## Abstract

We consider linear network coding functions that can be employed at the relays in wireless physical layer network coding, applied to a general multi-hop network topology. We introduce a general model of such a network, and discuss the algebraic basis of linear functions, deriving conditions for unambiguous decodability of the source data at the destination. We consider the use of integer rings, integer fields, binary extension fields and the ring of binary matrices as potential algebraic constructs, and show that the ring constructs provide more flexibility. We use the two-way relay channel and a network containing two sources and two relays to illustrate the concept and to demonstrate the effect of fading of the wireless channels. We show the capacity benefits of the more flexible rings.

Original language | English |
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Title of host publication | European Signal Processing Conference |

Publisher | European Signal Processing Conference, EUSIPCO |

Pages | 1153-1157 |

Number of pages | 5 |

Volume | 22 |

ISBN (Print) | 9780992862619 |

Publication status | Published - 2014 |

Event | 22nd European Signal Processing Conference, EUSIPCO 2014 - Lisbon, United Kingdom Duration: 1 Sept 2014 → 5 Sept 2014 |

### Conference

Conference | 22nd European Signal Processing Conference, EUSIPCO 2014 |
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Country/Territory | United Kingdom |

City | Lisbon |

Period | 1/09/14 → 5/09/14 |

## Keywords

- linear algebra
- Physical layer network coding (PNC)
- rings