Cyclic triorthogonal codes in prime dimension
Abstract
We study cyclic triorthogonal codes in prime dimension p, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd p, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to p, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for p=3 up to length $121$ and p=5,7 up to length $96$. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$. For qudits, we find codes such as [[19,1,3]]3 and [[7,1,3]]7, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of [[p,1,2⌊ p/6⌋+1]]p codes.
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BibTeX
@misc{prakash2026cyclic,
title = {{Cyclic triorthogonal codes in prime dimension}},
author = {Shiroman Prakash},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08012},
eprint = {2610.08012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08012},
}
RIS
TY - UNPB
TI - Cyclic triorthogonal codes in prime dimension
AU - Shiroman Prakash
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08012
UR - https://arxiv.org/abs/2610.08012
AB - We study cyclic triorthogonal codes in prime dimension p, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd p, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to p, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for p=3 up to length $121$ and p=5,7 up to length $96$. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$. For qudits, we find codes such as [[19,1,3]]_3 and [[7,1,3]]_7, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of [[p,1,2⌊ p/6⌋+1]]_p codes.
ER -