## Abstract

We consider the generalized Reed-Muller code R_{Fq} (ρ, m) of order ρ and length q^{m}, m > 1, over the field F_{q}, where q = p^{t} for prime p and t ≥ 1. In particular, we are interested in the case that t > 1 (so that q is not prime), and the order ρ is at least q. As shown by Ding and Key, under these conditions, unless ρ is very large (i.e., ρ > (m - 1) (q-1) + p^{t-1} - 2), the code is not spanned by its minimum-weight words. Furthermore, there was no known characterization of words with small weight that span the code. In this correspondence, we characterize a set of words that span the code, and show that their weight is upper-bounded by q^{⌈m(q-1)-ρ} ^{/q-q/p⌉}, which is at most quadratic in the weight of the minimum-weight words.

Original language | English |
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Pages (from-to) | 4039-4043 |

Number of pages | 5 |

Journal | IEEE Transactions on Information Theory |

Volume | 51 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2005 |

Externally published | Yes |

## Keywords

- Affine subspaces
- Generalized Reed-Muller code
- Multivariate polynomials
- Property testing