There's no need to do direct core work if you can't carry double your bodyweight for 30 seconds. If you can carry twice your bodyweight for 30 seconds, only then should you introduce direct and isolated core training into your programming.
If you can't pass the test, your core strength movements should be centered on carries and isometric full-body tension variations until you bring up your weak links to the point of a near 2x bodyweight carry.
The Core of the Matter
From the bros doing thousands of crunches a week to the fluffy functional training gurus, everyone has an answer for how to optimize core strength and aesthetics. But rarely will they ever tell you to improve your relative strength.
If you're having trouble activating your core, your shoulders ache, and your back is chronically flared up, you've got some weakness to address. And no amount of corrective exercises, mobility drills, or manual therapy can fix that. Thousands of crunches won't solve the problem either.
Luckily, one of the best tests and corrections for both mechanical and neurological full-body weakness is the loaded carry. By holding weight and walking for a given time, you can identify your weak links in the kinetic chain. They're what's holding back your ability to get stronger.
Test the loaded carry by holding near maximal loads – the goal is twice your body weight – and walking with them for 30 seconds. Since not many gyms have farmer's carry implements or dumbbells that go above 150 pounds, try snagging a trap bar. It's easier to load.
Deadlift the load off the floor, stabilize it, and walk with control of the shoulders, spine, and hips. Some people won't be able to deadlift twice their bodyweight off the floor in order to initiate the carry. And that's why it's a test, not everyone will pass. But remember, being weak in the deadlift is also indication that your full body capacity is also weak.
Don't test the loaded carry for distance. This may tempt you to go faster. Along with being inherently dangerous, running with the weight to get to a distance faster is also a form of cheating the test. Keep your shoulders, hips, and core in a neutral position while staying under control.