This is related to the question Is the Witness Size of Membership for Every NP Language Already Known?

Some natural $\mathsf{NP}$(-complete) problems have linear length witnesses: a satisfying assignment for $SAT$, a sequence of vertices for $HAMPATH$, etc.

Consider the complexity class "$\mathsf{NP}$ restricted to linear length witnesses". Formal definition of this complexity class, call it $\mathcal{C}$: $L\in\mathcal{C}$ if $\exists L'\in\mathsf{P}\colon (x\in L \iff \exists w\in\{0, 1\}^{O(|x|)}\colon (x, w)\in L')$.

Is this a known complexity class? What are its properties?