Super-logspace mapping reducibility
In Model Checking Recursive Programs with Numeric Data Types, Hague and Lin presented a NEXP-complete language under logspace reduction, called $ \mathtt{SUCCINCT~0\mbox{-}1~KNAPSACK} $ composed by strings 

\begin{equation*}
a^m \# a^k \# \theta,
\end{equation*} where $ \theta $ is a Boolean formula with variables $ x_1, \ldots, x_{m+k} $ satisfying that the string $ b \# a_1 \# \cdots \# a_{2^k-1} $ defined based on $ \theta $ (described below) is a member of $ \mathtt{KNAPSCAK} $, i.e. $ \sum_{i=1}^{2^k-1} a_i z_i = b $ for some $ z_1, \ldots,z_{2^k-1} \in \{0,1\} $, where $ b $ and each $ a_i $ are precisely $ 2^m $ bits (leading 0s permitted) binary numbers.

The string $ b \# a_1 \# \cdots \# a_{2^k-1} $ is defined based on $ \theta $ as follows. Let $ (i)_{2,k} $ be the $ k $-bit binary representation of $ i $.


The $ i^{th} $ bit of $ b $ is the value of $ \theta $ by setting $ x_1,\ldots,x_{k+m} $ to the digits of $ 0^k  (i)_{2,m} $ in order.
The $ i^{th} $ bit of $ a_j $ is the value of $ \theta $ by setting $ x_1,\ldots,x_{k+m} $ to the digits of $ (j)_{2,k}  (i)_{2,m} $ in order.




It is not hard to show that $ \mathtt{SUCCINCT~0\mbox{-}1~KNAPSACK} $ can be reduced to NP-Complete $ \mathtt{KNAPSCAK} $ via linearspace reduction.

Moreover, since any language in NEXP is logspace reducible to $ \mathtt{SUCCINCT~0\mbox{-}1~KNAPSACK} $, we can say that any language in NEXP is polyspace reducible to $ \mathtt{KNAPSACK} $. (Here we can combine logspace and linearspace reductions. Since the output string after logspace reduction can have polynomial length, we can use polynomial space overall.)