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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 encodedthrough a $(K_c, N)$ MDS storage code. A user wishes to retrieve one message,as efficiently as possible, while revealing no information about the desiredmessage index to any colluding set of up to $T$ servers. The fundamental limiton the efficiency of retrieval, i.e., the capacity of MDS-TPIR is known only atthe extremes where either $T$ or $K_c$ belongs to $\{1,N\}$. The focus of thiswork is a recent conjecture by Freij-Hollanti, Gnilke, Hollanti and Karpukwhich offers a general capacity expression for MDS-TPIR. We prove that theconjecture is false by presenting as a counterexample a PIR scheme for thesetting $(K, N, T, K_c) = (2,4,2,2)$, which achieves the rate $3/5$, exceedingthe conjectured capacity, $4/7$. Insights from the counterexample lead us tocapacity characterizations for various instances of MDS-TPIR including allcases with $(K, N, T, K_c) = (2,N,T,N-1)$, where $N$ and $T$ can be arbitrary.

Private Information Retrieval from MDS Coded Data with Colluding Servers: Settling a Conjecture by Freij-Hollanti et al.