Cyclotomic numbers (a,b)_{q-1} over F_{q^r} satisfy (a,b)_{q-1} ≤ ⌈k/2⌉ for all a,b except when q=2 and r≥3, with tighter bounds for prime r.
Lehmer, On cyclotomic numbers of order sixteen,Canad
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Cyclotomic Numbers of Order $q-1$ over $\mathbb{F}_{q^r}$
Cyclotomic numbers (a,b)_{q-1} over F_{q^r} satisfy (a,b)_{q-1} ≤ ⌈k/2⌉ for all a,b except when q=2 and r≥3, with tighter bounds for prime r.