---
title: Linear Relaxed Locally Decodable Codes
url: https://www.emergentmind.com/topics/linear-relaxed-locally-decodable-codes-rldcs
type: topic
---

# Linear Relaxed Locally Decodable Codes

Linear relaxed locally decodable codes (RLDCs) are error-correcting codes $C: F^k \to F^n$ defined over a finite field $F$, characterized by the existence of a randomized local decoder $D$ that, on input index $i\in[k]$ and oracle access to a (possibly corrupted) codeword $w\in F^n$, reads at most $q$ coordinates and recovers the $i$-th message bit either with high probability or aborts (outputs a special symbol $\bot$) if the word is detected to be corrupt. Linear RLDCs generalize locally decodable codes (LDCs) by allowing the decoder to output $\bot$, relaxing the strict requirement to always return a symbol from $F$, and thereby facilitating codes with dramatically improved blocklength-query tradeoffs in many parameter regimes.

## 1. Formal Definitions and Parameters

Let $C: F^k \to F^n$ denote a linear code, i.e., an $F$-linear map specified by a generator matrix $G \in F^{n\times k}$. A $(q, \delta, \sigma)$-RLDC comprises a randomized (possibly nonadaptive) decoder $D$ such that, for every $x \in F^k$ and index $i \in [k]$:

- $D$ reads at most $q$ coordinates of $w \in F^n$.
- **Completeness**: If $w = C(x)$, then $D^w(i)$ outputs $x_i$ with probability at least $\sigma$ (often $\sigma=1$ for linear RLDC constructions).
- **Relaxed soundness**: If $w$ is $\delta n$ away from some $C(x)$ (Hamming distance), then $D^w(i)$ outputs either $x_i$ or $\bot$ with probability at least $\sigma$.

In contrast, a $q$-LDC never outputs $\bot$, and satisfies strict soundness: $\Pr[D^w(i) \neq x_i] \leq s$ for some error parameter $s$. Strong soundness in the LDC regime bounds the error proportionally to the Hamming distance.

## 2. Blocklength and Query Complexity Tradeoffs

A defining feature of RLDCs is the blocklength achievable as a function of query complexity $q$ and message length $k$. For large $q$, RLDCs admit polynomial-length codes $n = k^{1 + O(1/q)}$, which is exponentially shorter than the subexponential blocklength $n = 2^{k^{o(1)}}$ of the best-known $q$-LDCs for constant $q$ [2009.07311].

The following construction and lower bound results are essential:

| Code Type & Query Complexity $q$ & Blocklength $n$ |
|-----------|----------------------|-----------------|
| RLDC (upper bound) | $O(q)$ | $n = k^{1 + O(1/q)}$ |
| RLDC (lower bound) | $O(q)$ | $n \ge k^{1 + \Omega(1/q)}$ |

The upper bound is realized via linear RLDCs built upon Reed–Muller codes and proximity proofs [2009.07311], whereas the lower bound harnesses robust daisies—a combinatorial object analogous to relaxed sunflowers—to demonstrate the necessity of the $k^{1+\Omega(1/q)}$ blocklength scaling [2511.21659]. The gap between upper and lower bounds is nearly closed for nonadaptive codes.

## 3. Equivalence and Separation Phenomena

For small $q$, RLDCs and LDCs may coincide:

- **$q=2$**: Any linear 2-query RLDC with constant decoding radius and soundness $s < 1/2$ is a 2-query LDC [2511.02633].
- **$q=3$**: Any linear (3, $\delta$, 1, $s$)-RLDC with $s < 1/2$ and nonadaptive decoding is also a 3-query LDC. The equivalence leverages a decomposition into “smooth” and “nonsmooth” query distributions; if the smooth part is too small, soundness is violated by appropriately constructed error patterns.

Separation between RLDCs and LDCs occurs at larger $q$:

- An explicit linear 15-query RLDC is provided with $n = k^{O(\log\log k)}$, perfect completeness, soundness $s=1/3$, and decoding radius $\delta = \Omega(1)$ that is provably not a $q'$-LDC for any constant $q'$ [2511.02633].

## 4. Soundness Error Thresholds

The key threshold theorem establishes a dichotomy:

- For each constant $q$, there exists $s(q) = 2^{-q/2}$ such that any linear $(q, \delta, 1, s)$-RLDC with nonadaptive decoder and $s < (1 - \alpha)s(q)$ for some $\alpha > 0$ must in fact be a $q$-LDC with arbitrary small soundness error $\epsilon > 0$ [2511.02633].

This threshold is tight up to constant factors as shown by sequential repetition arguments. If soundness $s$ exceeds $s(q)$, genuine separation between RLDCs and LDCs is possible, and linear RLDCs cannot exhibit strong soundness (proportional to error fraction) unless they are truly LDCs.

## 5. Constructions and Lower Bound Techniques

The polynomial-length RLDCs achieving $n = k^{1+O(1/q)}$ are built using high-dimensional Reed-Muller codes, consistency-testing random walks (CTRWs), and correctable proofs of proximity (PCPPs):

- Encoding: The message is mapped into Reed–Muller evaluations, augmented with PCPP-encoded constraints on subspaces and lines.
- Decoding: To recover $f(x)$, the decoding algorithm samples subspaces via CTRW, invokes PCPP verifiers, and outputs $\bot$ upon rejection. Each step incurs $O(q)$ queries and ensures that if the codeword is close to valid, either the correct symbol or $\bot$ is recovered with high probability [2009.07311].

The robust daisy method underpins the lower bound $n \ge k^{1+\Omega(1/q)}$ [2511.21659]. The analysis leverages spreadness of query distributions, extraction of kernels of size $o(n)$, and reduction to hitting-set properties, alongside information-theoretic sampling reductions.

## 6. Comparison with Locally Correctable Codes and Open Questions

Nearly identical equivalence and separation phenomena hold for RLCCs (relaxed locally correctable codes):

- For $q=2,3$, RLCCs coincide with LCCs for soundness errors below $1/2$.
- Explicit separations exist for $q=41$ and above [2511.02633].

Major open directions include determining the minimal $q$ for RLDC vs. LDC separation, improving the soundness error threshold to $2^{-\Theta(q)}$, extending results to nonlinear codes over large fields, and closing the gap between polynomial LDC lower bounds and known subexponential constructions for $q \geq 3$. High-dimensional expander techniques and adaptations for adaptive decoding processes remain unexplored possibilities.

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The theory of linear RLDCs thus establishes a near-optimal blocklength-query tradeoff in the relaxed regime, sharp equivalence thresholds, and explicit constructions and separations, contributing crucial insights into the broader landscape of error-correcting codes with local and relaxed decodability.

Source: https://www.emergentmind.com/topics/linear-relaxed-locally-decodable-codes-rldcs