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  1. Random linear codes (RLCs) are well known to have nice combinatorial properties and near-optimal parameters in many different settings. However, getting explicit constructions matching the parameters of RLCs is challenging, and RLCs are hard to decode efficiently. This motivated several previous works to study the problem of partially derandomizing RLCs, by applying certain operations to an explicit mother code. Among them, one of the most well studied operations is random puncturing, where a series of works culminated in the work of Guruswami and Mosheiff (FOCS’ 22), which showed that a random puncturing of a low-biased code is likely to possess almost all interesting local properties of RLCs. In this work, we provide an in-depth study of another, dual operation of random puncturing, known as random shortening, which can be viewed equivalently as random puncturing on the dual code. Our main results show that for any small , by starting from a mother code with certain weaker conditions (e.g., having a large distance) and performing a random (or even pseudorandom) shortening, the new code is -biased with high probability. Our results hold for any field size and yield a shortened code with constant rate. This can be viewed as a complement to random puncturing, and together, we can obtain codes with properties like RLCs from weaker initial conditions. Our proofs involve several non-trivial methods of estimating the weight distribution of codewords, which may be of independent interest. 
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  2. Longest Increasing Subsequence (LIS) is a fundamental problem in combinatorics and computer science. Previously, there have been numerous works on both upper bounds and lower bounds of the time complexity of computing and approximating , yet only a few on the equally important space complexity. In this paper, we further study the space complexity of computing and approximating LIS in various models. Specifically, we prove non-trivial space lower bounds in the following two models: (1) the adaptive query-once model or read-once branching programs, and (2) the streaming model where the order of streaming is different from the natural order. As far as we know, there are no previous works on the space complexity of LIS in these models. Besides the bounds, our work also leaves many intriguing open problems. 
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  3. 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. Locally Decodable Codes (LDCs) are error-correcting codes for which individual message symbols can be quickly recovered despite errors in the codeword. LDCs for Hamming errors have been studied extensively in the past few decades, where a major goal is to understand the amount of redundancy that is necessary and sufficient to decode from large amounts of error, with small query complexity. Despite exciting progress, we still don't have satisfactory answers in several important parameter regimes. For example, in the case of 3-query LDCs, the gap between existing constructions and lower bounds is superpolynomial in the message length. In this work we study LDCs for insertion and deletion errors, called Insdel LDCs. Their study was initiated by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), who gave a reduction from Hamming LDCs to Insdel LDCs with a small blowup in the code parameters. On the other hand, the only known lower bounds for Insdel LDCs come from those for Hamming LDCs, thus there is no separation between them. Here we prove new, strong lower bounds for the existence of Insdel LDCs. In particular, we show that 2-query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all q-query Insdel LDCs with constant q. For q ≥ 3 our bounds are exponential in the existing lower bounds for Hamming LDCs. Furthermore, our exponential lower bounds continue to hold for adaptive decoders, and even in private-key settings where the encoder and decoder share secret randomness. This exhibits a strict separation between Hamming LDCs and Insdel LDCs. Our strong lower bounds also hold for the related notion of Insdel LCCs (except in the private-key setting), due to an analogue to the Insdel notions of a reduction from Hamming LCCs to LDCs. Our techniques are based on a delicate design and analysis of hard distributions of insertion and deletion errors, which depart significantly from typical techniques used in analyzing Hamming LDCs. 
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