Improved Lower Bounds for all Odd-Query Locally Decodable Codes

We prove that for every odd \(q\geq 3\), any \(q\)-query binary, possibly non-linear locally decodable code (\(q\)-LDC) \(E:\{\pm1\}^k \rightarrow \{\pm1\}^n\) must satisfy \(k \leq \tilde{O}(n^{1-2/q})\). For even \(q\), this bound was established in a sequence of prior works. For \(q=3\), the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for \(2\)-LDCs. Their strategy hits an inherent bottleneck for \(q \geq 5\). Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called \(t\)-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size \(t\) (i.e., its co-degree) be equal to the same but arbitrary value \(d_t\) up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to \(d_t\). This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees. We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy \(t\)-approximate strong regularity for any \(t \leq q\). Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary \(q\)-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees.