## 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

## 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

- Combinatorial design code
- One-hot code
- Weight-two code
- One-versus-one (OVO) code
- Simplex code — All non-zero simplex codewords have a constant weight of \(q^{k-1}\).
- Balanced code

## Cousins

- Constant-excitation (CE) code — Constant-weight codes are classical analogues of qubit constant-excitation codes.
- Universally optimal \(q\)-ary code — See [4; Table 8.4] for constant-weight universally optimal \(q\)-ary codes.
- Divisible code — Codes whose codewords have a constant weight of \(m\) are automatically \(m\)-divisible. However, divisible codes are linear by definition while constant-weight codes do not have to be.
- Binary balanced spherical code — Binary balanced spherical codes are obtained from constant-weight binary codes.

## 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.

## Page edit log

- Victor V. Albert (2022-07-14) — most recent
- Micah Shaw (2022-05-30)

## Cite as:

“Constant-weight code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/constant_weight