Advances in Error-Correcting Codes

The field of error-correcting codes is witnessing significant developments, with a focus on improving decoding algorithms and understanding the properties of various code families. Researchers are exploring new metrics, such as the Lee and Euclidean metrics, to measure errors in codes, and developing polynomial-time algorithms for list decoding in these metrics. Additionally, there is a growing interest in understanding the local properties of codes, such as list decodability and list recoverability, and in constructing explicit codes that achieve optimal thresholds. Notably, recent works have established novel combinatorial bounds on list recoverability and have developed new techniques, such as discrete Brascamp-Lieb inequalities, to analyze the performance of codes. Furthermore, researchers are investigating the asymptotic properties of linear codes, including the number of equivalence classes, and are exploring connections to other areas, such as matroid theory. Some noteworthy papers in this area include: List Decoding Reed-Solomon Codes in the Lee, Euclidean, and Other Metrics, which gives a polynomial-time algorithm for list decoding Reed-Solomon codes in various metrics. Combinatorial Bounds for List Recovery via Discrete Brascamp-Lieb Inequalities, which establishes novel bounds on list recoverability for various code families. From Random to Explicit via Subspace Designs With Applications to Local Properties and Matroids, which develops a framework for studying local properties of subspace designable codes and applies it to matroid theory.

Sources

List Decoding Reed--Solomon Codes in the Lee, Euclidean, and Other Metrics

Combinatorial Bounds for List Recovery via Discrete Brascamp--Lieb Inequalities

From Random to Explicit via Subspace Designs With Applications to Local Properties and Matroids

Decoding Balanced Linear Codes With Preprocessing

The asymptotic number of equivalence classes of linear codes with given dimension

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