A weighted version of a result of Hamada on minihypers and on linear codes meeting the Griesmer bound

Abstract

Minihypers were introduced by Hamada to investigate linear codes meeting the Griesmer bound. Hamada (Bull Osaka Women’s Univ 24:1–47, 1985; Discrete Math 116:229–268, 1993) characterized the non-weighted minihypers having parameters $\{\sum_{i=1}^h v_{\lambda_i+1},\sum_{i=1}^h v_{\lambda_i};k-1,q\}$, with k−1 > λ₁ > λ₂ > ... > λ _h  ≥ 0, as the union of a λ₁-dimensional space, λ₂-dimensional space, ..., λ _h -dimensional space, which all are pairwise disjoint. We present in this article a weighted version of this result. We prove that a weighted $\{\sum_{i=1}^h v_{\lambda_i+1},\sum_{i=1}^h v_{\lambda_i};k-1,q\}$-minihyper ${\mathfrak{F}}$ , with k−1 > λ₁ > λ₂ > ... > λ _h  ≥ 0, is a sum of a λ₁-dimensional space, λ₂-dimensional space, ..., and λ _h -dimensional space.