On Embeddings of $\ell_1^k$ from Locally Decodable Codes

TL;DR

This paper establishes a connection between locally decodable codes and embeddings of ^k spaces into Banach spaces, providing new proofs for lower bounds on code lengths and exploring related tensor product properties.

Contribution

It introduces a novel method to embed ^k into Banach spaces using LDC decoders, offering alternative proofs for known bounds and linking coding theory with Banach space geometry.

Findings

Constructs ^k embeddings from LDC decoders

Provides alternative proofs for lower bounds on 2-query LDCs

Reproves bounds for larger q and discusses tensor product properties

Abstract

We show that any $q$-query locally decodable code (LDC) gives a copy of $\ell_1^k$ with small distortion in the Banach space of $q$-linear forms on $\ell_{p_1}^N\times\cdots\times\ell_{p_q}^N$, provided $1/p_1 + \cdots + 1/p_q \leq 1$ and where $k$, $N$, and the distortion are simple functions of the code parameters. We exhibit the copy of $\ell_1^k$ by constructing a basis for it directly from "smooth" LDC decoders. Based on this, we give alternative proofs for known lower bounds on the length of 2-query LDCs. Using similar techniques, we reprove known lower bounds for larger $q$. We also discuss the relation with an alternative proof, due to Pisier, of a result of Naor, Regev, and the author on cotype properties of projective tensor products of $\ell_p$ spaces.