Constant-weight code


A linear \(q\)-ary block code whose codewords all have the same Hamming weight \(w\). The complement of a binary constant-weight code is a constant-weight code obtained by interchanging zeroes and ones in the codewords. The set of all binary codewords of length \(n\) forms the Johnson space \(J(n,w)\) [1][2][3][4].

Constant-weight codes that contain all strings of some fixed Hamming weight are known as \(m\)-in-\(n\) or \({n \choose m}\) codes.


Radio-network frequency hopping [5].


Tables of binary constant-weight codes for \(n \leq 28\) [6] and \(n > 28\) [5].See book [7] for (Johnson) bounds on the size of constant-weight codes.


  • Divisible code — Codes whose codewords have a constant weight of \(m\) are automatically \(m\)-divisible.




Delsarte, Philippe. "An algebraic approach to the association schemes of coding theory." Philips Res. Rep. Suppl. 10 (1973): vi+-97.
P. Delsarte, “Association schemes and t-designs in regular semilattices”, Journal of Combinatorial Theory, Series A 20, 230 (1976) DOI
Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
D. H. Smith, L. A. Hughes, and S. Perkins, “A New Table of Constant Weight Codes of Length Greater than 28”, The Electronic Journal of Combinatorics 13, (2006) DOI
A. E. Brouwer et al., “A new table of constant weight codes”, IEEE Transactions on Information Theory 36, 1334 (1990) DOI
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
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Zoo Code ID: constant_weight

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“Constant-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_constant_weight, title={Constant-weight code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Constant-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.