diff --git a/pixi.toml b/pixi.toml
index 52e57fb3..5e68051e 100644
--- a/pixi.toml
+++ b/pixi.toml
@@ -28,7 +28,7 @@ mkdocs-jupyter = "*"
 [feature.tests.tasks]
 test = "pytest --pyargs sparse -n auto"
 test-mlir = { cmd = "pytest --pyargs sparse/mlir_backend -vvv", env = { SPARSE_BACKEND = "MLIR" } }
-test-finch = { cmd = "pytest --pyargs sparse/tests  -n auto", env = { SPARSE_BACKEND = "Finch" }, depends-on = ["precompile"] }
+test-finch = { cmd = "pytest --pyargs sparse/tests  -n auto -vvv", env = { SPARSE_BACKEND = "Finch", PYTHONFAULTHANDLER = "${HOME}/faulthandler.log" }, depends-on = ["precompile"] }
 
 [feature.tests.dependencies]
 pytest = ">=3.5"
diff --git a/sparse/mlir_backend/_levels.py b/sparse/mlir_backend/_levels.py
new file mode 100644
index 00000000..8f110dc1
--- /dev/null
+++ b/sparse/mlir_backend/_levels.py
@@ -0,0 +1,144 @@
+import ctypes
+import dataclasses
+import enum
+import itertools
+import re
+import typing
+
+import mlir.runtime as rt
+from mlir import ir
+from mlir.dialects import sparse_tensor
+
+import numpy as np
+
+from ._common import MlirType, fn_cache
+from ._dtypes import DType, asdtype
+
+_CAMEL_TO_SNAKE = [re.compile("(.)([A-Z][a-z]+)"), re.compile("([a-z0-9])([A-Z])")]
+
+
+def _camel_to_snake(name: str) -> str:
+    for exp in _CAMEL_TO_SNAKE:
+        name = exp.sub(r"\1_\2", name)
+
+    return name.lower()
+
+
+@fn_cache
+def get_nd_memref_descr(rank: int, dtype: type[DType]) -> type:
+    return rt.make_nd_memref_descriptor(rank, dtype.to_ctype())
+
+
+class LevelProperties(enum.Flag):
+    NonOrdered = enum.auto()
+    NonUnique = enum.auto()
+
+    def build(self) -> list[sparse_tensor.LevelProperty]:
+        return [getattr(sparse_tensor.LevelProperty, _camel_to_snake(p.name)) for p in type(self) if p in self]
+
+
+class LevelFormat(enum.Enum):
+    Dense = "dense"
+    Compressed = "compressed"
+    Singleton = "singleton"
+
+    def build(self) -> sparse_tensor.LevelFormat:
+        return getattr(sparse_tensor.LevelFormat, self.value)
+
+
+@dataclasses.dataclass
+class Level:
+    format: LevelFormat
+    properties: LevelProperties = LevelProperties(0)
+
+    def build(self):
+        sparse_tensor.EncodingAttr.build_level_type(self.format.build(), self.properties.build())
+
+
+@dataclasses.dataclass
+class StorageFormat:
+    levels: tuple[Level, ...]
+    order: typing.Literal["C", "F"] | tuple[int, ...]
+    pos_width: int
+    crd_width: int
+    dtype: type[DType]
+
+    @property
+    def storage_rank(self) -> int:
+        return len(self.levels)
+
+    @property
+    def rank(self) -> int:
+        return self.storage_rank
+
+    def __post_init__(self):
+        rank = self.storage_rank
+        self.dtype = asdtype(self.dtype)
+        if self.order == "C":
+            self.order = tuple(range(rank))
+            return
+
+        if self.order == "F":
+            self.order = tuple(range(rank))[::-1]
+            return
+
+        if sorted(self.order) != list(range(rank)):
+            raise ValueError(f"`sorted(self.order) != list(range(rank))`, {self.order=}, {rank=}.")
+
+        self.order = tuple(self.order)
+
+    @fn_cache
+    def get_mlir_type(self, *, shape: tuple[int, ...]) -> ir.RankedTensorType:
+        if len(shape) != self.rank:
+            raise ValueError(f"`len(shape) != self.rank`, {shape=}, {self.rank=}")
+        mlir_levels = [level.build() for level in self.levels]
+        mlir_order = list(self.order)
+        mlir_reverse_order = [0] * self.rank
+        for i, r in enumerate(mlir_order):
+            mlir_reverse_order[r] = i
+
+        dtype = self.dtype.get_mlir_type()
+        encoding = sparse_tensor.EncodingAttr.get(
+            mlir_levels, mlir_order, mlir_reverse_order, self.pos_width, self.crd_width
+        )
+        return ir.RankedTensorType.get(list(shape), dtype, encoding)
+
+    @fn_cache
+    def get_ctypes_type(self):
+        ptr_dtype = asdtype(getattr(np, f"uint{self.pos_width}"))
+        idx_dtype = asdtype(getattr(np, f"uint{self.crd_width}"))
+
+        def get_fields():
+            fields = []
+            compressed_counter = 0
+            for level, next_level in itertools.zip_longest(self.levels, self.levels[1:]):
+                if LevelFormat.Compressed == level.format:
+                    compressed_counter += 1
+                    fields.append((f"pointers_to_{compressed_counter}", get_nd_memref_descr(1, ptr_dtype)))
+                    if next_level is not None and LevelFormat.Singleton == next_level.format:
+                        fields.append((f"indices_{compressed_counter}", get_nd_memref_descr(2, idx_dtype)))
+                    else:
+                        fields.append((f"indices_{compressed_counter}", get_nd_memref_descr(1, idx_dtype)))
+
+            fields.append(("values", get_nd_memref_descr(1, self.dtype.np_dtype)))
+            return fields
+
+        class Format(ctypes.Structure, MlirType):
+            _fields_ = get_fields()
+
+            def get_mlir_type(self, *, shape: tuple[int, ...]):
+                return self.get_mlir_type(shape=shape)
+
+            def to_module_arg(self) -> list:
+                return [ctypes.pointer(ctypes.pointer(f) for f in self.get__fields_())]
+
+            def get__fields_(self) -> list:
+                return [getattr(self, field[0]) for field in self._fields_]
+
+        return Format
+
+    def __hash__(self):
+        return hash(id(self))
+
+    def __eq__(self, value):
+        return self is value