### Abstract

An efficient implementation of a parallel version of the Feng-Rao algorithm on a one-dimensional systolic array is presented in this paper by adopting an extended syndrome matrix. Syndromes of the same order, lying on a slant diagonal in the extended syndrome matrix, are scheduled to be examined by a series of cells simultaneously and, therefore, a high degree of concurrency of the Feng-Rao algorithm can be achieved. The time complexity of the proposed architecture is m + g + 1 by using a series of t + ⌊g-1/2⌋ + 1, nonhomogeneous but regular, effective processors, called PE cells, and g trivial processors, called D cells, where t is designed as the half of the Feng-Rao bound. Each D cell contains only delay units, while each PE cell contains one finite-field inverter and, except the first one, one or more finite-field multipliers. Cell functions of each PE cell are basically the same and the overall control circuit of the proposed array is quite simple. The proposed architecture requires, in total, t + ⌊g-1/2⌋ + 1 finite-field inverters and (t+⌊(g-1)/2⌋)(t+⌊(g-1)/2⌋+1)/2 finite-field multipliers. For a practical design, this hardware complexity is acceptable.

Original language | English |
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Pages (from-to) | 690-706 |

Number of pages | 17 |

Journal | IEEE Transactions on Computers |

Volume | 48 |

Issue number | 7 |

DOIs | |

State | Published - 1 Jul 1999 |

### Keywords

- Algebraic-geometric codes
- Error-correcting codes
- Feng-rao algorithm
- Systolic array

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## Cite this

*IEEE Transactions on Computers*,

*48*(7), 690-706. https://doi.org/10.1109/12.780877