Good Locally Testable Codes

• Good LTCs are error-correcting codes with constant rate, relative distance, and constant query complexity, verified by a randomized local tester.
• They employ high-dimensional expansion techniques and structures like 2D Cayley complexes to ensure robust local-to-global testability.
• These constructions overcome classical trade-offs, significantly advancing both classical and quantum error correction methodologies.

A good Locally Testable Code (LTC) is a family of error-correcting codes $\{C_n\}$ with the following properties: for some absolute constants $R, \delta, q, s>0$ (independent of the block-length $n$), each $C_n$ is a linear code of length $n$, dimension $\ge R n$, and minimum Hamming distance $\ge \delta n$, equipped with a randomized tester that, on input $x\in\mathbb{F}_q^n$, queries at most $q$ positions and either accepts or rejects, with the rejection probability lower-bounded (linearly) in the normalized Hamming distance from $x$ to $C_n$. The existence and explicit construction of such codes, sometimes called $c^3$-LTCs, was a central open problem in coding theory and combinatorics, resolved only recently.

1. Definitional Framework and Parameters

A $q$-query LTC over a finite field $\mathbb{F}_q$ is a code family $\{C_n \subset \mathbb{F}_q^n\}$ such that:

• Rate: $\operatorname{dim} C_n \ge R n$ for some $R>0$ (constant rate).
• Relative distance: $\min_{x\neq y\in C_n} d_H(x,y) \ge \delta n$ for some $\delta>0$ (constant relative distance).
• Local testability: There exists a tester $T$ that, on input $x\in\mathbb{F}_q^n$, selects (with prescribed randomness) at most $q$ coordinates and decides acceptance/rejection, with:
  • Completeness: $x\in C_n \implies \mathbb{P}[\text{accept}]=1$.
  • Soundness: For all $x\in\mathbb{F}_q^n$, $$\mathbb{P}[\text{reject}] \geq s \cdot \frac{1}{n} d_H(x, C_n)$$ for some $s > 0$.

The minimal $q$ such that a code family achieves all three properties is called its query complexity. Codes with constant $q$, $R$, and $\delta$ are termed asymptotically good LTCs, or $c^3$-LTCs.

2. Main Constructions of Good LTCs

Historically, known LTCs exhibited trade-offs: algebraic constructions (Hadamard, Reed–Muller) achieved local testability at the expense of vanishing rate, while high-rate codes achieved only superconstant or sublinear query complexity. Explicit construction of $c^3$-LTCs was achieved independently via several methodologies in 2021–2022, fundamentally relying on high-dimensional expanders and algebraic-combinatorial techniques.

2.1 Two-Dimensional Cayley Complex and Product-code Construction

In the construction of Dinur–Evra–Livne–Lubotzky–Mozes (Dinur et al., 2022), and concurrently in (Dinur et al., 2021, Panteleev et al., 2021), the code is defined over the 2D left-right Cayley complex $X = \mathrm{Cay}^2(G;A,B)$:

• Vertices $V=G$, edges $E$ correspond to left/right multiplications by symmetric generators $A,B\subset G$, and squares are the commutator cycles $[a,g,b]$.
• For each edge, local consistency is enforced by a base code (e.g., $C_A$ or $C_B$).
• Global codewords are functions $f: X(2) \to \mathbb{F}_q$ such that for every edge, the induced vector on incident squares is a codeword in the corresponding base code.
• The tester selects a random vertex, queries all incident squares ($q=|A|\cdot|B|=O(1)$), and checks that the collection forms a valid codeword of the tensor product code $C_A \otimes C_B$.

Key analysis steps utilize:

• Spectral-expansion of the Cayley graphs to ensure rapid mixing and small-set expansion.
• Distance and rate derived from the expansion and the distance of the base codes.
• Soundness analysis (robust local testability): rejection probability is linearly lower-bounded in normalized Hamming distance, via the propagation of disagreements along the high-dimensional expander skeleton, and an iterative local improvement procedure.

The parameters achieved are (for explicit, binary codes): for some absolute constants $R,\delta,s>0$,

• $\operatorname{dim} C_n \ge R n$,
• minimum distance at least $\delta n$,
• locality $q=O(1)$,
• rejection probability at least $s \cdot d_H(x,C_n)/n$ for all $x$.

This construction resolves the c³-LTC conjecture for linear codes (Dinur et al., 2022, Dinur et al., 2021, Panteleev et al., 2021).

2.2 Lossless Expanders and Balanced Product Complexes

Further avenues, e.g., (Lin et al., 2022), use chain complexes built from balanced products of bipartite lossless expanders endowed with free group action. Codes are defined as the kernel of a block-sparse parity-check matrix, where each row corresponds to a local constraint of bounded weight (the row/column degree of the expander). Expansion in the underlying graphs guarantees distance and soundness, making every code in the family a $(w,s)$-locally testable code with blocklength $n$, dimension $\Omega(n)$, relative distance $\Omega(1)$, and constant $q=w$.

