diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/LoopEmitter.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/LoopEmitter.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/LoopEmitter.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/LoopEmitter.cpp
@@ -388,7 +388,7 @@
              !highs[t][l]);
       const auto lvlTp = lvlTypes[t][l];
       // Handle sparse storage schemes.
-      if (isCompressedDLT(lvlTp)) {
+      if (isCompressedDLT(lvlTp) || isCompressedWithHiDLT(lvlTp)) {
         // Generate sparse primitives to obtain positions and coordinates.
         positionsBuffers[t][l] = genToPositions(builder, loc, tensor, l);
         coordinatesBuffers[t][l] =
@@ -557,6 +557,7 @@
     OpBuilder &builder, Location loc, TensorId tid, Level dstLvl, Value lo,
     Value hi, MutableArrayRef<Value> reduc, bool isParallel) {
   bool isSparseCond = isCompressedDLT(lvlTypes[tid][dstLvl]) ||
+                      isCompressedWithHiDLT(lvlTypes[tid][dstLvl]) ||
                       isSingletonDLT(lvlTypes[tid][dstLvl]);
 
   const auto reassoc = getCollapseReassociation(tid, dstLvl);
@@ -695,7 +696,7 @@
     auto lvlType = lvlTypes[t][l];
     // Must be a recognizable DLT.
     assert(isDenseDLT(lvlType) || isCompressedDLT(lvlType) ||
-           isSingletonDLT(lvlType));
+           isCompressedWithHiDLT(lvlType) || isSingletonDLT(lvlType));
 
     // This is a slice-driven loop on sparse level.
     if (!dependentLvlMap[t][l].empty() && !isDenseDLT(lvlType)) {
@@ -901,7 +902,8 @@
     // TODO: support coiteration with slice driven tensors.
     const auto lvlTp = lvlTypes[tid][lvl];
     assert(dependentLvlMap[tid][lvl].empty() && "TODO: not yet implemented");
-    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp)) {
+    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp) ||
+        isCompressedWithHiDLT(lvlTp)) {
       const auto reassoc = getCollapseReassociation(tid, lvl);
       for (unsigned i = 0, e = reassoc.size() - 1; i < e; i++) {
         if (!isUniqueDLT(lvlTypes[tid][reassoc[i]])) {
@@ -941,7 +943,8 @@
   for (auto [t, lvl] : llvm::zip(tids, lvls)) {
     const TensorId tid = t; // Why `t` can not be captured by lambda?
     const auto lvlTp = lvlTypes[tid][lvl];
-    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp)) {
+    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp) ||
+        isCompressedWithHiDLT(lvlTp)) {
       const auto reassoc = getCollapseReassociation(tid, lvl);
       assert(reassoc.size() == 1 || isUniqueCOOType(tensors[tid].getType()));
       for (unsigned i = 0, e = reassoc.size() - 1; i < e; i++) {
@@ -974,7 +977,8 @@
   for (auto [tid, lvl] : llvm::zip(tids, lvls)) {
     // Prepares for next level.
     const auto lvlTp = lvlTypes[tid][lvl];
-    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp)) {
+    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp) ||
+        isCompressedWithHiDLT(lvlTp)) {
       coords[tid][lvl] = genSparseCrd(builder, loc, tid, lvl);
       if (isSparseSlices[tid]) {
         auto [trans, pred] =
@@ -1023,7 +1027,8 @@
   if (!needsUniv) {
     for (auto [tid, lvl] : llvm::zip(tids, lvls)) {
       const auto lvlTp = lvlTypes[tid][lvl];
-      if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp)) {
+      if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp) ||
+          isCompressedWithHiDLT(lvlTp)) {
         const auto crd = coords[tid][lvl];
         if (min) {
           Value cmp = CMPI(ult, coords[tid][lvl], min);
@@ -1117,12 +1122,14 @@
     // Either the first level, or the previous level has been set.
