Abstract. One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., [Formula: see text]). [Karchmer, Raz, and Wigderson Super-logarithmic depth lower bounds via direct sum in communication complexity, in Proceedings of the Sixth Annual Structure in Complexity Theory Conference, IEEE Computer Society, Chicago, 1991, pp. 299–304] suggested approaching this problem by proving that the depth complexity of a composition of functions [Formula: see text] is roughly the sum of the depth complexities of [Formula: see text] and [Formula: see text]. They showed that the validity of this conjecture would imply that [Formula: see text]. The intuition that underlies the Karchmer, Raz, and Wigderson (KRW) conjecture is that the composition [Formula: see text] should behave like a “direct-sum problem”, in a certain sense, and, therefore, the depth complexity of [Formula: see text] should be the sum of the individual depth complexities. Nevertheless, there are two obstacles toward turning this intuition into a proof: first, we do not know how to prove that [Formula: see text] must behave like a direct-sum problem; second, we do not know how to prove that the complexity of the latter direct-sum problem is indeed the sum of the individual complexities. In this work, we focus on the second obstacle. To this end, we study a notion called “strong composition”, which is the same as [Formula: see text] except that it is forced to behave like a direct-sum problem. We prove a variant of the KRW conjecture for strong composition, thus overcoming the above second obstacle. This result demonstrates that the first obstacle above is the crucial barrier toward resolving the KRW conjecture. Along the way, we develop some general techniques that might be of independent interest.

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $P \nsubseteq NC^{1}$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested approaching this problem by proving that depth complexity of a composition of functions $f \diamond g$ is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that $\mathbf{P} \nsubseteq \mathbf{N C}^{1}$. The intuition that underlies the KRW conjecture is that the composition $f \diamond g$ should behave like a “direct-sum problem”, in a certain sense, and therefore the depth complexity of $f \diamond g$ should be the sum of the individual depth complexities. Nevertheless, there are two obstacles toward turning this intuition into a proof: first, we do not know how to prove that $f \diamond g$ must behave like a direct-sum problem; second, we do not know how to prove that the complexity of the latter direct-sum problem is indeed the sum of the individual complexities. In this work, we focus on the second obstacle. To this end, we study a notion called “strong composition”, which is the same as $f \diamond g$ except that it is forced to behave like a direct-sum problem. We prove a variant of the KRW conjecture for strong composition, thus overcoming the above second obstacle. This result demonstrates that the first obstacle above is the crucial barrier toward resolving the KRW conjecture. Along the way, we develop some general techniques that might be of independent interest.

Lifting theorems are theorems that bound the communication complexity of a composed function f∘gⁿ in terms of the query complexity of f and the communication complexity of g. Such theorems constitute a powerful generalization of direct-sum theorems for g, and have seen numerous applications in recent years. We prove a new lifting theorem that works for every two functions f,g such that the discrepancy of g is at most inverse polynomial in the input length of f. Our result is a significant generalization of the known direct-sum theorem for discrepancy, and extends the range of inner functions g for which lifting theorems hold.

Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of O(p²) under a random restriction that leaves each variable alive independently with probability p [SICOMP, 1998]. Using this result, he gave an Ω̃(n³) formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this work, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization - random projections. Based on our shrinkage results, we obtain an Ω̃(n³) formula size lower bound for an explicit function computed in AC⁰. This improves upon the best known formula size lower bounds for AC⁰, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function’s structure so that the function maintains structure even under projection - using such projections is necessary, as standard random restrictions simplify AC⁰ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on the proof of Håstad for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of p-random restrictions, our proof can be used as an exposition of Håstad’s result.

Lifting theorems are theorems that relate the query complexity of a function $f:\{0,1\}^{n}\to \{0,1\}$ to the communication complexity of the composed function $f\circ g^{n}$ for some “gadget” $g:\{ 0,1\}^{b}\times \{0,1\}^{b}\to \{0,1\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P\nsubseteq NC1). Karchmer, Raz, and Wigderson [13] suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions f◇g. They showed that the validity of this conjecture would imply that P\nsubseteq NC1. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function f.

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve three open problems: ; We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. : We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. : We give the strongest separation to-date between monotone Boolean formulas and monotone Boolean circuits. Namely, we show that the classical GEN problem, which has polynomial-size monotone Boolean circuits, requires monotone Boolean formulas of size 2Ω(n/polylog(n)). An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal. This is an extended abstract. The full version of the paper is available at https://arxiv.org/abs/2001.02144.

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t+1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. [Arkadev Chattopadhyay et al., 2017] and Wu et al. [Xiaodi Wu et al., 2017]. The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [Mika Göös et al., 2017], used the much larger indexing gadget. Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, Göös, Pitassi and Watson [Mika Göös et al., 2017] used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.

