q-Analogs of Covering Designs and Steiner Systems Based on Singular Linear Space

In this paper, \( q \)-analogs of covering designs and Steiner systems based on the subspaces of type \( (m,0) \) and the subspaces of type \( (m_1,0) \) in singular linear space \( \mathbb{F}_q^{(n+l)} \) over \( \mathbb{F}_q \) are presented, where \( m_1 < m \). Then the properties about \( q \)-analogs of covering designs and Steiner systems are discussed.