Index: streamexecutor/include/streamexecutor/KernelSpec.h
===================================================================
--- streamexecutor/include/streamexecutor/KernelSpec.h
+++ streamexecutor/include/streamexecutor/KernelSpec.h
@@ -121,12 +121,11 @@
       llvm::StringRef KernelName,
       const llvm::ArrayRef<CUDAPTXInMemorySpec::PTXSpec> SpecList);
 
-  /// Returns a pointer to the PTX code for the requested compute capability.
+  /// Returns a pointer to the PTX code for the greatest compute capability not
+  /// exceeding the requested compute capability.
   ///
-  /// Returns nullptr on failed lookup (if the requested compute capability is
-  /// not available). Matches exactly the specified compute capability. Doesn't
-  /// try to do anything smart like finding the next best compute capability if
-  /// the specified capability cannot be found.
+  /// Returns nullptr on failed lookup (if the requested version is not
+  /// available and no lower versions are available).
   const char *getCode(int ComputeCapabilityMajor,
                       int ComputeCapabilityMinor) const;
 
Index: streamexecutor/lib/KernelSpec.cpp
===================================================================
--- streamexecutor/lib/KernelSpec.cpp
+++ streamexecutor/lib/KernelSpec.cpp
@@ -31,12 +31,13 @@
 
 const char *CUDAPTXInMemorySpec::getCode(int ComputeCapabilityMajor,
                                          int ComputeCapabilityMinor) const {
-  auto PTXIter =
-      PTXByComputeCapability.find(CUDAPTXInMemorySpec::ComputeCapability{
+  auto Iterator =
+      PTXByComputeCapability.upper_bound(CUDAPTXInMemorySpec::ComputeCapability{
           ComputeCapabilityMajor, ComputeCapabilityMinor});
-  if (PTXIter == PTXByComputeCapability.end())
+  if (Iterator == PTXByComputeCapability.begin())
     return nullptr;
-  return PTXIter->second;
+  --Iterator;
+  return Iterator->second;
 }
 
 CUDAFatbinInMemorySpec::CUDAFatbinInMemorySpec(llvm::StringRef KernelName,
Index: streamexecutor/unittests/CoreTests/KernelSpecTest.cpp
===================================================================
--- streamexecutor/unittests/CoreTests/KernelSpecTest.cpp
+++ streamexecutor/unittests/CoreTests/KernelSpecTest.cpp
@@ -30,8 +30,9 @@
   const char *PTXCodeString = "Dummy PTX code";
   se::CUDAPTXInMemorySpec Spec("KernelName", {{{1, 0}, PTXCodeString}});
   EXPECT_EQ("KernelName", Spec.getKernelName());
+  EXPECT_EQ(nullptr, Spec.getCode(0, 5));
   EXPECT_EQ(PTXCodeString, Spec.getCode(1, 0));
-  EXPECT_EQ(nullptr, Spec.getCode(2, 0));
+  EXPECT_EQ(PTXCodeString, Spec.getCode(2, 0));
 }
 
 TEST(CUDAPTXInMemorySpec, TwoComputeCapabilities) {
@@ -40,9 +41,10 @@
   se::CUDAPTXInMemorySpec Spec(
       "KernelName", {{{1, 0}, PTXCodeString10}, {{3, 0}, PTXCodeString30}});
   EXPECT_EQ("KernelName", Spec.getKernelName());
+  EXPECT_EQ(nullptr, Spec.getCode(0, 5));
   EXPECT_EQ(PTXCodeString10, Spec.getCode(1, 0));
   EXPECT_EQ(PTXCodeString30, Spec.getCode(3, 0));
-  EXPECT_EQ(nullptr, Spec.getCode(2, 0));
+  EXPECT_EQ(PTXCodeString10, Spec.getCode(2, 0));
 }
 
 TEST(CUDAFatbinInMemorySpec, BasicUsage) {
@@ -89,8 +91,9 @@
   EXPECT_TRUE(MultiSpec.hasOpenCLTextInMemory());
 
   EXPECT_EQ(KernelName, MultiSpec.getCUDAPTXInMemory().getKernelName());
+  EXPECT_EQ(nullptr, MultiSpec.getCUDAPTXInMemory().getCode(0, 5));
   EXPECT_EQ(PTXCodeString, MultiSpec.getCUDAPTXInMemory().getCode(1, 0));
-  EXPECT_EQ(nullptr, MultiSpec.getCUDAPTXInMemory().getCode(2, 0));
+  EXPECT_EQ(PTXCodeString, MultiSpec.getCUDAPTXInMemory().getCode(2, 0));
 
   EXPECT_EQ(KernelName, MultiSpec.getCUDAFatbinInMemory().getKernelName());
   EXPECT_EQ(FatbinBytes, MultiSpec.getCUDAFatbinInMemory().getBytes());