Divisible code[1] 


A linear \(q\)-ary block code is \(\Delta\)-divisible if the Hamming weight of each of its codewords is divisible by divisor \(\Delta\). A \(2\)-divisible (\(4\)-divisible) code is called even (doubly-even) [2,3]. A code is called singly-even if all codewords are even and at least one has weight equal to 2 modulo 4.


See Ref. [4] for an introduction to triply-even binary linear codes and their construction from doubly-even codes.




  • Binary quadratic-residue (QR) code — Extended binary quadratic residue codes of length \(8m\) are self-dual doubly-even codes [3; pg. 82].
  • \([8,4,4]\) extended Hamming code — The extended Hamming code code is the smallest double-even self-dual code.
  • Constant-weight 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.
  • Self-dual linear code — Binary self-dual codes are singly-even and binary self-orthogonal codes that are not doubly-even are singly-even [7; Def. 4.1.6]. The minimum distance of doubly-even binary self-dual codes asymptotically satisfies \(d\leq0.1664n+o(n)\) [8].
  • Ternary Golay code — Extended ternary Golay code is 3-divisible ([9], pg. 296).
  • Griesmer code — If a \(p\)-ary Griesmer code with \(p\) prime is such that a power of \(p\) divides the distance, then the code is divisible by that power [10].
  • Two-weight code — Two-weight codes are \(m\)-divisible, where \(m\) is the greatest common factor of their two possible weights.
  • Quantum divisible code — Quantum divisible codes are constructed via the CSS construction using a divisible linear binary code.
  • Doubled color code — Doubled color codes are constructed using a generalization of the doubling transformation [4] that combine doubly-even codes to make triply-even codes.


H. N. Ward, “Divisible codes”, Archiv der Mathematik 36, 485 (1981) DOI
S. Kurz, “Divisible Codes”, (2022) arXiv:2112.11763
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
S. Bouyuklieva, "Self-dual codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
I. Krasikov and S. Litsyn, “Linear programming bounds for doubly-even self-dual codes”, IEEE Transactions on Information Theory 43, 1238 (1997) DOI
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
H. N. Ward, “Divisibility of Codes Meeting the Griesmer Bound”, Journal of Combinatorial Theory, Series A 83, 79 (1998) DOI
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Zoo Code ID: divisible

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“Divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/divisible
@incollection{eczoo_divisible, title={Divisible code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/divisible} }
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“Divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/divisible

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/weight/divisible.yml.