\([15,6,5]\) Petersen cycle code

\([15,6,5]\) Petersen cycle code[1]

Description

A \([15,6,5]\) cycle code whose parity-check matrix is the incidence matrix of the Petersen graph. The Petersen graph can be thought of as a dodecahedron with antipodes identified [2; Appx. A.2.1].

Gates

In the inner/outer-code framework for magic-state distillation, it is the smallest outer code with weight-three checks that is \((4,2)\)-sensitive, giving a fifth-order protocol on 15 magic states [2; Appx. A.2.1].

Notes

See [3; Example 10.11.7] for more details.

Cousins

Dodecahedron code— The Petersen graph can be thought of as a dodecahedron with antipodes identified [2; Appx. A.2.1].
Petersen spherical code— The Petersen spherical code is a spherical realization associated with the Petersen cycle code.

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC

Cycle codeProjective geometry Linear \(q\)-ary ECC

Parents

Cycle code

Small-distance block codeECC

\([15,6,5]\) Petersen cycle code

References

[1]: S. Hakimi and J. Bredeson, “Graph theoretic error-correcting codes”, IEEE Transactions on Information Theory 14, 584 (1968) DOI
[2]: J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[3]: D. Jungnickel, Graphs, Networks and Algorithms (Springer Berlin Heidelberg, 2008) DOI

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-03-16)

Cite as:

“\([15,6,5]\) Petersen cycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/petersen, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/graph/incidence/petersen.yml.