Local Decodable Information

• Local Decodable Information is the structured organization of data that allows specific message symbols to be recovered by accessing only a small subset of codeword symbols.
• It underpins coding theory, private information retrieval, and compression by balancing locality, rate, and error tolerance while sometimes necessitating exponential redundancy.
• Recent advances extend its applications to randomized encoding, handling insertion/deletion errors, and quantum decoding, thereby enhancing both theoretical capacity and practical system performance.

Searching arXiv for the cited papers and related terminology to ground the article. Local decodable information is the organization of information so that a demanded symbol, fragment, or global hypothesis can be recovered from a small portion of an encoded object, transmitted codeword, or locally weighted observation rather than by reading the entire representation. In coding theory, the canonical form is the locally decodable code, where each message symbol is recoverable from few codeword symbols; related formulations appear in private information retrieval, index coding, and locally decodable compression. In a distinct but closely related use, “local information” can denote spatially resolved reliability information that changes decoding weights, as in quantum error correction with nonuniform noise (Sun et al., 2018, Natarajan et al., 2019, Vatedka et al., 2019, Hanks et al., 2019).

1. Information-theoretic core: locality, rate, and capacity

A locally decodable code (LDC) maps $K$ independent source symbols $W_1,\dots,W_K$, each of entropy $L_w$ bits, to $M$ coded symbols $X_1,\dots,X_M$, each of entropy $L_x$ bits, such that every $W_k$ is recoverable from some decoding set of exactly $N$ coded symbols. The locality is $N$, the symbol rate is $R_s = L_w/L_x$, and the code rate is $W_1,\dots,W_K$0 (Sun et al., 2018).

Two structural subclasses isolate different notions of uniform usefulness. A perfectly smooth LDC requires that, for each message, every coded symbol appears equally often across its decoding sets. A universal LDC weakens this to the requirement that every coded symbol appear in some decoding set for every message. Perfect smoothness implies universality, but the main capacity and length results already hold at the universal level (Sun et al., 2018).

For fixed $W_1,\dots,W_K$1 and $W_1,\dots,W_K$2, the maximal achievable symbol rate over all code lengths is

$W_1,\dots,W_K$3

This is the capacity of perfectly smooth LDCs and of universal LDCs. It is an information-theoretic converse that applies to arbitrary, possibly non-linear encoders. The same work shows that capacity is attained only when the code length is at least

$W_1,\dots,W_K$4

and gives explicit perfectly smooth constructions with $W_1,\dots,W_K$5, so the exponential length requirement is tight (Sun et al., 2018).

The converse is structural as well as numerical. In any capacity-achieving universal LDC, the entropy inequalities along a full $W_1,\dots,W_K$6-ary decoding tree are all tight, which forces non-zero conditional entropy for each coded symbol, identical interference structure inside any decoding set for the non-demanded messages, distinct desired information about the target message, and an information-theoretic independence property across the $W_1,\dots,W_K$7 symbols in a decoding set. The construction that meets capacity indexes coded symbols by $W_1,\dots,W_K$8, partitions them into $W_1,\dots,W_K$9 groups according to $L_w$0, and aligns interference so that the desired sub-symbol can be solved from one symbol drawn from each group (Sun et al., 2018).

This capacity viewpoint reframes local decodable information as a packing problem: how many message bits can be stored per coded-symbol bit while guaranteeing recovery from only $L_w$1 coded symbols. The answer is exact, and it shows that high symbol rate, strict locality, and universal participation of all coded symbols force exponential redundancy in $L_w$2 (Sun et al., 2018).

2. Retrieval and broadcast formulations

The same locality principle appears in index coding, where each receiver wants one message and holds side information about others. A locally decodable index code allows receiver $L_w$3 to query only a subset $L_w$4 of the broadcast codeword; its locality is $L_w$5, the overall locality is $L_w$6, and the average locality is $L_w$7 (Natarajan et al., 2019).

For single-unicast problems, the minimum possible locality is $L_w$8. At that point, the optimal broadcast rate is exactly the fractional chromatic number of the interference graph: $L_w$9 where $M$0 is the interference graph obtained from the complement of the underlying undirected side-information graph (Natarajan et al., 2019). This identifies the extreme point at which each receiver reads exactly one codeword symbol per demanded message symbol.

Beyond that extreme, the rate–locality tradeoff becomes graph-specific. For a directed cycle on $M$1 receivers, the optimal vector linear tradeoff is

$M$2

Thus the minimum locality at which the optimal rate $M$3 is achievable is $M$4. For scalar linear codes on the broader class with $M$5, if the smallest directed cycle has length $M$6, then every rate-$M$7 scalar linear code must satisfy $M$8 and

$M$9

and both bounds are tight (Natarajan et al., 2019). For the directed 3-cycle, the optimal tradeoff over all codes, including non-linear codes, is

$X_1,\dots,X_M$0

(Natarajan et al., 2019).

