Related papers: On generalized covering radii of binary primitive …
We prove the lower bound R(M_m) \geq 3/2 m^2 - 2 on the border rank of m x m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of the geometric complexity theory (GCT) program. While…
Kernel Regularized Least Squares (KRLS) is a popular method for flexibly estimating models that may have complex relationships between variables. However, its usefulness to many researchers is limited for two reasons. First, existing…
We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. A la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem.…
We study certain combinatorial aspects of list-decoding, motivated by the exponential gap between the known upper bound (of $O(1/\gamma)$) and lower bound (of $\Omega_p(\log (1/\gamma))$) for the list-size needed to decode up to radius $p$…
The paper has a threefold purpose. The first purpose is to present an explicit description of expanded cyclic codes defined in $\GF(q^m)$. The proposed explicit construction of expanded generator matrix and expanded parity check matrix…
We generalize to any q a theorem about covering radius of linear codes proved by Helleseth, Klove and Mykkelvit. Then we determine the covering radius of first order generalized Reed-Muller codes in second order generalized Reed-Muller…
The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an…
In this paper, in order to characterize the critical error linear complexity spectrum (CELCS) for $2^n$-periodic binary sequences, we first propose a decomposition based on the cube theory. Based on the proposed $k$-error cube…
We study subfield-subcodes of Generalized Toric (GT) codes over $\mathbb{F}_{p^s}$. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes. We identify polynomial…
BCH codes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of BCH cyclic codes has been made, but little is known about the minimum distances of the…
In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new…
The linear complexity and the $k$-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By studying the linear complexity of binary…
In this paper, we explain a new Iterative Method-Fixed Point and develop its convergence theory for finding approximate solutions of nonlinear equations in the setting of Banach spaces. First, we discuss the convergence analysis of our…
Relax, Compensate and then Recover (RCR) is a paradigm for approximate inference in probabilistic graphical models that has previously provided theoretical and practical insights on iterative belief propagation and some of its…
In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However,…
One approach to testing general relativity (GR) introduces free parameters in the post-Newtonian (PN) expansion of the gravitational-wave (GW) phase. If systematic errors on these testing GR (TGR) parameters exceed the statistical errors,…
In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have a size $M=2^k$. For every integer $\ell \geq 3$, we construct a $(n=2^\ell, M=2^{k_{\ell}},…
BCH codes are an important class of linear codes and find extensive utilization in communication and disk storage systems.This paper mainly analyzes the negacyclic BCH code and cyclic BCH code of length $\frac{q^m-1}{2}$. For negacyclic BCH…
We compare the performance of short-length linear binary codes on the binary erasure channel and the binary-input Gaussian channel. We use a universal decoder that can decode any linear binary block code: Gaussian-elimination based…
Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close…