We consider the problem of determining the expected dimension of the star product of two uniformly random linear codes that are not necessarily of the same dimension. We achieve this by establishing a correspondence between the star product and the evaluation of bilinear forms, which we use to provide a lower bound on the expected star product dimension. We show that asymptotically in both the field size q and the dimensions of the two codes, the expected dimension reaches its maximum. Lastly, we discuss some implications related to private information retrieval, secure distributed matrix multiplication, quantum error correction, and the potential for exploiting the results in cryptanalysis.

翻译：我们研究了确定两个维度不一定相同的均匀随机线性码的星积的期望维度问题。通过建立星积与双线性形式求值之间的对应关系，我们实现了这一目标，并利用该关系给出了星积期望维度的下界。我们证明，在域大小q和两个码的维度均渐近增长时，期望维度达到其最大值。最后，我们讨论了与私有信息检索、安全分布式矩阵乘法、量子纠错相关的若干应用，以及在密码分析中利用这些结果的潜力。