This special section comprises eleven fully refereed papers whose extended abstracts were presented at the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016) in New Brunswick, New Jersey, October 9--11, 2016. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2016 proceedings. The regular conference program consisted of 85 papers chosen from among 307 submissions. These were selected by a program committee consisting of Eric Blais, Mark Braverman, Siu On Chan, Moses Charikar, Marek Cygan, Irit Dinur (chair), Andrew Drucker, Faith Ellen, Sariel Har-Peled, Prahladh Harsha, Alexandra Kolla, Swastik Kopparty, Robert Krauthgamer, Brendan Lucier, Or Meir, Raghu Meka, Daniele Micciancio, Moni Naor, Joe Neeman, Rasmus Pagh, Rocco Servedio, Yaoyun Shi, Ola Svensson, Thomas Vidick, and Daniel Wichs. The papers invited to this special section were also selected with the input of the program committee. The eleven papers in this section include a broad range of topics, including algorithms, combinatorics, circuit complexity, data structures, learning theory, parameterized complexity, and probability. We thank the authors and the anonymous referees for their efforts. In addition, we would like to thank SICOMP Editor-in-Chief Leonard Schulman and SIAM Senior Publications Coordinator Heather Blythe for their help in preparing this special section.

One of the central problems in complexity theory is to prove super-logarithmic depth bounds for circuits computing a problem in P, i.e., to prove that P is not contained in NC^1. As an approach for this question, Karchmer, Raz and Wigderson [Mauricio Karchmer et al., 1995] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in NC^{1}. Since proving this conjecture is currently considered an extremely difficult problem, previous works by Edmonds, Impagliazzo, Rudich and Sgall [Edmonds et al., 2001], Håstad and Wigderson [Johan Håstad and Avi Wigderson, 1990] and Gavinsky, Meir, Weinstein and Wigderson [Dmitry Gavinsky et al., 2014] considered weaker variants of the conjecture. In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds.

Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit local algorithms for correction and detection of errors. Those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword. The fundamental question about LCCs and LTCs is to determine the optimal tradeoff among their rate, distance, and query complexity. In this work, we construct the first LCCs and LTCs with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length n , constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log n )) (for LCCs) and (log n ) O (log log n ) (for LTCs). In addition to having small query complexity, our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by LCCs or LTCs. Specifically, over large (but constant size) alphabet, our codes approach the Singleton bound, that is, they have almost the best-possible relationship between their rate and distance. Over the binary alphabet, our codes meet the Zyablov bound. Such tradeoffs between the rate and the relative distance were previously not known for any o ( n ) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.

One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [Comput. Complexity, 5 (1995), pp. 191--204] suggested approaching this problem by proving the following conjecture: given two Boolean functions $f$ and $g$, the depth complexity of the composed function $g\diamond f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$. As a starting point for studying the composition of functions, they introduced a relation called “the universal relation” and suggested studying the composition of universal relations. This suggestion proved fruitful, and an analogue of the Karchmer--Raz--Wigderson (KRW) conjecture for the universal relation was proved by Edmonds et al. [Comput. Complexity, 10 (2001), pp. 210--246]. An alternative proof was given later by H\aastad and Wigderson [in Advances in Computational Complexity Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 13, AMS, Providence, RI, 1993, pp. 119--134]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still an open question. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation.

One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson suggested to approach this problem by proving that formula complexity behaves "as expected" with respect to the composition of functions f * g. They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds. The first step toward proving the KRW conjecture was made by Edmonds et al., who proved an analogue of the conjecture for the composition of "universal relations". In this work, we extend the argument of Edmonds et al. further to f * g where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in Hastad's work on random restrictions, our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer and Wigderson. In addition, our proof gives a new structural result, which roughly says that the naive way for computing f * g is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n^{3-o(1)} due to Hastad.

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1) and constant relative distance, whose query complexity is exp(Õ(√logn)) (for LCCs) and (logn)O(loglogn) (for LTCs). Previously such codes were known to exist only with Ω(nβ) query complexity (for constant β>0).

The PCP theorem [Arora et al. 1998; Arora and Safra 1998] says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, that is, how long is the PCP compared to the original NP-proof? The state-of-the-art work of Ben-Sasson and Sudan [2008] and Dinur [2007] shows that one can encode proofs of length n by PCPs of length n · poly log n that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to n ε , then one can construct PCPs of length O ( n ) for circuit-SAT, and PCPs of length O ( t log t ) for any language in NTIME( t ). More specifically, for any ε > 0, we present (nonuniform) probabilistically checkable proofs (PCPs) of length 2 O (1/ε) · n that can be checked using n ε queries for circuit-SAT instances of size n . Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity ( o ( n )). Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations that are crucial in earlier high-rate algebraic PCP constructions. Using this observation, we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to this article for the first time for every message length, building on an earlier construction for infinitely many message lengths by Stichtenoth [2006].

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P ⊈ NC1). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds.

The PCP theorem (Arora et. al., J. ACM 45(1, 3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art work of Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)) shows that one can encode proofs of length n by PCPs of quasi-linear length that can be verified using a constant number of queries. In this work, we show that if the query complexity is relaxed to polynomial, then one can construct PCPs of linear length for circuit-SAT, and PCPs of length O(tlog t) for any language in NTIME(t). Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity. Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed for the first time for every message length.

