Related papers: Folding, Tiling, and Multidimensional Coding
Consider an $s$-dimensional function being evaluated at $n$ points of a low discrepancy sequence (LDS), where the objective is to approximate the one-dimensional functions that result from integrating out $(s-1)$ variables. Here, the…
The $k$-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative $k$-dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding,…
We present design and implementation of a chain of particles that can be programmed to fold the chain into a given curve. The particles guide an external force to fold, therefore the particles are simple and amenable for miniaturization. A…
In real-world, many problems can be formulated as the alignment between two geometric patterns. Previously, a great amount of research focus on the alignment of 2D or 3D patterns, especially in the field of computer vision. Recently, the…
This paper introduces a new systematic algorithm for constructing periodic Euclidean weaving diagrams with combinatorial arguments. It is shown that such a weaving diagram can be considered as a specific type of four-regular periodic planar…
In this paper, we shed light on how an adaptive, efficient error coding in the transport layer helps ensure the application requirements. We recap the use of MDS codes and show that binary coding can significantly reduce the complexity and…
We introduce STRING: Separable Translationally Invariant Position Encodings. STRING extends Rotary Position Encodings, a recently proposed and widely used algorithm in large language models, via a unifying theoretical framework.…
A low-complexity coding scheme is developed to achieve the rate region of maximum likelihood decoding for interference channels. As in the classical rate-splitting multiple access scheme by Grant, Rimoldi, Urbanke, and Whiting, the proposed…
In language processing, transformers benefit greatly from text being condensed. This is achieved through a larger vocabulary that captures word fragments instead of plain characters. This is often done with Byte Pair Encoding. In the…
In nature the three-dimensional structure of a protein is encoded in the corresponding gene. In this paper we describe a new method for encoding the three-dimensional structure of a protein into a binary sequence. The feature of the method…
We introduce the new concept of computation coding. Similar to how rate-distortion theory is concerned with the lossy compression of data, computation coding deals with the lossy computation of functions. Particularizing to linear…
A new family of codes, called clustering-correcting codes, is presented in this paper. This family of codes is motivated by the special structure of data that is stored in DNA-based storage systems. The data stored in these systems has the…
We discuss the method of folding for discrete planar systems and use it to establish the existence or non-existence of cycles or chaos in planar systems of rational difference equations with variable coefficients. These include some systems…
Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set…
Inspired by protein folding, we explored the construction of three-dimensional structures and machines from one-dimensional chains of simple building blocks. This approach not only allows us to recreate the self-replication mechanism…
We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and…
We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches…
Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and…
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…
A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing a structured codebook on each torus layer. The resulting spherical code can be the…