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Braverman, Mark (Ed.)For an abelian group H acting on the set [𝓁], an (H,𝓁)lift of a graph G₀ is a graph obtained by replacing each vertex by 𝓁 copies, and each edge by a matching corresponding to the action of an element of H. Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O'Donnell [STOC 2021] achieving distance Ω̃(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance Ω(N/log(N)). However, both these constructions are nonexplicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019]. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ⩽ Sym(𝓁), constant degree d ≥ 3 and ε > 0, we construct explicit dregular expander graphs G obtained from an (H,𝓁)lift of a (suitable) base nvertex expander G₀ with the following parameters: ii) λ(G) ≤ 2√{d1} + ε, for any lift size 𝓁 ≤ 2^{n^{δ}} where δ = δ(d,ε), iii) λ(G) ≤ ε ⋅ d, for any lift size 𝓁 ≤ 2^{n^{δ₀}} for a fixed δ₀ > 0, when d ≥ d₀(ε), or iv) λ(G) ≤ Õ(√d), for lift size "exactly" 𝓁 = 2^{Θ(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasicyclic LDPC codes with wide range of circulant sizes. Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for 2lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depthfirst search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.more » « less

null (Ed.)The Gilbert–Varshamov bound nonconstructively establishes the existence of binary codes of distance 1/2−є/2 and rate Ω(є2). In a breakthrough result, TaShma [STOC 2017] constructed the first explicit family of nearly optimal binary codes with distance 1/2−є/2 and rate Ω(є2+α), where α → 0 as є → 0. Moreover, the codes in TaShma’s construction are єbalanced, where the distance between distinct codewords is not only bounded from below by 1/2−є/2, but also from above by 1/2+є/2. Polynomial time decoding algorithms for (a slight modification of) TaShma’s codes appeared in [FOCS 2020], and were based on the SumofSquares (SoS) semidefinite programming hierarchy. The running times for these algorithms were of the form NOα(1) for unique decoding, and NOє,α(1) for the setting of “gentle list decoding”, with large exponents of N even when α is a fixed constant. We derive new algorithms for both these tasks, running in time Õє(N). Our algorithms also apply to the general setting of decoding directsum codes. Our algorithms follow from new structural and algorithmic results for collections of ktuples (ordered hypergraphs) possessing a “structured expansion” property, which we call splittability. This property was previously identified and used in the analysis of SoSbased decoding and constraint satisfaction algorithms, and is also known to be satisfied by TaShma’s code construction. We obtain a new weak regularity decomposition for (possibly sparse) splittable collections W ⊆ [n]k, similar to the regularity decomposition for dense structures by Frieze and Kannan [FOCS 1996]. These decompositions are also computable in nearlinear time Õ(W ), and form a key component of our algorithmic results.more » « less

null (Ed.)The GilbertVarshamov bound (nonconstructively) establishes the existence of binary codes of distance 1/2ε and rate Ω(ε 2 ) (where an upper bound of O(ε 2 log(1/ε)) is known). TaShma [STOC 2017] gave an explicit construction of εbalanced binary codes, where any two distinct codewords are at a distance between 1/2ε/2 and 1/2+ε/2, achieving a near optimal rate of Ω(ε 2+β ), where β→ 0 as ε→ 0. We develop unique and list decoding algorithms for (a slight modification of) the family of codes constructed by TaShma, in the adversarial error model. We prove the following results for εbalanced codes with block length N and rate Ω(ε 2+β ) in this family: For all , there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time N Oε,β(1) . For any fixed constant β independent of ε, there is an explicit construction of codes which can be uniquely decoded up to an error of half the minimum distance in time (log(1/ε)) O(1) ·N Oβ(1) . For any , there are explicit εbalanced codes with rate Ω(ε 2+β ) which can be list decoded up to error 1/2ε ' in time N Oε,ε' ,β(1), where ε ' ,β→ 0 as ε→ 0. The starting point of our algorithms is the framework for list decoding directsum codes develop in Alev et al. [SODA 2020], which uses the SumofSquares SDP hierarchy. The rates obtained there were quasipolynomial in ε. Here, we show how to overcome the far from optimal rates of this framework obtaining unique decoding algorithms for explicit binary codes of near optimal rate. These codes are based on simple modifications of TaShma's construction.more » « less