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A new effective decoding algorithm is presented for arbitrary algebraic-geometric codes on the basis of solving a generalized key equation with the majority coset scheme of Duursma. It is an improvement of Ehrhard's algorithm, since the…
General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding…
High-rate space-time block codes (STBC with code rate > 1) in multi-input multi-output (MIMO) systems are able to provide both spatial multiplexing gain and diversity gain, but have high maximum likelihood (ML) decoding complexity. Since…
Research on polar codes has been constantly gaining attention over the last decade, by academia and industry alike, thanks to their capacity-achieving error-correction performance and low-complexity decoding algorithms. Recently, they have…
Surface codes exploit topological protection to increase error resilience in quantum computing devices and can in principle be implemented in existing hardware. They are one of the most promising candidates for active error correction, not…
This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x in R^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably…
Commonly-used clustering algorithms usually find ellipsoidal, spherical or other regular-structured clusters, but are more challenged when the underlying groups lack formal structure or definition. Syncytial clustering is the name that we…
We show that every correctable subsystem for an arbitrary noise operation can be recovered by a unitary operation, where the notion of recovery is more relaxed than the notion of correction insofar as it does not protect the subsystem from…
We present a framework that can exploit the tradeoff between the undetected error rate (UER) and block error rate (BLER) of polar-like codes. It is compatible with all successive cancellation (SC)-based decoding methods and relies on a…
Generalized concatenated codes were introduced in the 1970s by Zinoviev. There are many types of codes in the literature that are known by other names that can be viewed as generalized concatenated codes. Examples include matrix-product…
We introduce group crosscoders, an extension of crosscoders that systematically discover and analyse symmetrical features in neural networks. While neural networks often develop equivariant representations without explicit architectural…
We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message…
The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of…
We propose several improvements for Linear Programming (LP) decoding algorithms for High Density Parity Check (HDPC) codes. First, we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts…
A reduced complexity sequential decoding algorithm for polar (sub)codes is described. The proposed approach relies on a decomposition of the polar (sub)code being decoded into a number of outer codes, and on-demand construction of codewords…
Let $X=\C^n$. In this paper we present an algorithm that computes the de Rham cohomology groups $H^i_{dR}(U,\C)$ where $U$ is the complement of an arbitrary Zariski-closed set $Y$ in $X$. Our algorithm is a merger of the algorithm given by…
An essential goal in mechanistic interpretability to decode a network, i.e., to convert a neural network's raw weights to an interpretable algorithm. Given the difficulty of the decoding problem, progress has been made to understand the…
We describe an algorithm to decompose rational functions from which we determine the poset of groups fixing these functions.
In this paper, we introduce a novel class of pre-transformed polar codes, termed as deep polar codes. We first present a deep polar encoder that harnesses a series of multi-layered polar transformations with varying sizes. Our approach to…
Fracton topological phases have a large number of materialized symmetries that enforce a rigid structure on their excitations. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fracton phase enable us to…