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| Plotkin bound | |
|---|---|
| Name | Plotkin bound |
| Field | Coding theory |
| Devised by | Marshall Plotkin |
| Introduced | 1960s |
| Related | Hamming bound, Singleton bound, Elias bound, Gilbert–Varshamov bound, Johnson bound |
Plotkin bound is an upper bound in information theory and coding theory that limits the maximum size of a code with given length and minimum distance over a finite alphabet. The bound, attributed to Marshall Plotkin, plays a central role in the study of error‑correcting codes and interacts with classical results such as the Hamming bound, Singleton bound, and Elias bound. It is especially powerful for codes with relative distance exceeding certain thresholds and informs constructions investigated by researchers at institutions like Bell Labs, Princeton University, and Massachusetts Institute of Technology.
The Plotkin bound emerged in the context of mid‑20th century advances in digital communication and combinatorial design studied by figures associated with Shannon's legacy at Bell Labs and academic centers such as Harvard University and Stanford University. It relates code length, alphabet size, minimum Hamming distance, and code cardinality, and is often presented alongside bounds due to Hamming, Singleton, Gilbert–Varshamov, and Elias. The bound has been cited in literature connected to projects at AT&T, IBM Research, and conferences like IEEE International Symposium on Information Theory.
Let q denote the alphabet size associated with Reed–Solomon codes and other q‑ary constructions, let n be the block length used in channel coding problems studied by Claude Shannon, and let d be the minimum Hamming distance as in classical treatments by Richard Hamming. For a q‑ary code C of length n and minimum distance d, if d > n(1 - 1/q), the Plotkin bound provides an absolute upper bound on |C|, the code cardinality. In the binary case q = 2, for example, when d > n/2 the bound implies that |C| ≤ 2⌊d/(2d - n)⌋, a statement that complements analyses appearing in works by Binary Golay code researchers and in texts from Cambridge University Press and Springer.
Standard proofs of the Plotkin bound use averaging arguments, pairwise distance summation, and linear programming techniques encountered in proofs by authors affiliated with University of California, Berkeley and University of Illinois Urbana–Champaign. One common derivation begins by summing Hamming distances over all ordered pairs of distinct codewords and compares this sum to a trivial upper bound obtained from coordinate‑wise analysis; the contradiction yields the stated cardinality limit. Alternative derivations employ symmetrization and sphere‑packing ideas related to methods used in analyses at Princeton University and in dissertations supervised by faculty at Massachusetts Institute of Technology.
The Plotkin bound yields immediate corollaries about asymptotic tradeoffs between rate and relative distance that parallel results credited to Elias and exploited in the study of asymptotic bounds recorded at École Polytechnique Fédérale de Lausanne. It implies nonexistence results for large codes beyond certain relative distance thresholds, informs upper bounds on packing density used in research at Max Planck Institute for Informatics, and combines with the Johnson bound to sharpen limitations in constant‑weight code settings investigated at Bell Labs and AT&T Laboratories.
Practitioners use the Plotkin bound to assess feasibility of code families such as BCH codes, Reed–Solomon codes, and Bose–Chaudhuri–Hocquenghem codes when designing systems for standards developed by IEEE, ITU, and 3GPP. It guides selection of parameters in concatenated code schemes studied by scholars at California Institute of Technology and in concatenation frameworks popularized by researchers connected to Nokia and Qualcomm. The bound also underpins impossibility proofs used in algorithmic complexity settings at Massachusetts Institute of Technology and in cryptographic protocol analyses at Bell Labs.
Several generalizations of the Plotkin bound extend to constant‑weight codes, list‑decoding regimes, and codes over nonprime power alphabets; these have been developed in literature by teams at University of Cambridge, ETH Zurich, and University of Toronto. Related classical bounds include the Hamming bound, Singleton bound, Elias bound, and Gilbert–Varshamov bound, while modern relaxations and refinements appear in linear programming bounds credited to researchers associated with Princeton University and Courant Institute.
Tightness of the Plotkin bound is witnessed in small parameter regimes and in constructions derived from Hadamard matrices and extremal combinatorial designs studied at Institute for Advanced Study and Carnegie Mellon University. For binary codes with d > n/2, examples based on equidistant sets and constructions related to Reed–Muller codes demonstrate near‑achievement of the bound; other tight cases link to combinatorial objects analyzed at University of Oxford and Yale University. In many parameter regions the bound is not tight, and comparisons with the Gilbert–Varshamov bound and Elias bound identify gaps that remain active research topics at ETH Zurich and Stanford University.