diff --git a/src/core_test/CMakeLists.txt b/src/core_test/CMakeLists.txt
index 169e364..30c7ffb 100644
--- a/src/core_test/CMakeLists.txt
+++ b/src/core_test/CMakeLists.txt
@@ -46,6 +46,7 @@ set(KLSK_TEST_CODEC_SRC
     codec/raw_code.cc
     codec/short_code.cc
     codec/common_code.cc
+    codec/helper/codec.cc
 )
 
 add_executable(test_klotski_codec ${KLSK_TEST_CODEC_SRC})
diff --git a/src/core_test/codec/common_code.cc b/src/core_test/codec/common_code.cc
index 0191e0a..af3fe0f 100644
--- a/src/core_test/codec/common_code.cc
+++ b/src/core_test/codec/common_code.cc
@@ -2,7 +2,8 @@
 #include <gtest/gtest.h>
 #include <BS_thread_pool.hpp>
 
-#include "sample.h"
+#include "helper/codec.h"
+#include "helper/sample.h"
 #include "raw_code/raw_code.h"
 #include "all_cases/all_cases.h"
 #include "short_code/short_code.h"
@@ -34,15 +35,36 @@ TEST(CommonCode, operators) {
     EXPECT_EQ(tmp.str(), TEST_C_CODE_STR);
     EXPECT_EQ((uint64_t)common_code, TEST_C_CODE); // convert as uint64_t
 
+    EXPECT_NE(0, common_code); // uint64_t != CommonCode
+    EXPECT_NE(common_code, 0); // CommonCode != uint64_t
     EXPECT_EQ(TEST_C_CODE, common_code); // uint64_t == CommonCode
     EXPECT_EQ(common_code, TEST_C_CODE); // CommonCode == uint64_t
-    EXPECT_EQ(common_code, common_code); // CommonCode == CommonCode
 
-    EXPECT_NE(TEST_C_CODE + 1, common_code); // uint64_t != CommonCode
-    EXPECT_NE(common_code, TEST_C_CODE + 1); // CommonCode != uint64_t
-    EXPECT_NE(common_code, CommonCode::unsafe_create(TEST_C_CODE + 1)); // CommonCode != CommonCode
+    EXPECT_LE(TEST_C_CODE, common_code); // uint64_t <= CommonCode
+    EXPECT_LE(TEST_C_CODE - 1, common_code);
+    EXPECT_LT(TEST_C_CODE - 1, common_code); // uint64_t < CommonCode
+
+    EXPECT_LE(common_code, TEST_C_CODE); // CommonCode <= uint64_t
+    EXPECT_LE(common_code, TEST_C_CODE + 1);
+    EXPECT_LT(common_code, TEST_C_CODE + 1); // CommonCode < uint64_t
+
+    EXPECT_GE(TEST_C_CODE, common_code); // uint64_t >= CommonCode
+    EXPECT_GE(TEST_C_CODE + 1, common_code);
+    EXPECT_GT(TEST_C_CODE + 1, common_code); // uint64_t > CommonCode
+
+    EXPECT_GE(common_code, TEST_C_CODE); // CommonCode >= uint64_t
+    EXPECT_GE(common_code, TEST_C_CODE - 1);
+    EXPECT_GT(common_code, TEST_C_CODE - 1); // CommonCode > uint64_t
+
+    EXPECT_EQ(common_code, common_code); // CommonCode == CommonCode
+    EXPECT_NE(common_code, CommonCode::unsafe_create(0)); // CommonCode != CommonCode
 
+    EXPECT_LE(common_code, common_code); // CommonCode <= CommonCode
+    EXPECT_LE(common_code, CommonCode::unsafe_create(TEST_C_CODE + 1));
     EXPECT_LT(common_code, CommonCode::unsafe_create(TEST_C_CODE + 1)); // CommonCode < CommonCode
+
+    EXPECT_GE(common_code, common_code); // CommonCode >= CommonCode
+    EXPECT_GE(CommonCode::unsafe_create(TEST_C_CODE + 1), common_code);
     EXPECT_GT(CommonCode::unsafe_create(TEST_C_CODE + 1), common_code); // CommonCode > CommonCode
 }
 
