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List Decoding of Direct Sum Codes

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List Decoding of Direct Sum Codes List Decoding of Direct Sum Codes Vedat Levi Alev∗ Fernando Granha Jeronimoy Dylan Quintanaz Shashank Srivastava§ Madhur Tulsiani{ Abstract 1 Introduction We consider families of codes obtained by "lifting" a base We consider the problem of list decoding binary codes C code through operations such as k-XOR applied to "local obtained by starting with a binary base code C and am- views" of codewords of C, according to a suitable k-uniform 0 hypergraph. The k-XOR operation yields the direct sum plifying its distance by "lifting" C to a new code C us- encoding used in works of [Ta-Shma, STOC 2017] and [Dinur ing an expanding or pseudorandom structure. Exam- and Kaufman, FOCS 2017]. ples of such constructions include direct products where We give a general framework for list decoding such n 0 k nk one "lifts" (say) C ⊆ F2 to C ⊆ (F2) with each po- lifted codes, as long as the base code admits a unique de- 0 coding algorithm, and the hypergraph used for lifting satis- sition in y 2 C being a k-tuple of bits from k positions fies certain expansion properties. We show that these proper- in z 2 C. Another example is direct sum codes where k ties are indeed satisfied by the collection of length k walks on C0 ⊆ Fn and each position in y is the parity of a k-tuple a sufficiently strong expanding graph, and by hypergraphs 2 corresponding to high-dimensional expanders. Instantiating of bits in z 2 C. Of course, for many applications, it is our framework, we obtain list decoding algorithms for direct interesting to consider a small “pseudorandom” set of sum liftings corresponding to the above hypergraph fami- k-tuples, instead of considering the complete set of size lies. Using known connections between direct sum and direct nk. product, we also recover (and strengthen) the recent results This kind of distance amplification is well known of Dinur et al. [SODA 2019] on list decoding for direct prod- + uct liftings. in coding theory [ABN 92, IW97, GI01, TS17] and it Our framework relies on relaxations given by the Sum- can draw on the vast repertoire of random and pseu- of-Squares (SOS) SDP hierarchy for solving various con- dorandom expanding objects [HLW06, Lub18]. Such straint satisfaction problems (CSPs). We view the problem constructions are also known to have several applica- of recovering the closest codeword to a given (possibly cor- tions to the theory of Probabilitically Checkable Proofs rupted) word, as finding the optimal solution to an instance + of a CSP. Constraints in the instance correspond to edges (PCPs) [IKW09, DS14, DDG 15, Cha16, Aro02]. How- of the lifting hypergraph, and the solutions are restricted to ever, despite having several useful properties, it might lie in the base code C. We show that recent algorithms for not always be clear how to decode the codes resulting (approximately) solving CSPs on certain expanding hyper- from such constructions, especially when constructed graphs by some of the authors also yield a decoding algo- using sparse pseudorandom structures. An impor- rithm for such lifted codes. tant example of this phenomenon is Ta-Shma’s explicit We extend the framework to list decoding, by requiring construction of binary codes of arbitrarily large dis- the SOS solution to minimize a convex proxy for negative tance near the (non-constructive) Gilbert-Varshamov entropy. We show that this ensures a covering property for bound [TS17]. Although the construction is explicit, ef- the SOS solution, and the "condition and round" approach ficient decoding is not known. Going beyond unique- used in several SOS algorithms can then be used to recover decoding algorithms, it is also useful to have efficient the required list of codewords. list-decoding algorithms for complexity-theoretic ap- plications [Sud00, Gur01, STV01, Tre04]. The question of list decoding such pseudoran- ∗Supported by NSERC Discovery Grant 2950-120715, NSERC Ac- dom constructions of direct-product codes was consid- celerator Supplement 2950-120719, and partially supported by NSF ered by Dinur et al. [DHK+19], extending a unique- awards CCF-1254044 and CCF-1718820. University of Waterloo. decoding result of Alon et al. [ABN+92]. While Alon [email protected]. ySupported in part by NSF grants CCF-1254044 and CCF-1816372. et al. proved that the code is unique-decodable when University of Chicago. [email protected]. the lifting hypergraph (collection of k-tuples) is a good zUniversity of Chicago. [email protected] "sampler", Dinur et al. showed that