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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. 

Two seven-coordinate mononuclear Co(ii) complexes are proved to have field-induced magnetic relaxation and one of them possesses capped octahedral geometry.

 

Abstract The accurate, continuous analysis of healthcare-relevant gases such as nitrogen oxides (NO x ) in a humid environment remains elusive for low-cost, stretchable gas sensing devices. This study presents the design and demonstration of a moisture-resistant, stretchable NO x gas sensor based on laser-induced graphene (LIG). Sandwiched between a soft elastomeric substrate and a moisture-resistant semipermeable encapsulant, the LIG sensing and electrode layer is first optimized by tuning laser processing parameters such as power, image density, and defocus distance. The gas sensor, using a needlelike LIG prepared with optimal laser processing parameters, exhibits a large response of 4.18‰ ppm −1 to NO and 6.66‰ ppm −1 to NO 2 , an ultralow detection limit of 8.3 ppb to NO and 4.0 ppb to NO 2 , fast response/recovery, and excellent selectivity. The design of a stretchable serpentine structure in the LIG electrode and strain isolation from the stiff island allows the gas sensor to be stretched by 30%. Combined with a moisture-resistant property against a relative humidity of 90%, the reported gas sensor has further been demonstrated to monitor the personal local environment during different times of the day and analyze human breath samples to classify patients with respiratory diseases from healthy volunteers. Moisture-resistant, stretchable NO x gas sensors can expand the capability of wearable devices to detect biomarkers from humans and exposed environments for early disease diagnostics.