@@ -54,12 +76,10 @@ TEST(CommonCode, exporter) {
     EXPECT_EQ(common_code.to_short_code(), TEST_S_CODE);
 
     auto code_shorten = common_code.to_string(true);
-    EXPECT_EQ(CommonCode::from_string(code_shorten), common_code); // l-value
-    EXPECT_EQ(CommonCode::from_string(std::move(code_shorten)), common_code); // r-value
+    EXPECT_EQ(CommonCode::from_string(code_shorten), common_code);
 
     auto code_normal = common_code.to_string(false);
-    EXPECT_EQ(CommonCode::from_string(code_normal), common_code); // l-value
-    EXPECT_EQ(CommonCode::from_string(std::move(code_normal)), common_code); // r-value
+    EXPECT_EQ(CommonCode::from_string(code_normal), common_code);
 }
 
 TEST(CommonCode, initializate) {
@@ -67,11 +87,17 @@ TEST(CommonCode, initializate) {
     auto short_code = ShortCode::unsafe_create(TEST_S_CODE);
     auto common_code = CommonCode::unsafe_create(TEST_C_CODE);
 
+    // operator==
+    CommonCode c1 = common_code;
+    EXPECT_EQ(c1, TEST_C_CODE); // l-value
+    CommonCode c2 = CommonCode {common_code};
+    EXPECT_EQ(c2, TEST_C_CODE); // r-value
+
     // CommonCode(...)
     EXPECT_EQ(CommonCode(raw_code), TEST_C_CODE);
     EXPECT_EQ(CommonCode(short_code), TEST_C_CODE);
     EXPECT_EQ(CommonCode(common_code), TEST_C_CODE); // l-value
-    EXPECT_EQ(CommonCode(CommonCode(common_code)), TEST_C_CODE); // r-value
+    EXPECT_EQ(CommonCode(CommonCode {common_code}), TEST_C_CODE); // r-value
 
     // CommonCode::create(uint64_t)
     EXPECT_TRUE(CommonCode::create(TEST_C_CODE).has_value());
@@ -80,17 +106,13 @@ TEST(CommonCode, initializate) {
 
     // CommonCode::unsafe_create(uint64_t)
     EXPECT_EQ(CommonCode::unsafe_create(TEST_C_CODE), TEST_C_CODE);
+    EXPECT_EQ(CommonCode::unsafe_create(TEST_C_CODE_ERR), TEST_C_CODE_ERR);
 
-    // CommonCode::from_string(const std::string &)
+    // CommonCode::from_string(std::string_view)
     EXPECT_TRUE(CommonCode::from_string(TEST_C_CODE_STR).has_value());
     EXPECT_FALSE(CommonCode::from_string(TEST_C_CODE_STR_ERR).has_value());
     EXPECT_EQ(CommonCode::from_string(TEST_C_CODE_STR), TEST_C_CODE);
 
-    // CommonCode::from_string(std::string &&)
-    EXPECT_TRUE(CommonCode::from_string(TEST_C_CODE_STR_RV).has_value());
-    EXPECT_FALSE(CommonCode::from_string(TEST_C_CODE_STR_ERR_RV).has_value());
-    EXPECT_EQ(CommonCode::from_string(TEST_C_CODE_STR_RV), TEST_C_CODE);
-
     // CommonCode::from_raw_code(RawCode)
     EXPECT_EQ(CommonCode::from_raw_code(raw_code), TEST_C_CODE);
 
@@ -107,15 +129,10 @@ TEST(CommonCode, initializate) {
     EXPECT_FALSE(CommonCode::from_short_code(TEST_S_CODE_ERR).has_value());
     EXPECT_EQ(CommonCode::from_short_code(TEST_S_CODE), TEST_C_CODE);
 
