Private information retrieval from MDS coded data with colluding servers: Settling a conjecture by Freij-Hollanti et al

A (K, N, T, K
_c
) instance of the MDS-TPIR problem is comprised of K messages and N distributed servers. Each message is separately encoded through an (N, K
_c
) MDS storage code. A user wishes to retrieve one message, as efficiently as possible, while revealing no information about the desired message index to any colluding set of up to T servers. The fundamental limit on the efficiency of retrieval, i.e., the capacity of MDS-TPIR is known only at the extremes where either T or K
_c
belongs to {1, N}. The focus of this work is a recent conjecture by Freij-Hollanti, Gnilke, Hollanti and Karpuk which offers a general capacity expression for MDS-TPIR. We prove that the conjecture is false by presenting as a counterexample a PIR scheme for the setting (K, N, T, K
_c
) = (2,4, 2, 2), which achieves the rate 3/5, exceeding the conjectured capacity, 4/7.