T-Code: 3-Erasure Longest Lowest-Density MDS Codes

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(3¿ + 1, 3, 2)s. We present a class of NRB(3¿+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.