Related papers: Generic Decoding in the Sum-Rank Metric
We study Sigma-Delta quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the…
Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and…
In the constraint programming framework, state-of-the-art static and dynamic decomposition techniques are hard to apply to problems with complete initial constraint graphs. For such problems, we propose a hybrid approach of these techniques…
Quantum error correction (QEC) enables reliable computation on noisy hardware by encoding logical information across many physical qubits and periodically measuring parities to detect errors. A decoder is the classical algorithm that uses…
Representing images by compact codes has proven beneficial for many visual recognition tasks. Most existing techniques, however, perform this coding step directly in image feature space, where the distributions of the different classes are…
Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However,…
We describe a novel extension of subspace codes for noncoherent networks, suitable for use when the network is viewed as a communication system that introduces both dimension and symbol errors. We show that when symbol erasures occur in a…
Jamming in hard-particle packings has been the subject of considerable interest in recent years. In a paper by Torquato and Stillinger [J. Phys. Chem. B, 105 (2001)], a classification scheme of jammed packings into hierarchical categories…
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on…
Projective metrics on vector spaces over finite fields, introduced by Gabidulin and Simonis in 1997, generalize classical metrics in coding theory like the Hamming metric, rank metric, and combinatorial metrics. While these specific metrics…
This paper presents a novel algorithm for constructing a sum-of-squares (SOS) decomposition for positive semi-definite polynomials with rational coefficients. Unlike previous methods that typically yield SOS decompositions with…
We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank…
We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding…
In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed-Solomon codes. Whereas Reed-Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless,…
Most of the calculations in standard sphere decoders are redundant, in the sense that they either calculate quantities that are never used or calculate some quantities more than once. A new method, which is applicable to lattices as well as…
Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes -- the generalized covering radius. The generalized covering-radius hierarchy of a…
The aim of this work is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we give degree formulas for the generalized Hamming weights of…
Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank…
We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC…
We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat…