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Characterization of a class of complete permutation quadrinomials over π½22m
Abstract
Let q=2m, Q=2k, and 1β€ kβ€ mβ1. We characterize complete permutation polynomials (CPPs) over π½q2 of the form f(x)=c0xQ+1+c1xQ+q+c2xqQ+1 +c3xq(Q+1), ciβπ½q2. We prove that no such CPP exists when k>1, and recover the known characterization in the cubic case k=1. This completes the classification throughout the stated exponent range. The proof uses the known permutation classification to reduce completeness to linear perturbations of product and monomial models. The required nonpermutation results follow from direct elementary arguments based on the quadratic structure over π½2.
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BibTeX
@misc{li2026characterization,
title = {{Characterization of a class of complete permutation quadrinomials over $\mathbbF_2^2m$}},
author = {Yanjun Li and Maosheng Xiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30126},
eprint = {2608.30126},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30126},
}
RIS
TY - UNPB
TI - Characterization of a class of complete permutation quadrinomials over π½_2^2m
AU - Yanjun Li
AU - Maosheng Xiong
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.30126
UR - https://arxiv.org/abs/2608.30126
AB - Let q=2^m, Q=2^k, and 1β€ kβ€ mβ1. We characterize complete permutation polynomials (CPPs) over π½_q^2 of the form f(x)=c_0x^Q+1+c_1x^Q+q+c_2x^qQ+1 +c_3x^q(Q+1), c_iβπ½_q^2. We prove that no such CPP exists when k>1, and recover the known characterization in the cubic case k=1. This completes the classification throughout the stated exponent range. The proof uses the known permutation classification to reduce completeness to linear perturbations of product and monomial models. The required nonpermutation results follow from direct elementary arguments based on the quadratic structure over π½_2.
ER -