A characterization of low-weight words that span generalized Reed-Muller codes

Tali Kaufman, Dana Ron

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the generalized Reed-Muller code RFq (ρ, m) of order ρ and length qm, m > 1, over the field Fq, where q = pt 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) + pt-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 languageEnglish
Pages (from-to)4039-4043
Number of pages5
JournalIEEE Transactions on Information Theory
Volume51
Issue number11
DOIs
StatePublished - Nov 2005

Keywords

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

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