Improved Lower Bounds for all Odd-Query Locally Decodable Codes

TL;DR

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 ⩽ q works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices.

Abstract

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.

Figures (3)

• Figure 1: This example shows, for $q = 5$, a scenario where sets $S$ as shown have $\geqslant d_3$ edges where one of the colors is $i$.
• Figure 2: In this example, the hyperedges $C, C'$ induce an edge between $S, T$. Here, $q = 5, t = 2$. Vertices of the form $(x, 1)$ are colored green, while vertices of the form $(x, 2)$ are colored blue. Note that $S\oplus T = (C\setminus Q_\theta)\times\{1\}\bigcup\ (C'\setminus Q_\theta)\times\{2\}$. Also note how $S$ contains $2 = \left\lceil\frac{q - t}{2}\right\rceil$ green elements and $1 = \left\lfloor\frac{q - t}{2}\right\rfloor$ blue element from $S\oplus T$, while $T$ contains $1 = \left\lfloor\frac{q - t}{2}\right\rfloor$ green element and $2 = \left\lceil\frac{q - t}{2}\right\rceil$ blue elements from $S\oplus T$.
• Figure 3: Example of a contributing monomial to $\mathsf{Deg}(s,s')$. Here, $q = 5, t = 2$. As in \ref{['fig:KikuchiGraph']}, vertices of the form $(x, 1)$ are colored green, while vertices of the form $(x, 2)$ are colored blue. The elements of $Q_\theta$ are colored yellow. Note that $q - t = 3$ is odd, and $S$ indicated by $(s,s')$ contains $2 = \left\lceil\frac{q - t}{2}\right\rceil$ green elements from $\left(C\setminus Q_\theta\right)\times\{1\}$, and $1 = \left\lfloor\frac{q - t}{2}\right\rfloor$ blue element from $\left(C'\setminus Q_\theta\right)\times\{2\}$.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 2.1
• Definition 2.3: Weak Rainbow Even Covers
• Remark 2.5: Prior works
• Definition 2.6: Good index $t$
• Definition 2.7: Approximate Strong Regularity
• Lemma 2.8: Weak Rainbow Bound for Approximately Strongly Regular Hypergraphs
• proof : Proof of \ref{['lem:even-cover-regular-hypergraph']}
• Claim 2.9
• proof : Proof of \ref{['thm:linear-LDCs']}