Divisible code[1]

Description

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

Parent

Cousins

  • Quantum divisible code — Quantum divisible codes are constructed via the CSS construction using a divisible linear binary code.
  • Reed-Muller (RM) code — An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece's theorem [3][4].

Zoo code information

Internal code ID: divisible

Your contribution is welcome!

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Zoo Code ID: divisible

Cite as:
“Divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/divisible
BibTeX:
@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} }
Permanent link:
https://errorcorrectionzoo.org/c/divisible

References

[1]
H. N. Ward, “Divisible codes”, Archiv der Mathematik 36, 485 (1981). DOI
[2]
Sascha Kurz, “Divisible Codes”. 2112.11763
[3]
R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971). DOI
[4]
R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972). DOI

Cite as:

“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/divisible.yml.