Private information retrieval provides a second retrieval-theoretic interpretation. In the $X_1,\dots,X_M$1 model, the user downloads one answer of size $X_1,\dots,X_M$2 from each of $X_1,\dots,X_M$3 replicated databases, and the rate is $X_1,\dots,X_M$4. The universal-LDC correspondence implies

$X_1,\dots,X_M$5

and any capacity-achieving $X_1,\dots,X_M$6 or $X_1,\dots,X_M$7 scheme must have minimum upload cost $X_1,\dots,X_M$8 per database. Relaxing privacy to repudiability does not improve either the capacity or the minimum upload cost under the maximum-download metric (Sun et al., 2018).

These broadcast and retrieval formulations make locality a communication-theoretic quantity: it measures how much of the globally encoded information each receiver or querier must actually access.

3. Compression and randomized encoding

In source coding, local decodable information is the ability to read or modify a small fragment of compressed data without touching the entire file. For i.i.d. memoryless sources $X_1,\dots,X_M$9, one universal compression scheme achieves rate $L_x$0, vanishing block error probability, and average local decoding and update costs that are linear in fragment size. Specifically, for contiguous fragments of length $L_x$1,

$L_x$2

and analogously for updates,

$L_x$3

In particular,

$L_x$4

with total encoding and decoding complexity $L_x$5. A second scheme achieves the same near-entropy rate with worst-case single-symbol locality $L_x$6 (Vatedka et al., 2019).

Randomized encoding changes the LDC landscape in a different way. In the shared-randomness model for Hamming errors, there are efficient randomized-encoding LDCs with $L_x$7, constant rate, and query complexity

$L_x$8

For oblivious channels in the Hamming model, the same rate is achievable with

$L_x$9

For edit errors, both the shared-randomness and oblivious-channel constructions achieve

$W_k$0

at constant rate (Cheng et al., 2020). In the flexible-failure formulation, there are also constructions with $W_k$1 and query complexity $W_k$2 in the Hamming shared-randomness model, $W_k$3 for Hamming oblivious channels, and $W_k$4 for edit errors (Cheng et al., 2020).

A converse in the same work shows that randomized encoding does not help against stronger adversaries under a symmetry condition: if a randomized-encoding LDC for Hamming errors exists with message length $W_k$5, block length $W_k$6, error tolerance $W_k$7, and $W_k$8 queries, then there also exists a standard deterministic-encoding LDC with message length $W_k$9, the same $N$0 and $N$1, and decoding success $N$2 (Cheng et al., 2020). The gain therefore comes from hiding encoder randomness or restricting the channel.

4. Insertions, deletions, and hard limits

Insertion and deletion errors destroy coordinate alignment, so local decodability requires a local synchronization mechanism in addition to ordinary error correction. A general compiler transforms any Hamming LDC or LCC into a binary insdel LDC or LCC with only constant-factor loss in rate and error radius and with polylogarithmic blowup in locality. Concretely, if the outer Hamming code has locality $N$3, then the resulting insdel code has locality

$N$4

for LDCs and

$N$5

for LCCs, codeword length $N$6 with $N$7, and insdel tolerance $N$8 (Block et al., 2020). The compiler uses block decomposition, buffers, indexed inner codewords, and a noisy binary search that recovers the logical Hamming block corresponding to a queried insdel location.

The existence of such compilers does not imply that constant-query insdel local decoding is possible. In fact, constant-query deletion LDCs do not exist at all. For every $N$9, every constant $N$0, and every alphabet $N$1, there is a constant $N$2 such that for all message lengths $N$3, no $N$4-query deletion LDC $N$5 can tolerate an $N$6-fraction of deletions, for any block length $N$7. By a reduction of Blocki et al., the same impossibility extends to constant-query deletion LCCs (Gupta, 2023). This sharply separates Hamming locality from deletion locality: constant-query local decodable information exists under substitutions but fails under deletions.

Between these two facts lies a large constructive region. Applying the Hamming-to-InsDel compiler to private-key Hamming LDCs and to resource-bounded Hamming LDCs yields private-key and resource-bounded InsDel LDCs with constant rate, constant error tolerance, and polylogarithmic locality. The compiled one-time private construction has locality

$N$8

constant information rate, constant insdel tolerance $N$9, and negligible failure; the multi-time private version and the resource-bounded version retain the same qualitative parameters under their respective assumptions (Block et al., 2021).

The resulting picture is asymmetric. General insdel locality admits powerful compilers and high-rate non-classical constructions, but the deletion-only impossibility result rules out constant-query local decoding in the classical worst-case setting.

5. Amortization and computational relaxations

Amortization replaces “queries per recovered symbol” by “average queries per recovered symbol over a batch.” An amortized LDC permits decoding a set or interval of message coordinates with total query budget proportional to the batch size. This changes the parameter frontier even in classical examples. For the Hadamard code, if the decoder wants every symbol in a set $R_s = L_w/L_x$0 with $R_s = L_w/L_x$1, it can share one query across all requested coordinates and achieve amortized locality

$R_s = L_w/L_x$2

with failure probability at most $R_s = L_w/L_x$3. Thus Hadamard achieves amortized locality below $R_s = L_w/L_x$4, which is impossible in the non-amortized worst-case model (Blocki et al., 14 Feb 2025).

