发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Rademacher矩阵和实数域上的随机线性码，不依赖c^3-LTCs，证明了双侧乘积扩展码的存在性，并开发了控制Gram矩阵范数的新技术，为乘积扩展张量码和上边界扩展码的研究提供了新起点。
AI 中文摘要
我们证明了双侧乘积扩展码的存在性，与Kalachev和Panteleev（FOCS，2025）的早期结果不同，我们的证明不依赖于渐近最优局部可测试码（$c^3$-LTCs）的显式构造。对于任意固定数量的分量码和维数，其速率远离0和1，我们证明在足够大的素数域上，独立的随机线性码以趋于1的概率具有恒定的双侧乘积扩展。权衡之处在于，我们的证明要求特征随块长度增长，而[KP25]采用$\mathbb{F}_2$的扩展。为了替代$c^3$-LTCs的使用，我们开发了几种我们认为本身具有独立价值的新技术。我们不直接在有限域上工作，而是在实数上工作，并使用随机Rademacher矩阵作为分量码的生成矩阵。由此，我们证明$\varepsilon$-闭集的可扩展性可以归结为控制精心选择的Gram矩阵的稀疏限制的算子范数。应用迹幂方法（trace power method）来限制该范数，归结为限制二部图上某些闭合游走的可能标记数量，我们利用算子的稀疏性以及Anderson和Zeitouni（Probab. Theory Relat. Fields., 2006）关于弱Wigner词等价类数量的界限（这些界限最初应用于带状矩阵）来限制该数量。由于我们的技术不依赖于显式的$c^3$-LTCs，我们相信它们为证明乘积扩展张量码的存在性提供了一个有前景的起点，其中分量码具有非平凡的自动同构群，以及可用于非立方复形（如单纯复形）的局部码的上边界扩展码。
英文摘要
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.
Comments65 pages
