Symmetric Properties and Subspace Degradations of Linear Operator Channels over Finite Fields

Motivated by the communication through a network employing linear network coding, linear operator channels (LOCs) over finite fields are studied with arbitrarily distributed transfer matrices. Some intrinsic symmetric properties of LOCs are revealed and are used to simplify transition matrix computation and input distribution optimization. Subspace coding for LOCs is studied with the help of the symmetric properties. Our results demonstrate that using constant-dimensional subspace coding are good enough for many typical parameters. For LOCs satisfying certain constraints, the optimal subspace coding is constant-dimensional. Simple method is derived to find an optimal constantdimensional input distribution, as well as the maximum achievable rate using constant-dimensional subspace coding. Index Terms linear operator channel, linear network coding, subspace coding