Constant-weight block code[1]
Description
A block code whose codewords all have the same number of nonzero coordinates. Code constructions exist for codes over fields [2] or rings [3].
The set of all weight-\(w\) \(q\)-ary strings of length \(n\) forms the nonbinary Johnson space, a finite symmetric space \(G/H\) with \(G = S_{q-1} \wr S_n\) [1][4; Sec. 8.8][5; Table 3]. The number of such strings is \({n \choose w} (q-1)^w\).
Cousins
- Constant-excitation (CE) code— Constant-weight codes are classical analogues of qubit constant-excitation codes.
- Linear code over \(\mathbb{Z}_q\)— Constant-weight linear codes over \(\mathbb{Z}_q\) have been classified [3].
- \(q\)-ary code over \(\mathbb{Z}_q\)— Optimal constant-weight codes over \(\mathbb{Z}_q\) can be constructed [6] from a generalization of combinatorial designs to \(q\)-ary alphabets [7,8].
- Combinatorial design— Optimal constant-weight codes over \(\mathbb{Z}_q\) can be constructed [6] from a generalization of combinatorial designs to \(q\)-ary alphabets [7,8].
- \(q\)-ary simplex code— All nonzero simplex codewords have a constant weight of \(q^{m-1}\).
Primary Hierarchy
Parents
The set of all weight-\(w\) \(q\)-ary strings of length \(n\) forms the nonbinary Johnson space, a finite symmetric space \(G/H\) with \(G = S_{q-1} \wr S_n\) [1][4; Sec. 8.8][5; Table 3]. The number of such strings is \({n \choose w} (q-1)^w\). This reduces to the Johnson space for \(q=2\).
Constant-weight block code
Children
The set of all weight-\(w\) binary strings of length \(n\) forms the Johnson space \(J(n,w)\), a finite two-point homogeneous space \(G/H\) with \(G = S_n\) and \(H = S_w \times S_{n-q}\) [9–13][5; Table 2]. This is a special case of the nonbinary Johnson space for \(q=2\).
References
- [1]
- H. Tarnanen, M. J. Aaltonen, and J.-M. Goethals, “On the Nonbinary Johnson Scheme”, European Journal of Combinatorics 6, 279 (1985) DOI
- [2]
- 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
- [3]
- J. A. Wood, “The structure of linear codes of constant weight”, Transactions of the American Mathematical Society 354, 1007 (2001) DOI
- [4]
- T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli, Harmonic Analysis on Finite Groups (Cambridge University Press, 2008) DOI
- [5]
- C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin, “Invariant Semidefinite Programs”, International Series in Operations Research & Management Science 219 (2011) arXiv:1007.2905 DOI
- [6]
- T. Etzion, “Optimal constant weight codes over Zk and generalized designs”, Discrete Mathematics 169, 55 (1997) DOI
- [7]
- H. Hanani, “On Some Tactical Configurations”, Canadian Journal of Mathematics 15, 702 (1963) DOI
- [8]
- S.-T. Xia and F.-W. Fu, “Undetected error probability of q-ary constant weight codes”, Designs, Codes and Cryptography 48, 125 (2007) DOI
- [9]
- Delsarte, Philippe. “An algebraic approach to the association schemes of coding theory.” Philips Res. Rep. Suppl. 10 (1973): vi+-97.
- [10]
- P. Delsarte, “Association schemes and t-designs in regular semilattices”, Journal of Combinatorial Theory, Series A 20, 230 (1976) DOI
- [11]
- Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
- [12]
- V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
- [13]
- C. Bachoc, “Semidefinite programming, harmonic analysis and coding theory”, (2010) arXiv:0909.4767
Page edit log
- Victor V. Albert (2025-10-28) — most recent
Cite as:
“Constant-weight block code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/q-ary_constant_weight