-    // CommonCode::from_short_code(const std::string &)
+    // CommonCode::from_short_code(std::string_view)
     EXPECT_TRUE(CommonCode::from_short_code(TEST_S_CODE_STR).has_value());
     EXPECT_FALSE(CommonCode::from_short_code(TEST_S_CODE_STR_ERR).has_value());
     EXPECT_EQ(CommonCode::from_short_code(TEST_S_CODE_STR), TEST_C_CODE);
-
-    // CommonCode::from_short_code(std::string &&)
-    EXPECT_TRUE(CommonCode::from_short_code(TEST_S_CODE_STR_RV).has_value());
-    EXPECT_FALSE(CommonCode::from_short_code(TEST_S_CODE_STR_ERR_RV).has_value());
-    EXPECT_EQ(CommonCode::from_short_code(TEST_S_CODE_STR_RV), TEST_C_CODE);
 }
 
 TEST(CommonCode, code_verify) {
diff --git a/src/core_test/codec/helper/codec.cc b/src/core_test/codec/helper/codec.cc
new file mode 100644
index 0000000..4b59f9a
--- /dev/null
+++ b/src/core_test/codec/helper/codec.cc
@@ -0,0 +1,27 @@
+#include <BS_thread_pool.hpp>
+
+#include "codec.h"
+
+void head_parallel(std::function<void(uint64_t head)> &&func) {
+    BS::thread_pool pool;
+    // TODO: skip invalid head
+    pool.detach_sequence(0, 16, [func = std::move(func)](const uint64_t head) {
+        if (head == 3 || head == 7 || head == 11 || head == 15) {
+            return;
+        }
+        func(head);
+    });
+    pool.wait();
+}
+
+std::vector<uint64_t> all_common_codes() {
+    // TODO: using `std::ranges`
+    std::vector<uint64_t> common_codes;
+    common_codes.reserve(ALL_CASES_NUM_);
+    for (uint64_t head = 0; head < 16; ++head) {
+        for (auto range : AllCases::instance().fetch()[head]) {
+            common_codes.emplace_back(head << 32 | range);
+        }
+    }
+    return common_codes;
+}
diff --git a/src/core_test/codec/helper/codec.h b/src/core_test/codec/helper/codec.h
new file mode 100644
index 0000000..4d55c9c
--- /dev/null
+++ b/src/core_test/codec/helper/codec.h
@@ -0,0 +1,14 @@
+#pragma once
+
+#include <functional>
+
+#include "all_cases/all_cases.h"
+
+using klotski::cases::AllCases;
+using klotski::cases::ALL_CASES_NUM_;
+
+/// Build all valid CommonCodes.
+std::vector<uint64_t> all_common_codes();
+
+/// Spawn all valid klotski headers in parallel.
+void head_parallel(std::function<void(uint64_t head)> &&func);
diff --git a/src/core_test/codec/helper/sample.h b/src/core_test/codec/helper/sample.h
new file mode 100644
index 0000000..ee03abf
--- /dev/null
+++ b/src/core_test/codec/helper/sample.h
@@ -0,0 +1,36 @@
+#pragma once
+
+#include <string>
+#include <cstdint>
+
+// ----------------------------------------------------------------------------------------- //
+
+/// Valid klotski RawCode
+constexpr uint64_t TEST_R_CODE = 0x0603'EDF5'CAFF'F5E2;
+
+/// Valid klotski ShortCode
+constexpr uint32_t TEST_S_CODE = 4091296;
+const std::string TEST_S_CODE_STR = "4WVE1"; // TODO: using `std::string_view`
+#define TEST_S_CODE_STR_RV std::string(TEST_S_CODE_STR) // TODO: remove r-value
+
+/// Valid klotski CommonCode
+constexpr uint64_t TEST_C_CODE = 0x1'A9BF'0C00;
+const std::string TEST_C_CODE_STR = "1A9BF0C00";
+#define TEST_C_CODE_STR_RV std::string(TEST_C_CODE_STR) // TODO: remove r-value
+
+// ----------------------------------------------------------------------------------------- //
+
+/// Invalid klotski RawCode
+constexpr uint64_t TEST_R_CODE_ERR = 0x1603'ED00'CAFF'F5E2;
+
+/// Invalid klotski ShortCode
+constexpr uint32_t TEST_S_CODE_ERR = 1234564323;
+const std::string TEST_S_CODE_STR_ERR = "ZZZZZZ";
+#define TEST_S_CODE_STR_ERR_RV std::string(TEST_S_CODE_STR_ERR) // TODO: remove r-value
+
+/// Invalid klotski CommonCode
+constexpr uint64_t TEST_C_CODE_ERR = 0x3'A9BF'0C00;
+const static std::string TEST_C_CODE_STR_ERR = "0123456789";
+#define TEST_C_CODE_STR_ERR_RV std::string(TEST_C_CODE_STR_ERR) // TODO: remove r-value
+
+// ----------------------------------------------------------------------------------------- //
diff --git a/src/core_test/codec/raw_code.cc b/src/core_test/codec/raw_code.cc
index f533f96..836ec9e 100644
--- a/src/core_test/codec/raw_code.cc
+++ b/src/core_test/codec/raw_code.cc
@@ -1,8 +1,9 @@
 #include <gtest/gtest.h>
 #include <BS_thread_pool.hpp>
 
