# Divisible code[1]

## Description

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, \(8\)-divisible) code is called even (doubly-even, triply-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. More generally, a code is \(m\)-even if it is \(2^{m}\)-divisible.

## Notes

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

## Parent

## Children

- \([48,24,12]\) self-dual code — The \([48,24,12]\) self-dual code is the only self-dual doubly-even code at its parameters [5].
- Single parity-check (SPC) code — Binary SPCs are two-divisible.
- Reed-Muller (RM) code — An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece's theorem [6,7].

## Cousins

- 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 [8; Def. 4.1.6]. The minimum distance of doubly-even binary self-dual codes asymptotically satisfies \(d\leq0.1664n+o(n)\) [9].
- Ternary Golay code — Extended ternary Golay code is 3-divisible ([10], 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 [11].
- Two-weight code — Two-weight codes are \(m\)-divisible, where \(m\) is the greatest common factor of their two possible weights.
- Quantum divisible code — The \(X\)-type stabilizers of a level-\(\nu\) quantum divisible code form a \(\nu\)-even linear binary code.
- \([[49,1,5]]\) triorthogonal code — The \([[49,1,5]]\) triorthogonal code stabilizer generator matrix can be obtained from a triply-even linear binary code [12; Appx. B].

## References

- [1]
- H. N. Ward, “Divisible codes”, Archiv der Mathematik 36, 485 (1981) DOI
- [2]
- S. Kurz, “Divisible Codes”, (2023) arXiv:2112.11763
- [3]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [4]
- K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
- [5]
- S. K. Houghten et al., “The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code”, IEEE Transactions on Information Theory 49, 53 (2003) DOI
- [6]
- R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
- [7]
- R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
- [8]
- S. Bouyuklieva, "Self-dual codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [9]
- I. Krasikov and S. Litsyn, “Linear programming bounds for doubly-even self-dual codes”, IEEE Transactions on Information Theory 43, 1238 (1997) DOI
- [10]
- J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
- [11]
- H. N. Ward, “Divisibility of Codes Meeting the Griesmer Bound”, Journal of Combinatorial Theory, Series A 83, 79 (1998) DOI
- [12]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI

## Page edit log

- Victor V. Albert (2022-07-14) — most recent

## Cite as:

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