $α$ codewords of $\mathcal{C}$ spans a subspace of dimension at least $δ+k$ in $\mathbb{F}_q^n.$ In this paper, we derive new upper and lower bounds on the size of covering Grassmannian codes. T…

Enumerative Coding for Line Polar Grassmannians with applications to codes

A $k$-polar Grassmannian is the geometry having as pointset the set of all $k$-dimensional subspaces of a vector space $V$ which are totally isotropic for a given non-degenerate bilinear form $μ$ defined on $V.$ Hence it can be regarded as a subgeometry of the ordinary $k$-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume $k=2$ and $μ$ a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nic…

Polar Grassmannians and their Codes

We present a concise description of Orthogonal Polar Grassmann Codes and motivate their relevance. We also describe efficient encoding and decoding algorithms for the case of Line Grassmannians and introduce some open problems.