  This paper presents a generalization to multisimplicial sets of previously
defined $E_\infty$-coalgebra structures on the chains of simplicial and cubical
sets. We focus on the surjection chain complexes of McClure--Smith as a main
example and construct a zig-zag of complexity preserving quasi-isomorphisms of
$E_\infty$-coalgebras relating these to both the singular chains on
configuration spaces and the Barratt--Eccles chain complexes.
