>Grain free does not mean carb free, they simply mostly replace the grain with potato.
Brainless has it, here. Most manufacturers have cottoned onto the fact that people are looking for 'grain free' foods, so they've simply switched their carb source to potato instead of rice. It doesn't mean there is any more meat, necessarily.
An ideal diet for a dog is high in protein, high in saturated fat (this is where energy should come from, not carbs), and low in carbs. Most commercial dog foods are the opposite of this: Mostly carbs, medium in fat and low in protein. Carbs can actually fuel cancer cells, which is why the diet for dogs with cancer is low in carbs.
In terms of a commercial food which is low in carbs (not only low in grain), I'd recommend firstly Ziwipeak. Which is really expensive, but if you have a toy or medium breed it should be possible to feed it - you feed much less of it than you would a food with carbs. Carbs bulk out food. The energy in Ziwipeak comes from fat. So it's not going to look like as much, in a dog's bowl. The biggest bag you can get of it is only 5kg, but that can last as long as a 12kg bag of foods with carbs.
There is also K9 Natural, which is freeze-dried raw. You reconstitute it with water. It looks like an excellent food but you'd need to win the lottery to afford it.
Next to Ziwipeak and K9 Natural, there's Orijen. Orijen does have some carbs from vegetables and potato, but it has a v high protein content and meat is the first 5 ingredients - so definitely not using potato to substitute rice in the usual dog food make-up. Orijen is also affordable, just about. If you join their breeders' club, you get it slightly cheaper too. And often there are 'damaged packaging' discounts in the special offers section on the Bern Pet Foods website.