In cryptographic settings, amortization enables the “trifecta” of constant rate, constant error tolerance, and constant amortized locality. In the private-key setting, and in the resource-bounded setting, there are amortized LDCs for decoding consecutive intervals $R_s = L_w/L_x$5 with $R_s = L_w/L_x$6 that achieve exactly those three properties (Blocki et al., 14 Feb 2025). A subsequent Hamming-to-InsDel compiler for amortized LDCs shows that this phenomenon persists under insertions and deletions, provided the Hamming decoder has a consecutive-interval querying structure. The compiler preserves rate and error tolerance up to constants and preserves amortized locality up to a factor

$R_s = L_w/L_x$7

where $R_s = L_w/L_x$8 is the Hamming block size and $R_s = L_w/L_x$9 is the queried interval length. Combined with an ideal Hamming amortized LDC satisfying this query structure, it yields ideal InsDel amortized LDCs in private-key and resource-bounded settings with constant amortized locality, constant rate, and constant error tolerance (Blocki et al., 3 Jul 2025).

Computational asymmetry also improves non-amortized locality. In the random oracle model, if a channel class admits a $W_1,\dots,W_K$00-safe function, then one can bootstrap private LDCs to explicit constant-rate LDCs with locality polylogarithmic in the security parameter. For example, there are binary $W_1,\dots,W_K$01 coding schemes with constant rate and locality $W_1,\dots,W_K$02 or $W_1,\dots,W_K$03, depending on the target success parameters, against various resource-constrained channels (Blocki et al., 2019). A related relaxed direction gives binary relaxed locally correctable and relaxed locally decodable codes with constant information rate and poly-logarithmic locality in computationally bounded channels, using collision-resistant hash functions and local expander graphs (Blocki et al., 2018).

Amortization and computational restrictions therefore identify two distinct mechanisms for increasing the amount of information that is locally decodable: one spreads query cost over many recovered symbols, and the other weakens the channel enough that hidden structure can be exploited by the decoder but not by the adversary.

6. Quantum local variation and HDX local list decoding

In quantum error correction, “local information” can mean something different: site-specific reliability data rather than small-query access to a codeword. For repetition and surface codes under a phenomenological noise model, the local information is the spatially varying error probability $W_1,\dots,W_K$04 attached to edges of the space–time lattice. Standard minimum-weight perfect matching uses homogeneous weights proportional to $W_1,\dots,W_K$05, whereas the locally informed decoder uses

$W_1,\dots,W_K$06

This lets the decoder distinguish error chains of similar geometric length but different local likelihoods (Hanks et al., 2019).

The numerical effect is a reduction in logical error rate for fixed code distance, or equivalently a reduction in required distance for a fixed target logical error. For the repetition code, “improvements on the order of 10% are observed for relative widths of order 0.4–0.5,” and for $W_1,\dots,W_K$07 the improvements are of order $W_1,\dots,W_K$08–$W_1,\dots,W_K$09 for the largest simulated distances. For the surface code at $W_1,\dots,W_K$10 and $W_1,\dots,W_K$11, the relative improvement grows with $W_1,\dots,W_K$12 but is roughly half as large (Hanks et al., 2019). Here local decodable information is not stored redundancy but local calibration data that changes the posterior geometry of decoding.

A more abstract extension appears in approximate locally list-decodable codes built from high-dimensional expanders. For every $W_1,\dots,W_K$13 and $W_1,\dots,W_K$14, there is a binary approximate LLDC decodable from $W_1,\dots,W_K$15 errors with rate $W_1,\dots,W_K$16, query complexity $W_1,\dots,W_K$17, and polylogarithmic-time local algorithms. There is also a constant-rate version for infinitely many $W_1,\dots,W_K$18, with binary rate

$W_1,\dots,W_K$19

and query complexity

$W_1,\dots,W_K$20

The same work proves a lower bound

$W_1,\dots,W_K$21

for any weakly locally computable approximate LLDC of rate $W_1,\dots,W_K$22 (Dikstein et al., 30 Jan 2026). The decoder is a polylogarithmic-round propagation procedure on an HDX, and the central combinatorial innovation is strongly explicit local routing that outputs random paths in polylogarithmic time and sub-logarithmic depth (Dikstein et al., 30 Jan 2026).

Across these regimes, local decodable information has a stable core and a changing implementation. The stable core is restricted access: recovery or inference must proceed from a local view. The changing implementation is whether locality is imposed on codeword queries, broadcast observations, compressed data probes, dynamic updates, routed paths in an HDX, or spatially varying likelihoods in a quantum decoder. The modern literature shows that this locality can be quantified exactly, pushed to capacity in some models, ruled out sharply in others, and rehabilitated by amortization or computational asymmetry when classical worst-case locality fails.