     /// FIXME: See the [CLARIFY_POSITS_LVL] note in the header.
     assert(srcLvl == 0 || posits[tid][srcLvl - 1]);
-    if (!isCompressedDLT(lvlTp) && !isSingletonDLT(lvlTp))
+    if (isDenseDLT(lvlTp))
       continue;
-    if (isCompressedDLT(lvlTp)) {
+    if (isCompressedDLT(lvlTp) || isCompressedWithHiDLT(lvlTp)) {
       const Value mem = positionsBuffers[tid][srcLvl];
 
-      const Value pLo = srcLvl == 0 ? c0 : posits[tid][srcLvl - 1];
+      Value pLo = srcLvl == 0 ? c0 : posits[tid][srcLvl - 1];
+      if (isCompressedWithHiDLT(lvlTp))
+        pLo = builder.create<arith::MulIOp>(loc, pLo, C_IDX(2));
       posits[tid][srcLvl] = genIndexLoad(builder, loc, mem, pLo);
 
       const Value pHi = ADDI(pLo, c1);
@@ -1321,7 +1328,8 @@
   Value one = C_IDX(1);
   for (auto [tid, dstLvl] : llvm::zip(loopInfo.tids, loopInfo.lvls)) {
     const auto lvlTp = lvlTypes[tid][dstLvl];
-    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp)) {
+    if (isCompressedDLT(lvlTp) || isSingletonDLT(lvlTp) ||
+        isCompressedWithHiDLT(lvlTp)) {
       const auto reassoc = getCollapseReassociation(tid, dstLvl);
       assert(reassoc.size() == 1 || isUniqueCOOType(tensors[tid].getType()));
       for (unsigned i = 0, e = reassoc.size() - 1; i < e; i++) {
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorCodegen.cpp
@@ -532,6 +532,8 @@
   const Level lvlRank = stt.getLvlRank();
   for (Level l = 0; l < lvlRank; l++) {
     const auto dlt = stt.getLvlType(l);
+    if (isCompressedWithHiDLT(dlt))
+      llvm_unreachable("TODO: Not yet implemented");
     if (isCompressedDLT(dlt)) {
       // Compressed dimensions need a position cleanup for all entries
       // that were not visited during the insertion pass.
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorStorageLayout.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorStorageLayout.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorStorageLayout.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/SparseTensorStorageLayout.cpp
@@ -145,7 +145,7 @@
     // As a result, the compound type can be constructed directly in the given
     // order.
     const auto dlt = lvlTypes[l];
-    if (isCompressedDLT(dlt)) {
+    if (isCompressedDLT(dlt) || isCompressedWithHiDLT(dlt)) {
       RETURN_ON_FALSE(fieldIdx++, SparseTensorFieldKind::PosMemRef, l, dlt);
       RETURN_ON_FALSE(fieldIdx++, SparseTensorFieldKind::CrdMemRef, l, dlt);
     } else if (isSingletonDLT(dlt)) {
diff --git a/mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp b/mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp
--- a/mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp
@@ -794,7 +794,8 @@
       const TensorId tid = env.makeTensorId(t.getOperandNumber());
       for (LoopId i = 0; i < numLoops; i++) {
         const auto dltI = env.dlt(tid, i);
-        if (isCompressedDLT(dltI) || isSingletonDLT(dltI)) {
+        if (isCompressedDLT(dltI) || isCompressedWithHiDLT(dltI) ||
+            isSingletonDLT(dltI)) {
           for (LoopId j = 0; j < numLoops; j++)
             if (isUndefDLT(env.dlt(tid, j))) {
               adjM[i][j] = true;
@@ -1410,7 +1411,7 @@
           DimLevelType dlt, bool /*unused*/) {
         assert(ldx == env.merger().loop(b));
         Value clause;
-        if (isCompressedDLT(dlt) || isSingletonDLT(dlt)) {
+        if (isCompressedDLT(dlt) || isSingletonDLT(dlt) || isCompressedWithHiDLT(dlt)) {
           assert(lvl.has_value());
           const Value crd = env.emitter().getCoords()[tid][*lvl];
           const Value lvar = env.getLoopVar(ldx);
diff --git a/mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp b/mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp
--- a/mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp
+++ b/mlir/lib/Dialect/SparseTensor/Utils/Merger.cpp
@@ -418,7 +418,7 @@
     // Slice on dense level has `locate` property as well, and can be optimized.
