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1. Unique Decoding of Explicit -balanced Codes Near the Gilbert-Varshamov Bound

   https://doi.org/10.1109/FOCS46700.2020.00048  

   Jeronimo, Fernando Granha; Quintana, Dylan; Srivastava, Shashank; Tulsiani, Madhur (November 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS))

   null (Ed.)

   The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance 1/2-ε and rate Ω(ε 2 ) (where an upper bound of O(ε 2 log(1/ε)) is known). Ta-Shma [STOC 2017] gave an explicit construction of ε-balanced binary codes, where any two distinct codewords are at a distance between 1/2-ε/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 Ta-Shma, in the adversarial error model. We prove the following results for ε-balanced 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 ε-balanced codes with rate Ω(ε 2+β ) which can be list decoded up to error 1/2-ε ' 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 direct-sum codes develop in Alev et al. [SODA 2020], which uses the Sum-of-Squares 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 Ta-Shma's construction. 

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2. A Complete Linear Programming Hierarchy for Linear Codes

   https://doi.org/10.4230/LIPIcs.ITCS.2022.51  

   Coregliano, Leonardo Nagami (January 2022, Leibniz international proceedings in informatics)

   Braverman, Mark (Ed.)

   A longstanding open problem in coding theory is to determine the best (asymptotic) rate R₂(δ) of binary codes with minimum constant (relative) distance δ. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte’s linear programs. To date these results remain the best known lower and upper bounds on R₂(δ) with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size A^{Lin}₂(n,d) of an optimum linear binary code (in fact, over any finite field) of a given blocklength n and distance d. This hierarchy has several notable features: 1) It is a natural generalization of the Delsarte LPs used in the first MRRW bound. 2) It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis. 3) It is complete in the sense that the optimum code size can be retrieved from level O(n²). 4) It provides an answer in the form of a hierarchy (in larger dimensional spaces) to the question of how to cut Delsarte’s LP polytopes to approximate the true size of linear codes. We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of 𝓁 words. Our method also generalizes to translation schemes under mild assumptions. 

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3. Leibniz International Proceedings in Informatics (LIPIcs):50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

   https://doi.org/10.4230/LIPIcs.ICALP.2023.41  

   Cheng, Kuan; Jin, Zhengzhong; Li, Xin; Wei, Zhide; Zheng, Yu (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   This work continues the study of linear error correcting codes against adversarial insertion deletion errors (insdel errors). Previously, the work of Cheng, Guruswami, Haeupler, and Li [Kuan Cheng et al., 2021] showed the existence of asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, or achieve rate arbitrarily close to 1/2 even over the binary alphabet. As shown in [Kuan Cheng et al., 2021], these bounds are also the best possible. However, known explicit constructions in [Kuan Cheng et al., 2021], and subsequent improved constructions by Con, Shpilka, and Tamo [Con et al., 2022] all fall short of meeting these bounds. Over any constant size alphabet, they can only achieve rate < 1/8 or correct < 1/4 fraction of errors; over the binary alphabet, they can only achieve rate < 1/1216 or correct < 1/54 fraction of errors. Apparently, previous techniques face inherent barriers to achieve rate better than 1/4 or correct more than 1/2 fraction of errors. In this work we give new constructions of such codes that meet these bounds, namely, asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, and binary asymptotically good linear insdel codes that can achieve rate arbitrarily close to 1/2. All our constructions are efficiently encodable and decodable. Our constructions are based on a novel approach of code concatenation, which embeds the index information implicitly into codewords. This significantly differs from previous techniques and may be of independent interest. Finally, we also prove the existence of linear concatenated insdel codes with parameters that match random linear codes, and propose a conjecture about linear insdel codes. 

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4. Coded Distributed Computing: Performance Limits and Code Designs

   https://doi.org/10.1109/ITW44776.2019.8989097  

   Jamali, Mohammad Vahid; Soleymani, Mahdi; Mahdavifar, Hessam (August 2019, 2019 IEEE Information Theory Workshop (ITW))

   We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into $$k$$ smaller tasks, encoded using an $(n,k)$ linear code, and performed over $$n$$ distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length $$n$$ used over erasure channels. Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist good binary linear codes that attain, asymptotically, the best performance any linear code, not necessarily binary, can achieve. We also investigate the performance of coded distributed computing systems using polar and Reed-Muller (RM) codes that can benefit from low-complexity decoding, and superior performance, respectively, as well as explicit constructions. The proposed framework in this paper can enable efficient designs of distributed computing systems given the rich literature in the channel coding theory. 

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5. Near-linear time decoding of Ta-Shma’s codes via splittable regularity

   https://doi.org/10.1145/3406325.3451126  

   Jeronimo, Fernando Granha; Srivastava, Shashank; Tulsiani, Madhur (June 2021, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing)

   null (Ed.)

   The Gilbert–Varshamov bound non-constructively establishes the existence of binary codes of distance 1/2−є/2 and rate Ω(є2). In a breakthrough result, Ta-Shma [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 Ta-Shma’s construction are є-balanced, 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) Ta-Shma’s codes appeared in [FOCS 2020], and were based on the Sum-of-Squares (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 direct-sum codes. Our algorithms follow from new structural and algorithmic results for collections of k-tuples (ordered hypergraphs) possessing a “structured expansion” property, which we call splittability. This property was previously identified and used in the analysis of SoS-based decoding and constraint satisfaction algorithms, and is also known to be satisfied by Ta-Shma’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 near-linear time Õ(|W |), and form a key component of our algorithmic results. 

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