when the hyper- §TTIC. [email protected] graph has additional structure (which they called be- { Supported by NSF grants CCF-1254044 and CCF-1816372. TTIC. ing a "double sampler") then the code is also list de- [email protected] Copyright ⃝c 2020 by SIAM Unauthorized reproduction of this article is prohibited codable. They also posed the question of understand- splitting can easily be related to that of the underlying ing structural properties of the hypergraph that might expander graph. In both cases, we take the function yield even unique decoding algorithms for the direct g to be k-XOR which corresponds to the direct sum sum based liftings. lifting. We also obtain results for direct product codes We develop a generic framework to understand via a simple (and standard) reduction to the direct sum properties of the hypergraphs under which the lifted case. code C0 admits efficient list decoding algorithms, as- Our Results. Now we provide a quantitative version suming only efficient unique decoding algorithms for of our main result. For this, we split the main result the base code C. Formally, let X be a downward-closed into two cases (due to their difference in parameters): hypergraph (simplicial complex) defined by taking the HDXs and length k walks on expander graphs. We start downward closure of a k-uniform hypergraph, and let with the former expanding object. k g : F2 ! F2 be any boolean function. X(i) denotes the collection of sets of size i in X and X(≤ d) the col- THEOREM 1.1. (HDX (INFORMAL)) Let b0 < 1/2 be lection of sets of size at most d. We consider the lift a constant and b 2 (0, b0). Suppose X(≤ d) is a g- 0 g X(1) 0 X(k) HDX on n vertices with g ≤ (log(1/b))−O(log(1/b)) and C = lift ( )(C), where C ⊆ F2 and C ⊆ F2 , and X k = (( ( ))2 2) each bit of y 2 C0 is obtained by applying the func- d W log 1/b /b . C ⊂ Fn ≥ tion g to the corresponding k bits of z 2 C. We study For every linear code 1 2 with relative distance X(k) properties of g and X under which this lifting admits 1/2 − b0, there exists a direct sum lifting Ck ⊂ F2 with W (1) an efficient list decoding algorithm. k = O (log(1/b)) and relative distance ≥ 1/2 − b b0 We consider two properties of this lifting, robust- satisfying the following: ness and tensoriality, which we will be formally defined later. We will show that these properties are sufficient - [Efficient List Decoding] If y˜ is (1/2 − b)-close to Ck, to yield decoding algorithms. The first property (ro- then we can compute the list of all the codewords of Ck −O(1) bustness) essentially requires that for any two words that are (1/2 − b)-close to y˜ in time nb · f (n), X(1) where f (n) is the running time of a unique decoding in F2 at a moderate distance, the lifting amplifies the distance between them. While the second property is algorithm for C1. of a more technical nature and is inspired by the Sum- 1 - [Rate] The rate rk of Ck is rk = r1 · jX(1)j / jX(k)j, of-Squares (SOS) SDP hierarchy used for our decoding where r is the rate of C . algorithms, it is implied by some simpler combinato- 1 1 rial properties. Roughly speaking, this combinatorial A consequence of this result is a method of decod- property requires that the graph on (say) X(k/2) de- ing the direct product lifting on a HDX via a reduction fined by connecting s, t 2 X(k/2) if s \ t = Æ and to the direct sum case. s [ t 2 X(k), is a sufficiently good expander (and sim- ilarly for graphs on X(k/4), X(k/8) and so on). This COROLLARY 1.1. (HDX (INFORMAL)) Let #0 < 1/2 be property requires that the k-tuples can be (recursively) a constant and # > 0. Suppose X(≤ d) is a g-HDX split into disjoint pieces such that at each step the graph on n vertices with g ≤ (log(1/#))−O(log(1/#)) and d = obtained between the pairs of pieces is a good ex- W((log(1/#))2/#2). n pander. We refer to this property as splittability. For every linear code C1 ⊂ F2 with relative distance Expanding Structures. We instantiate the above ≥ 1/2 − #0, there exists a direct product encoding C` ⊂ framework with two specific structures: the collection ` X(`) (F2) with ` = O(log(1/#)) that can be efficiently list of k-sized hyperedges of a high-dimensional expander decoded up to distance 1 − #. (HDX) and the collection of length k walks of an ex- pander graph. HDXs are downward-closed hyper- REMARK 1.1. List decoding the direct product lifting was graphs satisfying certain expansion properties. We will first established by Dinur et al. in[DHK+19] using an quantify this expansion using Dinur and Kaufman’s expanding object introduced by them, namely, double sam- notion of a g-HDX [DK17].
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