-#include "sample.h"
 #include "utils/common.h"
+#include "helper/codec.h"
+#include "helper/sample.h"
 #include "raw_code/raw_code.h"
 #include "all_cases/all_cases.h"
 #include "common_code/common_code.h"
diff --git a/src/core_test/codec/sample.h b/src/core_test/codec/sample.h
deleted file mode 100644
index 03c94e5..0000000
--- a/src/core_test/codec/sample.h
+++ /dev/null
@@ -1,54 +0,0 @@
-#pragma once
-
-#include <string>
-#include <cstdint>
-
-#include "all_cases/all_cases.h"
-
-// ----------------------------------------------------------------------------------------- //
-
-// Valid RawCode
-const uint64_t TEST_R_CODE = 0x0603'EDF5'CAFF'F5E2;
-
-// Valid ShortCode
-const uint32_t TEST_S_CODE = 4091296;
-const std::string TEST_S_CODE_STR = "4WVE1";
-#define TEST_S_CODE_STR_RV std::string(TEST_S_CODE_STR) // r-value
-
-// Valid CommonCode
-const uint64_t TEST_C_CODE = 0x1'A9BF'0C00;
-const std::string TEST_C_CODE_STR = "1A9BF0C00";
-#define TEST_C_CODE_STR_RV std::string(TEST_C_CODE_STR) // r-value
-
-// ----------------------------------------------------------------------------------------- //
-
-// Invalid RawCode
-const static uint64_t TEST_R_CODE_ERR = 0x1603'ED00'CAFF'F5E2;
-
-// Invalid ShortCode
-const static uint32_t TEST_S_CODE_ERR = 1234564323;
-const static std::string TEST_S_CODE_STR_ERR = "ZZZZZZ";
-#define TEST_S_CODE_STR_ERR_RV std::string(TEST_S_CODE_STR_ERR) // r-value
-
-// Invalid CommonCode
-const static uint64_t TEST_C_CODE_ERR = 0x3'A9BF'0C00;
-const static std::string TEST_C_CODE_STR_ERR = "0123456789";
-#define TEST_C_CODE_STR_ERR_RV std::string(TEST_C_CODE_STR_ERR) // r-value
-
-// ----------------------------------------------------------------------------------------- //
-
-using klotski::cases::AllCases;
-using klotski::cases::ALL_CASES_NUM_;
-
-inline std::vector<uint64_t> all_common_codes() {
-    std::vector<uint64_t> common_codes;
-    common_codes.reserve(ALL_CASES_NUM_);
-    for (uint64_t head = 0; head < 16; ++head) {
-        for (auto range : AllCases::instance().fetch()[head]) {
-            common_codes.emplace_back(head << 32 | range);
-        }
-    }
-    return common_codes;
-}
-
-// ----------------------------------------------------------------------------------------- //
diff --git a/src/core_test/codec/short_code.cc b/src/core_test/codec/short_code.cc
index 4c6cf4c..a993f83 100644
--- a/src/core_test/codec/short_code.cc
+++ b/src/core_test/codec/short_code.cc
@@ -2,8 +2,9 @@
 #include <gtest/gtest.h>
 #include <BS_thread_pool.hpp>
 
-#include "sample.h"
-#include "exposer.h"
+#include "helper/codec.h"
+#include "helper/sample.h"
+#include "utility/exposer.h"
 #include "all_cases/all_cases.h"
 #include "short_code/short_code.h"
 #include "common_code/common_code.h"