3. High-Dimensional Expansion Framework

A central insight is that local testability in these codes reduces to high-dimensional coboundary or cosystolic expansion properties. The crucial features are:

• Expander graphs (Ramanujan, 1-sided lossless, Cayley) serve as the 1- and 2-skeletons.
• Square complexes (and, in quantum settings, 3- or 4-complexes) lift local code testability to the global code via combinatorial expansion.
• The local-to-global argument proceeds through a robust version of the classical Sipser–Spielman expander code analysis: local disagreement propagates across the expander, forcing global disagreement unless the input agrees with some codeword.

In the algebraic setting, cosystolic expansion of sheaves on high-dimensional cell complexes (First et al., 28 Mar 2024, First et al., 2022), or high-dimensional expanding systems (Kaufman et al., 2021), yields conditions under which the corresponding cocycle code family is a good LTC. The key is expansion in all links of the complex, giving local-to-global testability.

4. Lower Bounds, Limitations, and Alphabet Dependencies

Negative results constrain the possibilities for LTCs:

• Dense LTCs (i.e., codes with dense tester hypergraphs) cannot be asymptotically good: any 3-query LTC with tester-density $d \gg 1$ has rate at most $O(d^{-1/2})$; with $q>3$, rate at most $O(1/d)$ for density $d \gg n^{q-2}$ (Dinur et al., 2010). This demonstrates that good LTCs must be built from sparse constraint structures.
• For affine-invariant codes with constant-query testers, the rate must vanish as $O((\log N)^{q-2}/N)$, providing tight lower bounds for this structural class (Bhattacharyya et al., 2015).

Recent progress achieved a precise dichotomy in alphabet-query complexity tradeoffs. There are fundamental obstructions for 2-query LTCs with binary alphabet (Ben-Sasson–Goldreich–Sudan impossibility), but constructions with $q=3$ queries circumvent this, achieving good LTCs over binary alphabets (First et al., 18 Dec 2025). On any alphabet of size $>2$, good 2-query LTCs now exist. For $\mathbb{F}$-linear codes, dimension $>1$ allows 2-query LTCs, while for dimension $1$ (i.e., field alphabet) at least $q=3$ is necessary.

5. Methodological Table: Modern Approaches to Good LTCs

Method	Key Reference(s)	Alphabet size	Rate & Distance	Query Complexity	Framework
2D Cayley Complex	(Dinur et al., 2022, Dinur et al., 2021, Panteleev et al., 2021)	binary/finite field	$>0$	$O(1)$	High-dimensional expansion, group theory
Balanced Product Expanders	(Lin et al., 2022)	binary	$>0$	$O(1)$	Lossless expanders, combinatorial
Sheaf–Expander Lifts	(First et al., 28 Mar 2024, First et al., 2022)	large/binary	$>0$	$2$ (or $3$)	Cosystolic expansion, sheaf theory
Polynomial Amplification	(Kopparty et al., 2015)	binary	$>0$	$n^{o(1)}$	Distance amplification, tensor codes

6. Principal Open Problems and Generalizations

The c³-problem for classical LTCs—existence of infinite families with all three parameters constant—is resolved. Strong technical barriers remain in exploring:

• Explicit quantum-LTCs with polylogarithmic soundness and constant queries (Dinur et al., 12 Feb 2024).
• Tighter bounds for parameters and constructive, practical codes.
• Extending expansion/testability equivalence to higher-dimensional lifted product complexes, affecting both classical and quantum error correction.
• Precise quantitative tradeoffs for alphabet size, code linearity, and query complexity (First et al., 18 Dec 2025).

7. Historical Context and Significance

Prior to 2021, all known good LTCs failed to simultaneously achieve constant rate, constant distance, and constant query locality. Early constructions relied on algebraic codes with high redundancy or large alphabets. The breakthrough via high-dimensional combinatorics, balanced product expanders, and the use of topological sheaf theory represents a significant advance, unifying a decade's progress in algebraic, combinatorial, and geometric coding theory (Dinur et al., 2022, Dinur et al., 2021, Panteleev et al., 2021, Lin et al., 2022).

This line of work is directly connected to the construction of quantum LDPC and LTC codes, expanding the applicability of expansion methods and opening avenues for further research in robustness, decoding, and practical implementation.

---

References:

• "Good Locally Testable Codes" (Dinur et al., 2022).
• "Locally Testable Codes with constant rate, distance, and locality" (Dinur et al., 2021).
• "Asymptotically Good Quantum and Locally Testable Classical LDPC Codes" (Panteleev et al., 2021).
• "c³-Locally Testable Codes from Lossless Expanders" (Lin et al., 2022).
• "Dense locally testable codes cannot have constant rate and distance" (Dinur et al., 2010).
• "Lower bounds for constant query affine-invariant LCCs and LTCs" (Bhattacharyya et al., 2015).
• "Good Locally Testable Codes with Small Alphabet and Small Query Size" (First et al., 18 Dec 2025).
• "Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes" (First et al., 28 Mar 2024).
• "On Good $2$-Query Locally Testable Codes from Sheaves on High Dimensional Expanders" (First et al., 2022).