Trade secrets and patents are very different intellectual properties. You can’t have both on the same invention. A patent gives the owner the right to bar anyone from making, using or selling the subject of the claims in the region where the patents are registered. The patent becomes a public document and, therefore, cannot be a trade secret.
Trades secrets do not get published or registered by any agency. Protection of trade secrets goes back to Common Law of centuries ago. Most states recognize their existence and some have legislation directly related to them. The Uniform Trade Secrets Act (UTSA) has created some uniformity in the laws among the states.