Existence of generalized bent functions in the exceptional $q\equiv2\pmod4$, odd-dimensional case
2026-07-27 • Information Theory
Information Theory
AI summaryⓘ
The authors solve a longstanding problem about special mathematical functions called generalized bent functions, focusing on a tricky case that was left open for nearly 40 years. They found a way to build these functions explicitly when certain numbers called Mersenne primes are involved. Additionally, they proved that these functions' related mathematical values cannot be simple roots of unity, answering a recent question negatively. Their work fills a gap left by previous researchers Kumar, Scholtz, and Welch.
generalized bent functionsMersenne primesFourier coefficientsroots of unitycharacter tablesmodular arithmeticdiscrete Fourier transformfinite abelian groups
Authors
Jianing Li, Shenxing Zhang
Abstract
We resolve an open problem of Kumar, Scholtz, and Welch (1985) by constructing generalized bent functions from $(\mathbb{Z}/q\mathbb{Z})^d$ to $\mathbb{Z}/q\mathbb{Z}$ in the exceptional case $q\equiv2\pmod4$ with $d$ odd, the case their paper left without a construction and which four decades of subsequent work had addressed only through nonexistence results. Concretely, for every odd integer $d\geq3$ such that $p=2^d-1$ is a Mersenne prime, we construct an explicit generalized bent function from $(\mathbb{Z}/2p\mathbb{Z})^d$ to $\mathbb{Z}/2p\mathbb{Z}$. In particular, this produces a function of type $[3,14]$. We further show that the Fourier coefficients of these generalized bent functions can not be a root of unity, which gives a negative answer to a recent question of Armario, Egan, Kharaghani, and Ó~Catháin about bent vectors for character tables.