### Abstract

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R -linear codes in RnRn extends to a monomial transformation of RnRn that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius.

This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.

This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.

Original language | Undefined/Unknown |
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Pages (from-to) | 177-193 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory Series A |

Volume | 125 |

DOIs | |

Publication status | Published - 2014 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Frobenius ring
- principal ideal ring
- linear code
- extension theorem
- Möbius function

## Cite this

Greferath, M., Honold, T., Mc Fadden, C., Wood, J. A., & Zumbrägel, J. (2014). MacWilliams' extension theorem for bi-invariant weights over finite principal ideal rings.

*Journal of Combinatorial Theory Series A*,*125*, 177-193. https://doi.org/10.1016/j.jcta.2014.03.005