This note analyzes a linear recursion that arises in the computation of the subpacketization level for the multi-message PIR scheme of Banawan and Ulukus. We derive an explicit representation for the normalized subpacketization level $L$, whose smallest integer multiple yields the subpacketization level of the scheme, in terms of the number of servers $N$, the total number of messages $K$, and the number of demand messages $D$. The resulting formula shows that $L$ is a polynomial in $N$ with nonnegative coefficients, and its leading term is $N^{K-D+1}/D$.

翻译：本文分析了Banawan和Ulukus多消息PIR方案中计算分包化层级时出现的线性递归关系。我们推导了归一化分包化层级$L$的显式表达式，该层级的整数倍即为方案的分包化层级，其表达式由服务器数量$N$、消息总数$K$及需求消息数$D$表示。所得公式表明$L$是$N$的非负系数多项式，其首项为$N^{K-D+1}/D$。