We consider polynomial codes for private distributed matrix multiplication (PDMM/SDMM). Existing codes for PDMM are either specialized for the outer product partitioning (OPP), or inner product partitioning (IPP), or are valid for the more general grid partitioning (GP). We design extension operations that can be applied to a large class of OPP code designs to extend them to the GP case. Applying them to existing codes improves upon the state-of-the-art for certain parameters. Additionally, we show that the GP schemes resulting from extension fulfill additional combinatorial constraints, potentially limiting their performance. We illustrate this point by presenting a new GP scheme that does not adhere to these constraints and outperforms the state-of-the-art for a range of parameters.

翻译：本文研究用于私有分布式矩阵乘法（PDMM/SDMM）的多项式码。现有PDMM方案要么专用于外积划分（OPP），要么专用于内积划分（IPP），或者适用于更一般的网格划分（GP）。我们设计了扩展操作，可将一大类OPP码设计方案扩展至GP情形。将其应用于现有方案可在特定参数下超越当前最优方案。此外，我们证明通过扩展得到的GP方案满足额外的组合约束，这可能限制其性能。我们通过提出一种不遵循这些约束的新GP方案来说明这一点，该方案在一系列参数上优于当前最优方案。