Linear Relaxed Locally Decodable Codes

• Linear RLDCs are error-correcting codes that relax strict decoding by permitting a decoder to output an abort symbol, enabling significantly improved blocklength for a given query complexity.
• They utilize constructions based on Reed–Muller evaluations, PCPPs, and consistency-testing random walks to achieve blocklengths of n = k^(1+O(1/q)) and nearly matching lower bounds.
• Analysis shows equivalence with standard LDCs for low query counts while demonstrating explicit separations at higher queries, sparking further research into error-correcting code fundamentals.

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 $q$ (Asadi et al., 2020).

The following construction and lower bound results are essential:

Code Type & Query Complexity $q$ & Blocklength $n$
RLDC (upper bound)
RLDC (lower bound)

The upper bound is realized via linear RLDCs built upon Reed–Muller codes and proximity proofs (Asadi et al., 2020), 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 (Goldberg et al., 26 Nov 2025). 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 (Grigorescu et al., 4 Nov 2025).
• $q=3$: 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 $q'$ (Grigorescu et al., 4 Nov 2025).

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 $\epsilon > 0$ (Grigorescu et al., 4 Nov 2025).

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 (Asadi et al., 2020).

The robust daisy method underpins the lower bound $n \ge k^{1+\Omega(1/q)}$ (Goldberg et al., 26 Nov 2025). 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 (Grigorescu et al., 4 Nov 2025).

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.