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| locally decodable codes | |
|---|---|
| Name | Locally decodable codes |
| Field | Information theory, Theoretical computer science |
| Introduced | 1990s |
| Notable | Madhu Sudan, Venkatesan Guruswami, Johan Hastad, Alexander Vardy |
locally decodable codes
Locally decodable codes are error-correcting codes that allow recovery of individual message symbols by probing a small number of coordinates of a possibly corrupted codeword. Originating from work in the 1990s and developed through interactions among complexity theory and information theory, these codes connect to foundational results by researchers associated with institutions such as Massachusetts Institute of Technology, Microsoft Research, Princeton University, University of California, Berkeley, and awards like the Gödel Prize and the Nevalinna Prize. They underpin cryptographic protocols and sublinear algorithms explored in venues such as STOC and FOCS.
Locally decodable codes (LDCs) were introduced to bridge error correction with sublinear-time decoding and probabilistically checkable proofs connected to lines of work by researchers at Harvard University, Carnegie Mellon University, Columbia University, and University of Chicago. Early constructions built on ideas from list decoding and algebraic geometry codes developed by teams at Bell Labs, AT&T Labs, and laboratories affiliated with IBM Research. LDCs influenced subsequent developments in distributed storage systems used by companies like Google and Amazon (company).
Formally, an LDC is a code C: Σ^k → Γ^n with a decoding algorithm that, given index i ∈ [k] and access to a received word y ∈ Γ^n with up to δn errors, makes q queries to y and outputs the i-th message symbol with probability at least 1−ε. This model was formalized in papers by researchers at Stanford University, Cornell University, and Yale University and relates to notions studied in the literature on probabilistically checkable proofs from groups at Rutgers University and University of California, San Diego. Variants include local list-decoding and smooth decoders introduced in collaborations involving scholars at University of Warwick and McGill University.
Classic constructions of LDCs use algebraic techniques such as Reed–Muller and Reed–Solomon codes developed historically by researchers affiliated with Bell Labs and institutions like University of Illinois Urbana-Champaign. The Hadamard code, connected to work by Alonzo Church-era foundations and modern expositions at California Institute of Technology, gives a 2-query LDC with exponential length. Constructions with subexponential length use concatenation frameworks inspired by coding theory advances at École Polytechnique Fédérale de Lausanne and University of Toronto and combinatorial designs related to research at Princeton University. More advanced algebraic-geometric constructions leverage tools from studies at Institut des Hautes Études Scientifiques and University of Cambridge.
Key parameters include block length n, message length k, query complexity q, error tolerance δ, and success probability 1−ε. Trade-offs among these parameters were explored in influential work by researchers at Massachusetts Institute of Technology and Harvard University and presented at conferences like ICALP and SODA. Lower query complexity often forces larger block length; for instance, 2-query LDCs require exponential n in k as shown in results from groups at Princeton University and Rutgers University, whereas higher-query constructions from teams at University of California, Los Angeles yield shorter codes.
LDCs have applications to private information retrieval protocols studied at Bell Labs and Microsoft Research, to fault-tolerant distributed storage systems deployed by Google and Facebook, and to complexity-theoretic separations investigated at ETH Zurich and University of Pennsylvania. They are used in sublinear algorithms and streaming contexts considered at IBM Research and in cryptographic primitives developed by researchers at RSA Security and universities such as Stanford University.
Fundamental lower bounds for LDCs include exponential-length results for constant-query decoders proved using techniques from quantum information theory and combinatorics by researchers at Perimeter Institute and University of Waterloo. Results showing limitations of 2-query LDCs and trade-offs for linear codes were established in collaborative work involving groups at University of Maryland and Technion – Israel Institute of Technology. Hardness results connect to communication complexity lower bounds whose history traces to work at AT&T Labs and Bell Labs.
LDCs relate to locally testable codes explored by researchers at Princeton University and University of California, Berkeley, to list-decodable codes connected to research at Columbia University and Rutgers University, and to multiplicity codes developed at institutions including Microsoft Research and EPFL. They also interface with pseudorandomness and extractor constructions investigated at Institute for Advanced Study and Weizmann Institute of Science, and with quantum error-correcting codes studied at Perimeter Institute and Caltech.