On multiplicative matrix channels over finite chain rings

Motivated by nested-lattice-based physical-layer network coding, this paper considers communication in multiplicative matrix channels over finite chain rings. Such channels are defined by the law Y = AX, where X and Y are the input and output matrices, respectively, and A is called the transfer matrix. We assume that the instances of the transfer matrix are unknown to the transmitter, but available at the receiver. As contributions, we obtain a closed-form expression for the channel capacity, and we propose a coding scheme that can achieve this capacity with polynomial time complexity. Our results extend the corresponding ones for finite fields.

[1]  Michael Gastpar,et al.  Compute-and-Forward: Harnessing Interference Through Structured Codes , 2009, IEEE Transactions on Information Theory.

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

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

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

[5]  Muriel Médard,et al.  An algebraic approach to network coding , 2003, TNET.

[6]  H. O. Foulkes Abstract Algebra , 1967, Nature.

[7]  Shenghao Yang,et al.  Capacity Analysis of Linear Operator Channels Over Finite Fields , 2014, IEEE Transactions on Information Theory.

[8]  A. A. Nechaev,et al.  Finite Rings with Applications , 2008 .

[9]  Suhas N. Diggavi,et al.  On the capacity of non-coherent network coding , 2009, 2009 IEEE International Symposium on Information Theory.

[10]  William C. Brown,et al.  Matrices over commutative rings , 1993 .

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

[12]  Frank R. Kschischang,et al.  Communication Over Finite-Chain-Ring Matrix Channels , 2013, IEEE Transactions on Information Theory.

[13]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

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

[15]  Thomas Honold,et al.  Linear Codes over Finite Chain Rings , 1999, Electron. J. Comb..

[16]  Soung Chang Liew,et al.  Physical-layer network coding: Tutorial, survey, and beyond , 2011, Phys. Commun..

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

[18]  Frank R. Kschischang,et al.  A Rank-Metric Approach to Error Control in Random Network Coding , 2007, IEEE Transactions on Information Theory.

[19]  B. R. McDonald Finite Rings With Identity , 1974 .

[20]  Frank R. Kschischang,et al.  An Algebraic Approach to Physical-Layer Network Coding , 2010, IEEE Transactions on Information Theory.

[21]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2007, IEEE Transactions on Information Theory.

[22]  Frank R. Kschischang,et al.  Universal Secure Network Coding via Rank-Metric Codes , 2008, IEEE Transactions on Information Theory.