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Extremal permutations of ℤn with many good slopes and applications to quantum Latin squares
Abstract
For n≥2, a good slope of a permutation f of ℤn=ℤ/nℤ is an element m∈ℤn for which x↦ f(x)+mx is a permutation. We prove the sharp upper bound φ(n) on the number of good slopes, where φ is Euler's totient function. We classify and count the equality cases for prime powers and products of two distinct odd primes. Non-affine equality cases exist for every composite n≥6. We also prove the sharp bound p0−1 for the size of a set of good slopes with invertible differences between distinct elements, where p0 is the least prime divisor of n. Finally, we establish gap results below φ(n) and apply our bounds to a construction of quantum Latin squares due to Ball and Simoens.
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BibTeX
@misc{bhunia2026extremal,
title = {{Extremal permutations of $\mathbbZ_n$ with many good slopes and applications to quantum Latin squares}},
author = {Dipak K. Bhunia},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01465},
eprint = {2611.01465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01465},
}
RIS
TY - UNPB
TI - Extremal permutations of ℤ_n with many good slopes and applications to quantum Latin squares
AU - Dipak K. Bhunia
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01465
UR - https://arxiv.org/abs/2611.01465
AB - For n≥2, a good slope of a permutation f of ℤ_n=ℤ/nℤ is an element m∈ℤ_n for which x↦ f(x)+mx is a permutation. We prove the sharp upper bound φ(n) on the number of good slopes, where φ is Euler's totient function. We classify and count the equality cases for prime powers and products of two distinct odd primes. Non-affine equality cases exist for every composite n≥6. We also prove the sharp bound p_0−1 for the size of a set of good slopes with invertible differences between distinct elements, where p_0 is the least prime divisor of n. Finally, we establish gap results below φ(n) and apply our bounds to a construction of quantum Latin squares due to Ball and Simoens.
ER -