The $\mathbf{IP}$ Theorem, which asserts that $\mathbf{IP}=\mathbf{PSPACE}$ [Lund et al., J. ACM, 39 (1992), pp. 859--868; Shamir, J. ACM, 39 (1992), pp. 878--880], is one of the major achievements of complexity theory. The known proofs of the theorem are based on the arithmetization technique, which transforms a quantified Boolean formula into a related polynomial. The intuition that underlies the use of polynomials is commonly explained by the fact that polynomials constitute good error-correcting codes. However, the known proofs seem tailored to the use of polynomials and do not generalize to arbitrary error-correcting codes. In this work, we show that the $\mathbf{IP}$ theorem can be proved by using general error-correcting codes and their tensor products. We believe that this establishes a rigorous basis for the aforementioned intuition and sheds further light on the $\mathbf{IP}$ theorem.

The PCP theorem (Arora et. al., J. ACM 45(1,3)) asserts the existence of proofs that can be verified by reading a very small part of the proof. Since the discovery of the theorem, there has been a considerable work on improving the theorem in terms of the length of the proofs, culminating in the construction of PCPs of quasi-linear length, by Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)). One common theme in the aforementioned PCP constructions is that they all rely heavily on sophisticated algebraic machinery. The aforementioned work of Dinur (J. ACM 54(3)) suggested an alternative approach for constructing PCPs, which gives a simpler and arguably more intuitive proof of the PCP theorem using combinatorial techniques. However, this combinatorial construction only yields PCPs of polynomial length, and is therefore inferior to the algebraic constructions in this respect. This gives rise to the natural question of whether the proof length of the algebraic constructions can be matched using the combinatorial approach. In this work, we provide a combinatorial construction of PCPs of length n · (log n)O(log log n ), coming very close to the state of the art algebraic constructions (whose proof length is n · (log n)O(1)). To this end, we develop a few generic PCP techniques which may be interesting in their own right. It should be mentioned that our construction does use low degree polynomials at one point. However, our use of polynomials is confined to the construction of error correcting codes with a certain simple multiplication property, and it is conceivable that such codes can be constructed without the use of polynomials.

A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof and in return is allowed to err with some bounded probability. The probability that the verifier accepts a proof of a false claim is called the soundness error and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with subconstant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work, we construct such PCP verifiers, i.e., PCPs that make only two queries and have subconstant soundness error. Our construction can be viewed as a combinatorial alternative to the “manifold vs. point” construction, which is the basis for all constructions in the literature for this parameter range. The “manifold vs. point” PCP is based on a low-degree test, while our construction is based on a direct product test. We also extend our construction to yield a decodable PCP (dPCP) with the same parameters. By plugging in this dPCP into the scheme of Dinur and Harsha (FOCS 2009), one gets an alternative construction of the result of Moshkovitz and Raz (FOCS 2008), namely a construction of two-query PCPs with small soundness error and small alphabet size. Our construction of a PCP is based on extending the derandomized direct product test of Impagliazzo, Kabanets, and Wigderson (STOC 09) to a derandomized parallel repetition theorem. More accurately, our PCP construction is obtained in two steps. We first prove a derandomized parallel repetition theorem for specially structured PCPs. Then, we show that any PCP can be transformed into one that has the required structure, by embedding it on a de-Bruijn graph.

A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts a false proof is called the soundness error, and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with sub-constant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work we construct such PCP verifiers, i.e., PCPs that make only two queries and have sub-constant soundness error. Our construction can be viewed as a combinatorial alternative to the "manifold vs. point'' construction, which is the only construction in the literature for this parameter range. The "manifold vs. point'' PCP is based on a low degree test, while our construction is based on a direct product test. Our construction of a PCP is based on extending the derandomized direct product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized parallel repetition theorem. More accurately, our PCP construction is obtained in two steps. We first prove a derandomized parallel repetition theorem for specially structured PCPs. Then, we show that any PCP can be transformed into one that has the required structure, by embedding it on a de-Bruijn graph.

The PCP theorem asserts the existence of proofs that can be verified by a verifier that reads only a very small part of the proof. The theorem was originally proved by Arora and Safra (J. ACM 45(1)) and Arora et al. (J. ACM 45(3)) using sophisticated algebraic tools. More than a decade later, Dinur (J. ACM 54(3)) gave a simpler and arguably more intuitive proof using alternative combinatorial techniques. One disadvantage of Dinur's proof compared to the previous algebraic proof is that it yields less efficient verifiers. In this work, we provide a combinatorial construction of PCPs with verifiers that are as efficient as the ones obtained by the algebraic methods. The result is the first combinatorial proof of the PCP theorem for (originally proved by Babai et al., STOC 1991), and a combinatorial construction of super-fast PCPs of Proximity for (first constructed by Ben-Sasson et al., CCC 2005).

An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. While the best known construction of locally testable codes (LTCs) by Ben-Sasson and Sudan [SIAM J. Comput., 38 (2008), pp. 551–607] and Dinur [J. ACM, 54 (2007), article 12] achieves very efficient parameters, it relies heavily on algebraic tools and on probabilistically checkable proof (PCP) machinery. In this work we present a new and arguably simpler construction of LTCs that is purely combinatorial, does not rely on PCP machinery, and matches the parameters of the best known construction. However, unlike the latter construction, our construction is not entirely explicit.