The Star Product of Uniformly Random Codes

2025-11-21

AI Summary

This work investigates the expected dimension of the star product of two uniformly random linear codes of possibly unequal dimensions. To address this problem, we establish an algebraic correspondence between the star product and the evaluation of a bilinear form, then combine probabilistic analysis with asymptotic methods over finite fields. Our main contribution is the first systematic characterization of the expected dimension of star products in the non-equal-dimensional regime. We prove that, as both the field size (q) and code length (n) tend to infinity, the expected dimension asymptotically achieves its theoretical upper bound; moreover, we derive a tight lower bound. This result overcomes the restrictive equal-dimension assumption prevalent in prior work. The findings provide foundational theoretical support for applications including private information retrieval, secure distributed matrix multiplication, and quantum error-correcting code design.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Determining expected dimension of star products for random linear codes
• Establishing correspondence between star products and bilinear forms
• Analyzing implications for information retrieval and quantum error correction

Innovation

• Star product of random linear codes analysis
• Bilinear forms correspondence for dimension bounds
• Asymptotic maximum dimension in field size