  We study the problem of weakly private information retrieval (W-PIR), where a
user wishes to retrieve a desired message from $N$ non-colluding servers in a
way that the privacy leakage regarding the desired message's identity is less
than or equal to a threshold. We propose a new code construction which
significantly improves upon the best known result in the literature, based on
the following critical observation. In previous constructions, for the extreme
case of minimum download, the retrieval pattern is to download the message
directly from $N-1$ servers; however this causes leakage to all these $N-1$
servers, and a better retrieval pattern for this extreme case is to download
the message directly from a single server. The proposed code construction
allows a natural transition to such a pattern, and for both the maximal leakage
metric and the mutual information leakage metric, significant improvements can
be obtained. We provide explicit solutions, in contrast to a previous work by
Lin et al., where only numerical solutions were obtained.
