Two-Sided Product Expanding Codes via Rademacher Matrices

2026-10-01

AI Summary

This study addresses the limitation that existing constructions of asymptotically optimal locally testable codes (LTCs) rely heavily on explicit $c^3$-LTCs, thereby restricting their broader applicability. To overcome this, the authors construct component codes over the real field using random Rademacher matrices. By combining the trace power method with weak Wigner word counting techniques to control the operator norm of sparse Gram matrices, they rigorously prove that independent random linear codes over large prime fields exhibit constant two-sided product expansion. This work eliminates the dependence on explicit LTCs, demonstrating that random linear codes satisfy this property with probability approaching one as the characteristic grows with the block length. Consequently, it establishes a novel theoretical foundation for research on tensor codes and non-cubical complexes.

Abstract

We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop 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 we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

Problem

• two-sided product expanding codes
• locally testable codes
• random linear codes
• tensor codes

Innovation

• Two-sided product expanding codes
• Rademacher matrices
• Trace power method
• Gram matrices
• Weak Wigner words