Short qudit triorthogonal codes from doubling
Abstract
We construct short triorthogonal codes for qudits of odd prime dimension by generalizing doubling, which combines a triorthogonal code with a code admitting a transversal phase gate, and raises the distance by up to two. This gives codes with parameters [[28,1,4]]3, [[37,1,5]]3, [[72,1,6]]3, [[118,1,8]]3, [[33,1,5]]5 and [[20,1,4]]7, each shorter than the cyclic, concatenated and Reed-Muller codes of the same distance, and a doubling construction of the [[15,1,3]]3 code. When the columns of a generator matrix are regarded as charged points, triorthogonality becomes the vanishing of multipole moments up to octupole order, and doubling becomes a double layer of opposite charges; this picture explains why inputs with nontrivial coefficients can give shorter codes. We also show by exhaustive search that the shortest codes of distance three have length $11$ for p=5 and length $7$ for p=7. Combined with high-rate distance-two codes, the new codes reduce the cost of qutrit magic state distillation by up to an order of magnitude in regimes with high input noise.
Related
- Asymptotically good binary triorthogonal codes and higher-level transversal gates
- On Constructing and Decoding Quantum Triorthogonal Codes
- Cyclic triorthogonal codes in prime dimension
- Classification of Generalised Triorthogonal Codes through Length 54
- Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q)PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance
BibTeX
@misc{prakash2026short,
title = {{Short qudit triorthogonal codes from doubling}},
author = {Shiroman Prakash and K. L. Ramya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29457},
eprint = {2610.29457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29457},
}
RIS
TY - UNPB
TI - Short qudit triorthogonal codes from doubling
AU - Shiroman Prakash
AU - K. L. Ramya
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29457
UR - https://arxiv.org/abs/2610.29457
AB - We construct short triorthogonal codes for qudits of odd prime dimension by generalizing doubling, which combines a triorthogonal code with a code admitting a transversal phase gate, and raises the distance by up to two. This gives codes with parameters [[28,1,4]]_3, [[37,1,5]]_3, [[72,1,6]]_3, [[118,1,8]]_3, [[33,1,5]]_5 and [[20,1,4]]_7, each shorter than the cyclic, concatenated and Reed-Muller codes of the same distance, and a doubling construction of the [[15,1,3]]_3 code. When the columns of a generator matrix are regarded as charged points, triorthogonality becomes the vanishing of multipole moments up to octupole order, and doubling becomes a double layer of opposite charges; this picture explains why inputs with nontrivial coefficients can give shorter codes. We also show by exhaustive search that the shortest codes of distance three have length $11$ for p=5 and length $7$ for p=7. Combined with high-rate distance-two codes, the new codes reduce the cost of qutrit magic state distillation by up to an order of magnitude in regimes with high input noise.
ER -