发表机构
Anhui University; Nanyang Technological University; Nanjing University of Aeronautics and Astronautics; Hefei University of Technology（安徽大学; 南洋理工大学; 南京航空航天大学; 合肥工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究广义Zetterberg码的广义覆盖半径，建立上界$2t+1$并给出下界，证明大$t$时达到最小值$t$，并确定二元码的第二、第三半径。
AI 中文摘要
广义覆盖半径衡量奇偶校验矩阵需要多少列才能同时生成多个校正子。其有限几何对应物是$(\rho,t)$-饱和集，其中每个$t$维子空间都包含在由至多$\rho$个指定向量生成的子空间中。我们研究了与广义Zetterberg码相关的范数一构型的这一覆盖问题。我们在每个非二元有限域上以及在显式二元范围内建立了上界$2t+1$，并通过计数子空间和构造子域障碍获得了互补的下界。对于显式的大$t$范围，这些构型是$t$强阻塞集，且第$t$个广义覆盖半径达到其最小可能值$t$。对于二元Zetterberg码，我们确定了每个扩展次数下的第二个广义覆盖半径，并证明对于无限子族，第三个半径为七。
英文摘要
Generalized covering radii measure how many columns of a parity-check matrix are needed to generate several syndromes simultaneously. Their finite-geometric counterparts are $(ρ,t)$-saturating sets, for which every $t$-dimensional subspace is contained in a subspace generated by at most $ρ$ prescribed vectors. We investigate this covering problem for the norm-one configurations associated with generalized Zetterberg codes. We establish the upper bound $2t+1$ over every nonbinary finite field and in an explicit binary range, together with complementary lower bounds obtained by counting subspaces and constructing subfield obstructions. For an explicit range of large $t$, these configurations are $t$-strong blocking sets, and the $t^{\rm th}$ generalized covering radius attains its minimum possible value $t$. For binary Zetterberg codes, we determine the second generalized covering radius in every extension degree and prove that the third radius is seven for an infinite subfamily.
