### Abstract

A key issue in compute-and-forward for physical layer network coding scheme is to determine a good function of the received messages to be reliably estimated at the relay nodes. We show that this optimization problem can be viewed as the problem of finding the closest point of Z[i] ^{n} to a line in the n-dimensional complex Euclidean space, within a bounded region around the origin. We then use the complex version of the LLL lattice basis reduction (CLLL) algorithm to provide a reduced complexity suboptimal solution as well as an upper bound to the minimum distance of the lattice point from the line. Using this bound we are able to find a lower bound to the ergodic rate and a union bound estimate on the error performance of a lattice constellation used for lattice network coding. We compare performance of the CLLL with a more complex iterative optimization method as well as with a simple quantized search. Simulations show how CLLL can trade some performance for a lower complexity.

Original language | English |
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Title of host publication | 2012 International Symposium on Network Coding, NetCod 2012 |

Pages | 131-136 |

Number of pages | 6 |

Publication status | Published - 1 Oct 2012 |

Event | 2012 International Symposium on Network Coding, NetCod 2012 - Cambridge, MA, United States Duration: 29 Jun 2012 → 30 Jun 2012 |

### Other

Other | 2012 International Symposium on Network Coding, NetCod 2012 |
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Country | United States |

City | Cambridge, MA |

Period | 29/6/12 → 30/6/12 |

### Fingerprint

### Keywords

- Clll al-gorithm
- Compute-and-forward
- Index terms-ergodic rate
- Quantized error
- Successive refinement

### ASJC Scopus subject areas

- Computer Networks and Communications

### Cite this

*2012 International Symposium on Network Coding, NetCod 2012*(pp. 131-136)

**On the ergodic rate for compute-and-forward.** / Sakzad, Amin; Viterbo, Emanuele; Hong, Yi; Boutros, Joseph.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2012 International Symposium on Network Coding, NetCod 2012.*pp. 131-136, 2012 International Symposium on Network Coding, NetCod 2012, Cambridge, MA, United States, 29/6/12.

}

TY - GEN

T1 - On the ergodic rate for compute-and-forward

AU - Sakzad, Amin

AU - Viterbo, Emanuele

AU - Hong, Yi

AU - Boutros, Joseph

PY - 2012/10/1

Y1 - 2012/10/1

N2 - A key issue in compute-and-forward for physical layer network coding scheme is to determine a good function of the received messages to be reliably estimated at the relay nodes. We show that this optimization problem can be viewed as the problem of finding the closest point of Z[i] n to a line in the n-dimensional complex Euclidean space, within a bounded region around the origin. We then use the complex version of the LLL lattice basis reduction (CLLL) algorithm to provide a reduced complexity suboptimal solution as well as an upper bound to the minimum distance of the lattice point from the line. Using this bound we are able to find a lower bound to the ergodic rate and a union bound estimate on the error performance of a lattice constellation used for lattice network coding. We compare performance of the CLLL with a more complex iterative optimization method as well as with a simple quantized search. Simulations show how CLLL can trade some performance for a lower complexity.

AB - A key issue in compute-and-forward for physical layer network coding scheme is to determine a good function of the received messages to be reliably estimated at the relay nodes. We show that this optimization problem can be viewed as the problem of finding the closest point of Z[i] n to a line in the n-dimensional complex Euclidean space, within a bounded region around the origin. We then use the complex version of the LLL lattice basis reduction (CLLL) algorithm to provide a reduced complexity suboptimal solution as well as an upper bound to the minimum distance of the lattice point from the line. Using this bound we are able to find a lower bound to the ergodic rate and a union bound estimate on the error performance of a lattice constellation used for lattice network coding. We compare performance of the CLLL with a more complex iterative optimization method as well as with a simple quantized search. Simulations show how CLLL can trade some performance for a lower complexity.

KW - Clll al-gorithm

KW - Compute-and-forward

KW - Index terms-ergodic rate

KW - Quantized error

KW - Successive refinement

UR - http://www.scopus.com/inward/record.url?scp=84866666719&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866666719&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781467318921

SP - 131

EP - 136

BT - 2012 International Symposium on Network Coding, NetCod 2012

ER -