In this paper, we study longest lowest-density MDS codes, a simple kind of multi-erasure array code with optimal redundancy and minimum update penalty. We prove some basic structure properties for longest lowest-density MDS codes. We define a perfect property for near-resolvable block designs (NRBs) and establish a bijection between 3-erasure longest lowest-density MDS codes (T-Codes) and perfect NRB(3k + 1, 3, 2)s. We present a class of NRB(3k + 1, 3, 2) s, and prove that it produces a family of T-Codes. This family is infinite assuming Artin s Conjecture. We also test some other NRBs and find some T-Code instances outside of this family.