This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Johnson bound | |
|---|---|
| Name | Johnson bound |
| Field | Coding theory |
| Introduced | 1962 |
| Introduced by | R. C. Johnson |
| Related | Elias bound, Hamming bound, Plotkin bound, Gilbert–Varshamov bound |
Johnson bound
The Johnson bound is a combinatorial upper bound used in coding theory to limit the size of constant-weight or general error-correcting codes given length, distance, and weight constraints. It connects parameters studied in Richard Hamming's work on the Hamming bound, Vladimir Gilbert's contributions to the Gilbert–Varshamov bound, and results related to the Plotkin bound and Elias bound, and is instrumental in analyses tied to the Binary Symmetric Channel and list decoding questions explored by researchers at institutions like Bell Labs and universities such as Massachusetts Institute of Technology and Princeton University.
The Johnson bound provides an upper limit on the maximum cardinality of a code with given length and minimum distance, often specialized to constant-weight codes studied in the context of binary codes and combinatorial designs linked to work at the Institute for Advanced Study and in conferences like the IEEE International Symposium on Information Theory. It is frequently invoked alongside classical bounds developed by figures including Elias (computer scientist), Richard M. Rothschild (for combinatorial constructions), and methods from researchers affiliated with Bell Labs and the University of Illinois Urbana-Champaign. The bound informs analyses of list-decoding radii, combinatorial packing problems, and constructions related to orthogonal arrays and block designs used by groups centered at institutions such as Cambridge University and Stanford University.
For a binary code of length n and minimum Hamming distance d, let A(n,d) denote the maximum size of such a code as studied by authors at Princeton University and in texts by researchers from California Institute of Technology. The Johnson bound gives an upper bound on A(n,d) or on the size of constant-weight codes A_w(n,d) by relating n, d, and the weight w via an inequality originally articulated in works connected to R. C. Johnson and later refined in expositions from Harvard University and ETH Zurich. In one common form, for constant-weight codes of weight w, the bound asserts that if t is a nonnegative integer satisfying certain quadratic constraints in n, d, and w (as in the combinatorial treatments from University of Cambridge and Columbia University), then - A_w(n,d) ≤ floor( n / (n - t) ) · A_w(n - 1, d), with t determined by the interplay of w and d in a manner parallel to constraints used in proofs by researchers at Bell Labs and in monographs from Springer Science+Business Media.
Proofs of the Johnson bound employ combinatorial double-counting, sphere-packing type arguments, and projection or shortening techniques familiar from expositions at Princeton University and Massachusetts Institute of Technology. Derivations often use Johnson graphs analyzed by combinatorialists at University of Cambridge and spectral methods associated with results from Yale University and University of Chicago. Alternative proofs draw on linear programming bounds influenced by the work of Delsarte and later developments at Université Paris-Saclay and incorporate eigenvalue interlacing arguments used in studies at University of Oxford. Researchers from University of California, Berkeley and Rice University have presented streamlined combinatorial proofs suitable for constant-weight settings and for translations to list-decoding contexts.
The Johnson bound is applied to bound list sizes in list-decoding analyses developed by authors at Microsoft Research and in foundational papers from Bell Labs. It constrains possible sizes of constant-weight codes used in optical communications research at AT&T Bell Labs and in experiments at NASA laboratories. The bound is instrumental in evaluating performance limits for codes used in standards influenced by work at International Telecommunication Union and in bounding parameters in combinatorial designs employed by teams at Los Alamos National Laboratory and Sandia National Laboratories. It also guides algorithmic choices for decoding procedures researched at Carnegie Mellon University and in coding-theoretic cryptography studied at MIT Lincoln Laboratory.
Generalizations of the Johnson bound include versions for q-ary alphabets developed at institutions like École Polytechnique and refinements connecting to the Elias bound and the Plotkin bound researched at Bell Labs and Princeton University. The bound is related to linear programming bounds by Delsarte and to combinatorial inequalities used in treatments from University of Waterloo and Télécom Paris. Further extensions connect to list-decoding capacity results and to bounds arising in extremal set theory studied by researchers at University of Minnesota and Rutgers University.
Typical numerical illustrations compare A(n,d) estimates from the Johnson bound with values from the Hamming and Gilbert–Varshamov bounds, as in lecture notes circulated at Harvard University and Stanford University. For moderate n (e.g., n around 50–200), researchers at University of Illinois Urbana-Champaign and University College London compute Johnson-bound-based upper limits that often tighten Hamming-bound estimates, especially for constrained constant-weight codes used in experiments at Bell Labs and in simulations at Los Alamos National Laboratory. Tables used in textbooks from Springer Science+Business Media juxtapose Johnson bounds with empirical code sizes from implementations at Nokia research labs and academic coding groups.
The bound originated in work by R. C. Johnson and contemporaries interacting with coding theorists at Bell Labs and academic centers like Princeton University and Massachusetts Institute of Technology. Subsequent refinements and widespread exposition were contributed by researchers affiliated with Harvard University, Cambridge University, and ETH Zurich, and propagated through conferences including the IEEE International Symposium on Information Theory. Its role in list decoding and combinatorial design theory was elaborated by scholars from Microsoft Research, University of California, Berkeley, and Carnegie Mellon University.