     if (simple[b] && !isSparseLvlWithNonTrivialIdxExp(b)) {
       const auto dlt = getDimLevelType(b);
-      if (!isCompressedDLT(dlt) && !isSingletonDLT(dlt)) {
+      if (!isCompressedDLT(dlt) && !isSingletonDLT(dlt) && !isCompressedWithHiDLT(dlt)) {
         if (reset)
           simple.reset(b);
         reset = true;
@@ -585,7 +585,7 @@
 bool Merger::hasAnySparse(const BitVector &bits) const {
   for (TensorLoopId b : bits.set_bits()) {
     const auto dlt = getDimLevelType(b);
-    if (isCompressedDLT(dlt) || isSingletonDLT(dlt))
+    if (isCompressedDLT(dlt) || isSingletonDLT(dlt) || isCompressedWithHiDLT(dlt))
       return true;
   }
   return hasSparseIdxReduction(bits);
diff --git a/mlir/test/Dialect/SparseTensor/sparse_foreach.mlir b/mlir/test/Dialect/SparseTensor/sparse_foreach.mlir
--- a/mlir/test/Dialect/SparseTensor/sparse_foreach.mlir
+++ b/mlir/test/Dialect/SparseTensor/sparse_foreach.mlir
@@ -138,4 +138,34 @@
     "test.use" (%v) : (f64) -> ()
   }
   return
-}
\ No newline at end of file
+}
+
+#BCOO = #sparse_tensor.encoding<{
+  dimLevelType = [ "dense", "compressed-hi-nu", "singleton" ],
+}>
+
+// CHECK-LABEL:   func.func @foreach_bcoo(
+// CHECK-SAME:      %[[VAL_0:.*]]: tensor<4x4x4xf64, #sparse_tensor.encoding<{{.*}}>>) {
+// CHECK-DAG:       %[[VAL_1:.*]] = arith.constant 4 : index
+// CHECK-DAG:       %[[VAL_2:.*]] = arith.constant 0 : index
+// CHECK-DAG:       %[[VAL_3:.*]] = arith.constant 1 : index
+// CHECK-DAG:       %[[VAL_4:.*]] = arith.constant 2 : index
+// CHECK-DAG:       %[[VAL_5:.*]] = sparse_tensor.positions %[[VAL_0]] {level = 1 : index} : tensor<4x4x4xf64, #sparse_tensor.encoding<{{.*}}>> to memref<?xindex>
+// CHECK-DAG:       %[[VAL_6:.*]] = sparse_tensor.values %[[VAL_0]] : tensor<4x4x4xf64, #sparse_tensor.encoding<{{.*}}>> to memref<?xf64>
+// CHECK:           scf.for %[[VAL_7:.*]] = %[[VAL_2]] to %[[VAL_1]] step %[[VAL_3]] {
+// CHECK:             %[[VAL_8:.*]] = arith.muli %[[VAL_7]], %[[VAL_4]] : index
+// CHECK:             %[[VAL_9:.*]] = arith.addi %[[VAL_8]], %[[VAL_3]] : index
+// CHECK:             %[[VAL_10:.*]] = memref.load %[[VAL_5]]{{\[}}%[[VAL_9]]] : memref<?xindex>
+// CHECK:             scf.for %[[VAL_11:.*]] = %[[VAL_2]] to %[[VAL_10]] step %[[VAL_3]] {
+// CHECK:               %[[VAL_12:.*]] = memref.load %[[VAL_6]]{{\[}}%[[VAL_11]]] : memref<?xf64>
+// CHECK:               "test.use"(%[[VAL_12]]) : (f64) -> ()
+// CHECK:             }
+// CHECK:           }
+// CHECK:           return
+func.func @foreach_bcoo(%A: tensor<4x4x4xf64, #BCOO>) {
+  sparse_tensor.foreach in %A : tensor<4x4x4xf64, #BCOO> do {
+  ^bb0(%1: index, %2: index, %3: index,  %v: f64) :
+    "test.use" (%v) : (f64) -> ()
+  }
+  return
+}