## Abstract

This semi-tutorial paper introduces the partial-inverse problem for polynomials and develops its application to decoding Reed-Solomon codes and some related codes. The most natural algorithm to solve the partial-inverse problem is very similar to, but more general than, the Berlekamp-Massey algorithm. Two additional algorithms are obtained as easy variations of the basic algorithm: the first variation is entirely new, while the second variation may be viewed as a version of the Euclidean algorithm. Decoding Reed-Solomon codes (and some related codes) can be reduced to the partial-inverse problem, both via the standard key equation and, more naturally, via an alternative key equation with a new converse. Shortened and singly-extended Reed-Solomon codes are automatically included. Using the properties of the partial-inverse problem, two further key equations with attractive properties are obtained. The paper also points out a variety of options for interpolation.

Original language | English |
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Article number | 7576713 |

Pages (from-to) | 6737-6756 |

Number of pages | 20 |

Journal | IEEE Transactions on Information Theory |

Volume | 62 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2016 |

## Keywords

- Berlekamp-Massey algorithm
- Euclidean algorithm
- Reed-Solomon codes
- key equation
- partial-inverse algorithm
- partial-inverse problem
- polynomial remainder codes