A Family of Optimal Linear Functional-Repair Regenerating Storage Codes

We construct a family of linear optimal functional-repair regenerating storage codes with parameters {m,(n,k),(r,α,β)}={(2r-α+1)α/2,(r+1,r),(r,α,1)} for any integers r,α with 1≤α≤r, over any field when α∈{1,r-1,r}, and over any finite field Fq with q≥r-1 otherwise. These storage codes are Minimum-Storage Regenerating (MSR) when α=1, Minimum-Bandwidth Regenerating (MBR) when α=r, and represents extremal points of the (convex) attainable cut-set region different from the MSR and MBR points in all other cases. It is known that when 2≤α≤r-1, these parameters cannot be realized by exact-repair storage codes. Each of these codes come with an explicit and relatively simple repair method, and repair can even be realized as help-by-transfer (HBT) if desired. The coding states of codes from this family can be described geometrically as configurations of r+1 subspaces of dimension α in an m-dimensional vector space with restricted sub-span dimensions. A few "small" codes with these parameters are known: one for (r,α)=(3,2) dating from 2013 and one for (r,α)=(4,3) dating from 2024. Apart from these, our codes are the first examples of explicit, relatively simple, optimal functional-repair storage codes over a small finite field, with an explicit repair method and with parameters representing an extremal point of the attainable cut-set region distinct from the MSR and MBR points.