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
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