Relaxed Locally Decodable Codes (RLDCs)

• Relaxed Locally Decodable Codes (RLDCs) are error-correcting codes that use a randomized local decoder to recover any message symbol while allowing a reject symbol (⊥) to avoid errors.
• They achieve significantly shorter blocklengths than classical LDCs by permitting decoding failures instead of errors, with explicit constructions like 3-query RLDCs reaching blocklength ∼O(k²).
• RLDCs underpin advances in probabilistically checkable proofs, privacy-preserving data retrieval, and fault-tolerant storage, while their analysis leverages robust daisy structures and innovative algebraic techniques.

A relaxed locally decodable code (RLDC) is a family of error-correcting codes equipped with a randomized local decoder that, given a string close to a codeword, can recover any desired message symbol with high probability, querying only a small number of positions in the received string. Critically, unlike classical locally decodable codes (LDCs), RLDCs allow the decoder to output a special “reject” symbol (denoted ⊥) rather than an incorrect message symbol; the decoder is required only to avoid decoding errors, not decoding failures. RLDCs have enabled code constructions with far shorter blocklengths than LDCs for constant query complexity and are central to several advances in the theory of probabilistically checkable proofs, privacy in data retrieval, and fault-tolerant storage.

1. Formal Definition and Relaxed Decoding Model

Let $C : \{0,1\}^k \rightarrow \Sigma^n$ be an encoding function. A $q$-query RLDC with error-radius $\delta$, soundness parameter $\sigma$, and (possibly) a success-rate guarantee is defined by the existence of a randomized oracle decoder $D$ that, for any received string $w$ with $\mathrm{dist}(w, C(x)) \leq \delta n$, and any index $i\in[k]$, makes at most $q$ non-adaptive queries into $w$ and outputs $D^w(i) \in \{x_i, \bot\}$. The requirements are:

• Perfect completeness: For $w = C(x)$ (no corruption), $D^w(i) = x_i$ always.
• Relaxed soundness: For corrupted input $w$, with high probability over the decoder’s coins, $D^w(i)$ outputs either $x_i$ or ⊥, but never a wrong symbol with probability larger than $1-\sigma$.
• Typical success-rate: For any received word $w$ within decoding radius, for all but a small fraction of message symbols, the probability of correct recovery (without ⊥) is close to 1.

This model interpolates between traditional LDCs (no errors allowed, only correct decoding) and error-detecting codes permitting explicit aborts. RLDCs never output incorrect bits, merely failures (⊥), on sufficiently corrupted local views (Gur et al., 2019, Goldberg et al., 26 Nov 2025, Gur et al., 15 Dec 2025).

2. Parameter Regimes, Constructions, and Trade-offs

The primary focus in RLDC research is the trade-off between the number of queries $q$ and the blocklength $n$ for a given message length $k$. In contrast to LDCs, where $q$-query codes require superpolynomial length unless $q$ is large, RLDCs admit the following trade-off:

• For every constant $q \geq 2$, there exist $q$-query RLDCs with $n = k^{1+O(1/q)}$ [(Asadi et al., 2020), BGHSV06].
• The construction is typically based on combining algebraic error-correcting codes (e.g., Reed–Muller), probabilistically checkable proofs of proximity (PCPPs), and structured combinatorial objects to certify correctness of local views (Gur et al., 15 Dec 2025, Asadi et al., 2020).
• For $q=3$, the latest results provide explicit constructions of $3$-query RLDCs of blocklength $\tilde O(k^2)$ and constant alphabet size, while any $3$-query LDC must satisfy $n\ge \tilde\Omega(k^3)$—thus, a strict exponential gap (Gur et al., 15 Dec 2025).

A synopsis of the best parameters:

Code type	Blocklength $n$ for $q=3$	Error tolerance	Alphabet size	Reference
3-query RLDC	$\tilde O(k^2)$	constant	constant	(Gur et al., 15 Dec 2025)
3-query LDC	$\tilde\Omega(k^3)$	constant	constant	[Alrabiah+ STOC'23]
$q$-query RLDC	$k^{1+O(1/q)}$	constant	constant	(Asadi et al., 2020)
$q$-query LDC	$2^{k^{o(1)}}$ (best known)	constant	constant	(Grigorescu et al., 4 Nov 2025)

This strict gap between RLDCs and LDCs for $q\geq 3$ demonstrates the power of relaxing the decoder’s obligations.

3. Lower Bounds, Robust Daisies, and Phase Transitions

Recent work has established nearly-tight lower bounds for the minimum blocklength of RLDCs. For any $q$-query (linear) RLDC with constant error-tolerance and soundness, the blocklength must satisfy

$n \geq k^{1+\Omega(1/q)}$

This matches, up to constants, the best known upper bounds (Goldberg et al., 26 Nov 2025). The proof framework introduces the robust daisy, a generalization of the relaxed sunflower lemma. Robust daisies describe a pseudorandom combinatorial structure in the supports of query distributions of the local decoder. The small-set spread lemma is used to show that, after conditioning and puncturing a small kernel, the decoder's local views form a structure such that global decoding from $o(k)$ sampled positions would violate information theory unless $n$ is sufficiently large (Goldberg et al., 26 Nov 2025, Gur et al., 2019).

The phase transition for RLDCs occurs at $q=2$. For $q=2$, any RLDC must have exponential length, exactly as for LDCs. For $q\geq 3$, RLDCs can achieve polynomial length, marking a sharp divide (Block et al., 2022).

