Abstract
We propose an algebraic framework for end-To-end physical-layer network coding based on submodules transmission. Our approach is motivated by nested-lattice-based network coding schemes, that naturally induce end-To-end channels where the ambient space has the structure of a module over a principal ideal ring. The setup is compatible with previously proposed approaches for finite chain rings, and extends them to arbitrary principal ideal rings. We introduce a distance function between modules, and describe how it relates to information loss and errors. We also show that computing the distance between modules reduces to computing the length of certain ideals in the base ring. We then propose a definition of submodule error-correcting code, and establish two upper bounds for the cardinality of these codes. Finally, we present some constructions of submodule codes, showing that they have asymptotically optimal cardinality for certain choices of the parameters.
| Original language | English |
|---|---|
| Pages (from-to) | 4480-4495 |
| Number of pages | 16 |
| Journal | IEEE Transactions on Information Theory |
| Volume | 64 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - Jun 2018 |
| Externally published | Yes |
Funding
Manuscript received November 12, 2016; revised June 23, 2017 and October 5, 2017; accepted November 20, 2017. Date of publication December 4, 2017; date of current version May 18, 2018. This work was supported in part by the Swiss National Science Foundation under Grant 200021_150207 and in part by ESF COST Action under Grant IC1104. The authors are with the University of Neuchâtel, 2000 Neuchâtel, Switzerland (e-mail: [email protected]; [email protected]). Communicated by P. Sadeghi, Associate Editor for Coding Techniques. Digital Object Identifier 10.1109/TIT.2017.2778726
Keywords
- nested lattices
- Physical-layer network coding
- submodule (ring theory)
- submodule code
- submodule distance