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Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of \emph{coded trace reconstruction}, the design and analysis of high-rate efficiently encodable codes that can be…
Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a…
Predicting the three-dimensional (3D) structure of a protein from its primary sequence of amino acids is known as the protein folding (PF) problem. Due to the central role of proteins' 3D structures in chemistry, biology and medicine…
We establish rigorous error bounds for prime counting using a truncated Gaussian (TG) kernel in the explicit formula framework. Our main theorem proves that the approximation error remains globally below 1/2 for all sufficiently large…
We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of…
The genetic code is connection between 64 codons, which are building blocks of the genes, and 20 amino acids, which are building blocks of the proteins. In addition to coding amino acids, a few codons code stop signal, which is at the end…
We present a semidefinite program optimization approach to quantum error correction that yields codes and recovery procedures that are robust against significant variations in the noise channel. Our approach allows us to optimize the…
Error correction codes are an integral part of communication applications, boosting the reliability of transmission. The optimal decoding of transmitted codewords is the maximum likelihood rule, which is NP-hard due to the curse of…
Many proposals for quantum information processing are subject to detectable loss errors. In this paper, we show that topological error correcting codes, which protect against computational errors, are also extremely robust against losses.…
Using the shape space of codons and tRNAs we give a physical description of the genetic code evolution on the basis of the codon capture and ambiguous intermediate scenarios in a consistent manner. In the lowest dimensional version of our…
A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential…
Code generation models have achieved impressive performance. However, they tend to be brittle as slight edits to a prompt could lead to very different generations; these robustness properties, critical for user experience when deployed in…
Low-depth random circuit codes possess many desirable properties for quantum error correction but have so far only been analyzed in the code capacity setting where it is assumed that encoding gates and syndrome measurements are noiseless.…
Modelising the translation errors by suitable mathematical operators in the crystal basis model of the genetic code and requiring that codons prone to be misread encode the same amino-acid, the main features of the organisation in…
We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with Tanner graph girth 16. While conventional constructions using circulant permutation matrices are limited to girth…
Randomisation is a critical tool in designing distributed systems. The common coin primitive, enabling the system members to agree on an unpredictable random number, has proven to be particularly useful. We observe, however, that it is…
Recently, we introduced a new class of finite alphabet iterative decoders (FAIDs) for low-density parity-check (LDPC) codes. These decoders are capable of surpassing belief propagation in the error floor region on the Binary Symmetric…
We propose a general framework for decoding quantum error-correcting codes with generative modeling. The model utilizes autoregressive neural networks, specifically Transformers, to learn the joint probability of logical operators and…
A permutationally invariant n-bit code for quantum error correction can be realized as a subspace stabilized by the non-Abelian group S_n. The code corresponds to bases for the trivial representation, and all other irreducible…
The importance of the notion of symmetry in physics is well established: could it also be the case for the genetic code? In this spirit, a model for the Genetic Code based on continuous symmetries and entitled the "Crystal Basis Model" has…