On a problem of Woodin (Q1570017)

In \textit{A. Kanamori}'s book [The higher infinite, Springer-Verlag, Berlin (1994; Zbl 0813.03034)], the following question of Woodin's is mentioned: If \(\kappa\) is a strongly compact cardinal and \(2^\delta= \delta^+\) for every cardinal \(\delta< \kappa\), does this entail that GCH holds? In this paper the author shows that in the absence of full AC it is possible to get a negative answer to Woodin's question. Specifically he proves: Let \(V\models \text{``ZFC}+ \kappa\) is supercompact''. Then there is a partial ordering \(P\in V\) and a symmetric inner model \(N\), \(V\subseteq N\subseteq V^P\), so that \(N\models \text{``ZF}+ \forall\delta< \kappa\) \([DC_\delta]+ \kappa\) is a strong limit cardinal \(+\) \(\forall\delta< \kappa\) \([2^\delta= \delta^+]+ \kappa\) is supercompact + There is a sequence \(\langle A_\alpha: \alpha< \kappa^{++}\rangle\) of distinct subsets of \(\kappa\)''.