The Star Product of Uniformly Random Codes
Johan V. Dinesen, Ragnar Freij-Hollanti, Camilla Hollanti, Benjamin Jany, Alberto Ravagnani
TL;DR
This work analyzes the star product of two independent uniformly random linear codes over a finite field, establishing a direct link to the evaluation of bilinear forms and deriving an explicit lower bound for the expected star-product dimension. By translating the problem into counting zeros of bilinear forms and studying the kernel of a bilinear-map, the authors obtain exact expressions and asymptotic results, showing that the expected dimension attains its maximum both as the field size grows and under admissible growth of the code dimensions. The paper also discusses meaningful implications for private information retrieval, secure distributed matrix multiplication, and binary CSS-T quantum codes, highlighting how maximal star-product dimensions constrain rates, recovery thresholds, and code constructions. The findings open avenues for non-asymptotic refinements and potential cryptanalytic insights, with broader relevance to coded distributed systems and quantum error-correcting code design.
Abstract
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.