Product Expansion of Random Codes via Analytic Techniques
Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner
原文摘要
Beginning with the work of Dinur, Lin, and Vidick (FOCS, 2024), tensor codes with constant product expansion have been foundational to recent advances in quantum locally testable codes (qLTCs) based on cubical complexes. Informally, product expansion says that any low-weight parity check of the tensor code can be written as the sum of few parity checks of each component code in each dimension. Recent work of Kalachev and Panteleev (FOCS, 2025) showed that tensors of random linear codes have constant product expansion. In this work we initiate the study of random linear codes with prescribed automorphism groups. For every sufficiently large prime $k$, we construct pairs of rate $1/2$ MDS codes of length $2k$, over finite fields whose size depends on $k$, that have constant two-sided product expansion and are invariant under the simultaneous cyclic shift of two cycles of length $k$. This shift is the local action in the cube complexes of Rungtanapirom, Stix, and Vdovina (Internat. J. Algebra Comput., 2019). Along the way we give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of [KP25], does not rely on explicit constructions of asymptotically optimal locally testable codes. Our proof relies on several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we show how to relate product expansion to the operator norm of sparse restrictions of carefully chosen Gram matrices, which we bound using tools from random matrix theory. More generally, we propose a method for studying properties of random linear codes by lifting to random codes over $\mathbb{R}$ and $\mathbb{C}$, which we anticipate will be applicable more broadly to existential problems in coding theory.
cs.IT · cs.CC · cs.DM