Existence of generalized bent functions in the exceptional q≡2\pmod4, odd-dimensional case

2026/07/27 by Jianing Li, Shenxing Zhang
Computer Science · Mathematics · #math.CO #cs.IT #math.IT

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Abstract

We resolve an open problem of Kumar, Scholtz, and Welch (1985) by constructing generalized bent functions from (ℤ/qℤ)^d to ℤ/qℤ in the exceptional case q≡2\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≥3 such that p=2^d-1 is a Mersenne prime, we construct an explicit generalized bent function from (ℤ/2pℤ)^d to ℤ/2pℤ. 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.

Citations

• Nonexistence of two classes of generalized bent functions

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