Constant-weight code 

Description

A block code over a field or a ring 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 \(J(n,w)\) [1–4].

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

Protection

See Ref. [5] for upper bounds on \(K\) for \((n,K,d)_q\) constant-weight binary codes.

Realizations

Radio-network frequency hopping [6].

Notes

Tables of binary constant-weight codes for \(n \leq 28\) [7] and \(n > 28\) [6]. Other code constructions also exist for codes over fields [8] or rings [9].See book [10] for (Johnson) bounds on the size of constant-weight codes.

Parent

Children

Cousins

References

[1]
Delsarte, Philippe. "An algebraic approach to the association schemes of coding theory." Philips Res. Rep. Suppl. 10 (1973): vi+-97.
[2]
P. Delsarte, “Association schemes and t-designs in regular semilattices”, Journal of Combinatorial Theory, Series A 20, 230 (1976) DOI
[3]
Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
[4]
V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
[5]
K. Zeger, A. Vardy, and E. Agrell, “Upper bounds for constant-weight codes”, IEEE Transactions on Information Theory 46, 2373 (2000) DOI
[6]
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
[7]
A. E. Brouwer et al., “A new table of constant weight codes”, IEEE Transactions on Information Theory 36, 1334 (1990) DOI
[8]
Fang-Wei Fu, A. J. Han Vinck, and Shi-Yi Shen, “On the constructions of constant-weight codes”, IEEE Transactions on Information Theory 44, 328 (1998) DOI
[9]
T. Etzion, “Optimal constant weight codes over Zk and generalized designs”, Discrete Mathematics 169, 55 (1997) DOI
[10]
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

Cite as:
“Constant-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/constant_weight
BibTeX:
@incollection{eczoo_constant_weight,
  title={Constant-weight code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/constant_weight}
}
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“Constant-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/constant_weight

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/block/constant_weight.yml.