  In computer and system sciences, higher-order cellular automata (HOCA) are a
type of cellular automata that evolve over multiple time steps and generate
complex patterns, which have various applications such as secret sharing
schemes, data compression, and image encryption. In this paper, we introduce
HOCA to quantum many-body physics and construct a series of symmetry-protected
topological (SPT) phases of matter, in which symmetries are supported on a
great variety of subsystems embbeded in the SPT bulk. We call these phases
HOCA-generated SPT (HGSPT) phases. Specifically, we show that HOCA can generate
not only well-understood SPTs with symmetries supported on either regular
(e.g., line-like subsystems in the 2D cluster model) or fractal subsystems, but
also a large class of unexplored SPTs with symmetries supported on more choices
of subsystems. One example is \textit{mixed-subsystem SPT} that has either
fractal and line-like subsystem symmetries simultaneously or two distinct types
of fractal symmetries simultaneously. Another example is
\textit{chaotic-subsystem SPT} in which chaotic-looking symmetries are
significantly different from and thus cannot reduce to fractal or regular
subsystem symmetries. We also introduce a new notation system to characterize
HGSPTs. We prove that all possible subsystem symmetries in square lattice can
be locally simulated by an HOCA generated symmetry. As the usual two-point
strange correlators are trivial in most HGSPTs, we find that the nontrivial SPT
orders can be detected by what we call \textit{multi-point strange
correlators}. We propose a universal procedure to design the spatial
configuration of the multi-point strange correlators for a given HGSPT phase.
Specifically, we find deep connections between multi-point strange correlators
and the spurious topological entanglement entropy (STEE), both exhibiting long
range behavior in SRE states.
