Lower bounds for constant query affine-invariant LCCs and LTCs

Ben-Sasson and Sudan in [BS11] obtained a similar result as Theorem 1, when the code is assumed to be linear, i.e., when the codewords form a vector space. They showed that if ${\mathcal{C}}\subset\{{\mathbb{K}}^{n}\to{\mathbb{F}}\}$ is an $(r-1)$-query locally correctable or $r$-query locally testable linear, affine-invariant code, where ${\mathbb{K}}$ and ${\mathbb{F}}$ are finite fields of characteristic $p>0$ with ${\mathbb{K}}$ an extension of ${\mathbb{F}}$, then the dimension of ${\mathcal{C}}$ as a vector space over ${\mathbb{F}}$ is at most $(n\log_{p}|{\mathbb{K}}|)^{r-2}$. When ${\mathbb{K}}$ is fixed (as in [GKS13]’s construction of constant query LCCs/LTCs), the result of [BS11] is a very special case of our Theorem 1. On the other hand, [BS11]’s result also applies when the size of ${\mathbb{K}}$ is growing (as long as ${\mathbb{K}}$ extends ${\mathbb{F}}$), whereas ours does not.

There are several works which study lower bounds for constant query LCCs [KT00, GKST02, DS07, KdW03, BDYW11, BDSS11, Woo12, DSW14]. For general (non-affine-invariant) LCCs, tight lower bounds are known only for 2-query LCCs. Kerendis and deWolf [KdW03] prove that if ${\mathcal{C}}\subset\{\{0,1\}^{n}\to\Sigma\}$ is a 2-query LCC^§§§Their lower bound also holds for the weaker notion of locally decodable dode (LDC), then $|{\mathcal{C}}|\leq\exp(O(n|\Sigma|^{5})).$ This is tight for constant $\Sigma$ and achieved by the Hadamard code. For $r$-query LCCs where $r>2$, the lower bounds known are much weaker. The best known bounds, due to [KdW03, Woo07], show that if ${\mathcal{C}}\subset\{\{0,1\}^{n}\to\{0,1\}\}$ is an $r$-query LCC, then

Higher-order Fourier analysis was applied to other problems in coding theory in [BL15b, TW14].

Our arguments are based on standard techniques from higher-order Fourier analysis [Tao12], but they are new in this context. We show that if an affine-invariant code is an $r$-query LCC, then its codewords are far from each other in the $U^{r}$-norm, the Gowers norm of order $r$. Similarly, we show that the codewords of an affine-invariant $r$-query LTC are far from each other in the $U^{r-1}$-norm. Therefore, we can upper bound the number of LCC/LTC codewords in terms of the size of a net that is fine enough with respect to the Gowers norm of an appropriate order. We bound the size of such a net by explicitly constructing one using a standard decomposition theorem (analogous to Szemerédi’s regularity lemma): any bounded function $f:{\mathbb{K}}^{n}\to{\mathbb{C}}$ can be approximated, upto a small error in the Gowers norm, by a composition of a bounded number of low-degree non-classical polynomials [TZ12].

The way we argue that two codewords $f$ and $g$ of an $r$-query LCC are far in the Gowers norm is that if $\|f-g\|_{U^{r}}<\epsilon$, then for small enough $\epsilon$ (with respect to $r$, $|\Sigma|$ and correctness probability), the local corrector when applied to $f$ can act as if it is applied to $g$. The argument is, briefly, as follows. On the one hand, the codewords $f$ and $g$ must be far in Hamming distance, because the definition of LCC implies that there is a unique codeword close to any string. So, with constant probability over choice of $y\in{\mathbb{K}}^{n}$, the local corrector’s guess for $f(y)$ must differ from $g(y)$. On the other hand, we can lower bound by a constant the probability of the event that the corrector outputs $g(y)$ when it queries coordinates of $f$, because $f$ and $g$ are close in the $\|\cdot\|_{U^{r}}$ norm. This last calculation uses the affine invariance of the code and the generalized von Neumann inequality, which bounds by $\|f_{0}\|_{U^{k}}$ the expectation over $z_{1},\dots,z_{m}\in{\mathbb{K}}^{n}$ of the product $\prod_{i=0}^{k}f_{i}({\mathcal{L}}_{i}(z_{1},\dots,z_{m}))$, where the ${\mathcal{L}}_{i}$’s are arbitrary linear forms so that no two are linearly dependent and $f_{i}:{\mathbb{K}}^{n}\to{\mathbb{C}}$ are arbitrary functions with $|f_{i}|\leq 1$.

The argument for $r$-query LTCs is similar. Suppose $f$ and $g$ are close in the $\|\cdot\|_{U^{r-1}}$ norm. Consider the random function $H$ such that for every $x$ independently, $H(x)$ equals $f(x)$ with probability $1/2$ and $g(x)$ with probability $1/2$. $H$ itself is far from a codeword with high probability. But we show that since the local tester accepts $f$, it will also accept $H\circ\ell$ for a random invertible affine map $\ell:{\mathbb{K}}^{n}\to{\mathbb{K}}^{n}$ with good probability. This implies that with good probability, $H\circ\ell$ is close to a codeword and by affine-invariance, $H$ itself is close to a codeword which gives a contradiction. To draw this conclusion, we again use the generalized von Neumann inequality as well as a hybrid argument.

