Conn. Gen. Stat. § 26-249: Private ownership in cultivated shellfish.

Each person who plants or cultivates oysters, clams or mussels, in any place lawfully designated, shall own them and also all other oysters, clams or mussels on such place, and have the exclusive right of taking up and disposing of them and of using such place for the purpose of planting or cultivating oysters, clams or mussels therein, which place shall be transferable by written assignment; but nothing herein contained shall affect the rights of any owner of lands in which there may be saltwater creeks or inlets, or which may be opposite or contiguous to such navigable waters; nor authorize any shellfish commission or selectmen to designate, nor any person to mark, stake out or enclose, any natural oyster or clam bed, or infringe upon the free navigation of such waters, or interfere with the drawing of seines in any place established and customarily used for seine fishing.