Handbook of Error-Correcting Codes

2026-06-09

AI Summary

This work systematically examines the structures of classical and quantum error-correcting codes in storage and communication, elucidating their deep connections to mathematical and physical objects such as sphere packings, lattices, combinatorial designs, group theory, and quantum phases of matter. The project pioneers a handbook-style integration of the error-correction knowledge base, employing a taxonomic framework that unifies information theory, algebraic coding theory, and interdisciplinary mapping techniques. Codes are organized structurally according to symbol types, enabling coherent classification and cross-referencing. The resulting resource not only serves as a rigorous and comprehensive reference but also empowers researchers to trace interrelationships among codes and inspire novel discoveries, thereby addressing a critical gap in systematic synthesis and cross-domain linkage within the field.

Abstract

Barcode scans, clear phone calls, reliable data storage, satellite communication, and large-scale quantum computation are all made possible by error correction. We present a handbook version of The Error Correction Zoo, a curated reference of methods for protecting classical or quantum information from errors during storage and transmission. The handbook includes descriptions of these error-correcting codes and a classification according to the symbols they use. It also catalogues relations among codes and related objects such as sphere packings, lattices, designs, groups, and classical and quantum phases of matter. The collection is intended both as a rigorous reference and as a practical aid for tracing the web of code relationships and uncovering new connections.

Problem

• error-correcting codes
• quantum information
• classical information
• code classification
• code relations

Innovation

• error-correcting codes
• quantum information
• code classification
• sphere packings
• lattices