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

[1]  R. Koetter,et al.  An algebraic approach to network coding , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[2]  Bartolomeu F. Uchôa Filho,et al.  On the Capacity of Multiplicative Finite-Field Matrix Channels , 2011, IEEE Transactions on Information Theory.

[3]  Tracey Ho,et al.  A Random Linear Network Coding Approach to Multicast , 2006, IEEE Transactions on Information Theory.

[4]  K. Jain,et al.  Practical Network Coding , 2003 .

[5]  Shuo-Yen Robert Li,et al.  Linear network coding , 2003, IEEE Trans. Inf. Theory.

[6]  En-Hui Yang,et al.  Coding for linear operator channels over finite fields , 2010, 2010 IEEE International Symposium on Information Theory.

[7]  Colin Cooper,et al.  On the distribution of rank of a random matrix over a finite field , 2000, Random Struct. Algorithms.

[8]  Frank R. Kschischang,et al.  Communication Over Finite-Field Matrix Channels , 2008, IEEE Transactions on Information Theory.

[9]  En-Hui Yang,et al.  Linear Operator Channels over Finite Fields , 2010, ArXiv.

[10]  D. Lun,et al.  Methods for Efficient Network Coding , 2006 .

[11]  E. Yang,et al.  Optimality of subspace coding for linear operator channels over finite fields , 2010, 2010 IEEE Information Theory Workshop on Information Theory (ITW 2010, Cairo).

[12]  Suhas N. Diggavi,et al.  On the Capacity of Noncoherent Network Coding , 2011, IEEE Transactions on Information Theory.

[13]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2008, IEEE Trans. Inf. Theory.

[14]  Raymond W. Yeung,et al.  Information Theory and Network Coding , 2008 .

[15]  Baochun Li,et al.  How Practical is Network Coding? , 2006, 200614th IEEE International Workshop on Quality of Service.