Let ${\mathcal{X}}$ be a finite set called the set of coordinates and $\Sigma$ be an other finite set called the alphabet. Let $\Sigma^{\mathcal{X}}$ denote the set of all functions from ${\mathcal{X}}\to\Sigma$. A subset ${\mathcal{C}}\subset\Sigma^{\mathcal{X}}$ is called a code and its elements are called codewords.

Let ${\blacktriangle}_{\Sigma}=\{q:\Sigma\to{\mathbb{R}}_{\geq 0}:\sum_{i\in\Sigma}q(i)=1\}$ denote the probability simplex on $\Sigma$. We embed $\Sigma$ into ${\blacktriangle}_{\Sigma}$ by sending $i\in\Sigma$ to $e_{i}$ which is the $i^{th}$ coordinate vector in ${\mathbb{R}}^{\Sigma}$. This also lets us extend functions $f:{\mathcal{X}}\to\Sigma$ to ${\hat{f}}:{\mathcal{X}}\to{\blacktriangle}_{\Sigma}$ using the embedding. We call ${\hat{f}}$ the simplex extension of $f$. Now given $f,g\in\Sigma^{\mathcal{X}}$, we can write the Hamming distance between them as

where $\left\langle\cdot,\cdot\right\rangle$ is the standard inner product in ${\mathbb{R}}^{\Sigma}$.

Locally correctable and testable codes are defined formally in Sections 3 and 4 respectively.

Fix a finite field ${\mathbb{F}}_{p}$ of prime order $p$, and let ${\mathbb{K}}={\mathbb{F}}_{q}$ where $q=p^{t}$ for a positive integer $t$. ${\mathbb{K}}$ is then a vector space of dimension $t$ over ${\mathbb{F}}_{p}$. We denote by ${\rm Tr}:{\mathbb{K}}\to{\mathbb{F}}_{p}$ the trace function:

Also, we use $|\cdot|$ to denote the obvious map from ${\mathbb{F}}_{p}$ to $\{0,1,\dots,p-1\}$.

Given functions $f,g:{\mathbb{K}}^{n}\to{\mathbb{C}}$, we define their inner product as $\left\langle f,g\right\rangle={\mathbb{E}}_{x}[\overline{f(x)}g(x)]$ where $x$ is chosen uniformly from ${\mathbb{K}}^{n}$. We define $\|\cdot\|_{p}$-norm on such functions as $\|f\|_{p}={\mathbb{E}}_{x}[|f(x)|^{p}]^{1/p}$. We say a function $f:{\mathbb{K}}^{n}\to{\mathbb{C}}$ is bounded if $|f|\leq 1$. Let ${\mathbb{T}}$ denote the circle group ${\mathbb{R}}/{\mathbb{Z}}$ and $e:{\mathbb{T}}\to{\mathbb{C}}$ be the map given by $e(x)=\exp(2\pi ix)$.

Let $\alpha_{1},\cdots,\alpha_{t}\in{\mathbb{K}}$ be a basis for ${\mathbb{K}}$ when viewed as a vector space over ${\mathbb{F}}_{p}$. It is known [TZ12, BB15] that non-classical polynomials of degree $\leq d$ are exactly those functions $P:{\mathbb{K}}^{n}\to{\mathbb{T}}$ which have the following form:

for some $c_{d_{1,1},\dots,d_{n,t},k}\in\{0,1,\cdots,p-1\}$ and $\theta\in{\mathbb{T}}$. Next, we define the Gowers norm for arbitrary functions $f:{\mathbb{K}}^{n}\to{\mathbb{C}}$.

The Gowers norm is an actual norm when $r\geq 2$. It also satisfies a useful monotonicity property: for any function $f:{\mathbb{K}}^{n}\to{\mathbb{C}}$,

See [Tao12] for more on Gowers norm. Observe that if $f:{\mathbb{K}}^{n}\to{\mathbb{C}}$ is a non-classical phase polynomial of degree $<r$ then $\|f\|_{U^{r}}=1$. The inverse Gowers theorem is a partial converse to this. It shows that the Gowers norm of order $r$ of a function is in direct correspondence with its correlation with non-classical phase polynomials of degree $<r$. In particular:

A linear form on $m$ variables is a vector ${\mathcal{L}}=(w_{1},\cdots,w_{m})\in{\mathbb{K}}^{m}$ that is interpreted as a function ${\mathcal{L}}:({\mathbb{K}}^{n})^{m}\to{\mathbb{K}}^{n}$ via the map $(x_{1},\cdots,x_{m})\mapsto\sum_{i=1}^{m}w_{i}x_{i}$. A key reason that the Gowers norm is useful in applications is that if a function has small Gowers norm of the appropriate order, then it behaves pseudorandomly in a certain way with respect to linear forms.

See Appendix A for proof.

The goal of this section is to establish the following claim.

For the proof, we need the following definitions.

The definition of refinement immediately implies:

The next claim shows that any bounded function is “close” to being measurable by a polynomial factor of bounded complexity. Precisely:

We are now ready to prove Theorem 2.