We present a new class of private information retrieval (PIR) schemes that keep the identity of the file requested private in the presence of at most $t$ colluding servers, based on the recent framework developed for such $t$-PIR schemes using star products of transitive codes. These $t$-PIR schemes employ the class of Berman codes as the storage-retrieval code pairs. Berman codes, which are binary linear codes of length $n^m$ for any $n\geq 2$ and $m\geq 1$ being positive integers, were recently shown to achieve the capacity of the binary erasure channel. We provide a complete characterization of the star products of the Berman code pairs, enabling us to calculate the PIR rate of the star product-based schemes that employ these codes. The schemes we present have flexibility in the number of servers, the PIR rate, the storage rate, and the collusion parameter $t$, owing to numerous codes available in the class of Berman codes.

翻译：我们提出了一类新的私密信息检索（PIR）方案，该方案基于近期利用传递码星积构建的$t$-PIR框架，能够在至多$t$个合谋服务器存在的情况下保护所请求文件的身份隐私。这些$t$-PIR方案采用Berman码作为存储-检索码对。Berman码是长度为$n^m$的二元线性码（其中$n\geq 2$且$m\geq 1$为正整数），近期被证明能够达到二进制擦除信道的容量。我们完整刻画了Berman码对的星积性质，从而能够计算采用这些码的星积基方案的PIR速率。由于Berman码类中拥有大量可用码，我们所提出的方案在服务器数量、PIR速率、存储速率以及合谋参数$t$方面具有灵活性。