4. Equivalence and Separations: Linear Codes and Query Thresholds

A critical distinction is between "relaxed" and "full" local decodability. For linear codes:

• For $q=2$, linear RLDCs are equivalent to LDCs; no length savings are possible (Grigorescu et al., 4 Nov 2025).
• For $q=3$, recent work demonstrated that general RLDCs (not necessarily linear) are strictly more powerful than LDCs, as there exist 3-query RLDCs of length $\tilde{O}(k^2)$, whereas the LDC lower bound is $\tilde{\Omega}(k^3)$ (Gur et al., 15 Dec 2025).
• However, for linear RLDCs at $q=3$ with strong soundness parameters (e.g., error below $1/2$), every such RLDC is also an LDC (Grigorescu et al., 4 Nov 2025). The first explicit linear separation (an RLDC that is not an LDC) occurs at $q=15$.
• More generally, any linear $q$-query RLDC with soundness $s < s(q) = 2^{-q/2}$ is in fact a $q$-LDC (Grigorescu et al., 4 Nov 2025).

A summary of this threshold behavior:

$q$	RLDC vs LDC equivalence (linear codes)	Blocklength gap?	Separating construction exists?
2	Equivalent	No	No
3	Equivalent if $s < 1/2$	Yes (nonlinear RLDCs)	Yes (Gur et al., 15 Dec 2025)
$\geq 15$	Separation (even linear codes)	Yes	Yes (Grigorescu et al., 4 Nov 2025)

This demonstrates that relaxed decoding is a genuine relaxation only above certain thresholds in query complexity and/or soundness.

5. Algorithmic Techniques and Structural Insights

The core construction paradigm for RLDCs involves:

• Encoding the message using algebraic-geometry codes (e.g., Reed–Muller over extension fields).
• Adding redundancy and proximity proofs using PCPPs, often derived from high-dimensional expander graphs.
• Designing a local decoder that verifies, via a small random walk over planes and lines, the proximity of local codeword views, outputting the correct symbol if all checks succeed, or ⊥ otherwise (Asadi et al., 2020, Gur et al., 15 Dec 2025).
• Exploiting daisy or robust daisy structures in the supports of the query distributions to demonstrate lower bounds.

A central technical innovation is the employment of decodable PCPs (dPCPs) and their composition, allowing efficient transformation of efficient proximity proof systems into short RLDCs (Gur et al., 15 Dec 2025).

6. Extensions, Variant Models, and Open Problems

Recent lines of research include:

• Relaxed locally correctable codes (RLCCs), which allow local correction of any codeword coordinate with similar guarantees: the same existential bounds as for RLDCs hold (Asadi et al., 2020).
• Computationally-relaxed LDCs (crLDCs), which allow substantially improved rate and locality under cryptographic hardness assumptions (specifically, collision-resistant hash functions and signatures), achieving constant rate and polylogarithmic query complexity in the presence of computationally-bounded adversaries (Blocki et al., 2018, Block et al., 2023).
• RLDCs for insertion–deletion channels (edit distance) present a stark separation: weak RLDCs (no explicit per-index success guarantee) can be nearly as short as Hamming RLDCs, but strong RLDCs (with a uniform per-coordinate success-rate) must have exponential blocklength for constant queries (Block et al., 2022).

Major open questions as of 2025 include:

• Determining the optimal blocklength for $q$-query RLDCs at small constant $q$ (notably for $q=3,4$).
• Constructing RLDCs that achieve constant rate with polylogarithmic locality in the information-theoretic setting (currently known only in the computational setting).
• Identifying new combinatorial or algebraic techniques to close the remaining gaps between upper and lower bounds, especially for non-linear codes or specialized error models.
• Exploring connections and potential separations between RLDCs and other relaxed local correction models under various constraints on error, alphabet, or adversary power.

7. Applications and Significance

RLDCs have had major impact in:

• Construction of efficient PCPs and PCPPs, used in hardness of approximation, property testing, and proof systems (Gur et al., 15 Dec 2025).
• Privacy-preserving data retrieval schemes and distributed storage, where rapid local error-resilient access to individual data fragments is critical.
• Coding for computationally-bounded channels and cryptography, providing robust and efficient codes under practical computational assumptions (Blocki et al., 2018, Block et al., 2023).
• Coding for non-traditional error models (e.g., edit distance, DNA storage systems), demonstrating the flexibility of the relaxed local decoding framework (Block et al., 2022).

The sharp delineations provided by recent results—phase transitions, robust daisy-based lower bounds, and separation at small $q$—clarify the landscape of local decoding and identify precise complexity barriers for efficient information-theoretic error-correcting codes with locality constraints.

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Selected references:

• (Gur et al., 15 Dec 2025): "3-Query RLDCs are Strictly Stronger than 3-Query LDCs"
• (Goldberg et al., 26 Nov 2025): "Nearly Tight Lower Bounds for Relaxed Locally Decodable Codes via Robust Daisies"
• (Grigorescu et al., 4 Nov 2025): "Relaxed vs. Full Local Decodability with Few Queries: Equivalence and Separations for Linear Codes"
• (Asadi et al., 2020): "Relaxed Locally Correctable Codes with Improved Parameters"
• (Block et al., 2022): "On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors"
• (Blocki et al., 2018): "Relaxed Locally Correctable Codes in Computationally Bounded Channels"
• (Block et al., 2023): "Computationally Relaxed Locally Decodable Codes, Revisited"
• (Gur et al., 2019): "A Lower Bound for Relaxed Locally Decodable Codes"