Locally Decodable Codes with Randomized Encoding

We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message bit can be recovered with good probability by querying only a small number of positions in the corrupted codeword. This allows one to recover any message bit very efficiently in sub-linear or even logarithmic time. Besides this straightforward application, locally decodable codes have also found many other applications such as private information retrieval, secure multiparty computation, and average-case complexity. However, despite extensive research, the tradeoff between the rate of the code and the number of queries is somewhat disappointing. For example, the best known constructions still need super-polynomially long codeword length even with a logarithmic number of queries, and need a polynomial number of queries to achieve a constant rate. In this paper, we show that by using a randomized encoding, in several models we can achieve significantly better rate-query tradeoff. In addition, our codes work for both the standard Hamming errors, and the more general and harder edit errors.Comment: 23 page

Miniaturization of Branch-Line Coupler Using Composite Right/Left-Handed Transmission Lines with Novel Meander-shaped-slots CSSRR

A novel compact-size branch-line coupler using composite right/left-handed transmission lines is proposed in this paper. In order to obtain miniaturization, composite right/left-handed transmission lines with novel complementary split single ring resonators which are realized by loading a pair of meander-shaped-slots in the split of the ring are designed. This novel coupler occupies only 22.8% of the area of the conventional approach at 0.7 GHz. The proposed coupler can be implemented by using the standard printed-circuit-board etching processes without any implementation of lumped elements and via-holes, making it very useful for wireless communication systems. The agreement between measured and stimulated results validates the feasible configuration of the proposed coupler

Phantom energy of a quenched, prethermal quantum many-body scar state

Strongly interacting quantum systems can exhibit emergent excitations that differ qualitatively from their microscopic degrees of freedom. Here we study an emergent phenomenon that is intrinsic to such systems far from equilibrium: Namely, the transmutation of attractive interactions into repulsive interactions. We initialize an attractively interacting Bose gas in a highly excited and correlated nonthermal state, quench the confining potential, and measure how the kinetic and total energies evolve after the quench. Although the bare interactions are attractive, the low-energy degrees of freedom evolve as if they repel each other: Thus, their kinetic energy paradoxically decreases as the gas is compressed. We quantify the missing ``phantom'' energy by benchmarking our experimental results against generalized hydrodynamics (GHD) calculations. We present evidence that the missing kinetic energy is stored in very high-momentum modes.Comment: 5 pages, 4 figures with 15-page supplement including 9 figure

Thermalization near integrability in a dipolar quantum Newton's cradle

Isolated quantum many-body systems with integrable dynamics generically do not thermalize when taken far from equilibrium. As one perturbs such systems away from the integrable point, thermalization sets in, but the nature of the crossover from integrable to thermalizing behavior is an unresolved and actively discussed question. We explore this question by studying the dynamics of the momentum distribution function in a dipolar quantum Newton's cradle consisting of highly magnetic dysprosium atoms. This is accomplished by creating the first one-dimensional Bose gas with strong magnetic dipole-dipole interactions. These interactions provide tunability of both the strength of the integrability-breaking perturbation and the nature of the near-integrable dynamics. We provide the first experimental evidence that thermalization close to a strongly interacting integrable point occurs in two steps: prethermalization followed by near-exponential thermalization. Exact numerical calculations on a two-rung lattice model yield a similar two-timescale process, suggesting that this is generic in strongly interacting near-integrable models. Moreover, the measured thermalization rate is consistent with a parameter-free theoretical estimate, based on identifying the types of collisions that dominate thermalization. By providing tunability between regimes of integrable and nonintegrable dynamics, our work sheds light both on the mechanisms by which isolated quantum many-body systems thermalize, and on the temporal structure of the onset of thermalization.Comment: 6 figures, 9 pages main text; 12 appendices with 12 figure

Linear Insertion Deletion Codes in the High-Noise and High-Rate Regimes

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