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.  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 \(d\leq0.1664n+o(n)\) .
- Ternary Golay code — Extended ternary Golay code is 3-divisible (, 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 .
- 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  that combine doubly-even codes to make triply-even codes.
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- Victor V. Albert (2022-07-14) — most recent
“Divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/divisible