The Department of Agriculture shall administer the registration of Connecticut-bred horses and the breeding records of stallions standing in this state. To qualify as a Connecticut-bred horse, the dam of such Connecticut horse shall have dropped such foal in this state and the registration certificate of such foal issued by the appropriate official breed registry shall be filed with the Department of Agriculture. To qualify as a sire of a Connecticut-bred award winner, the stallion shall have been based in the state at the time of the conception of such foal, shall have been registered with the Department of Agriculture and shall have stood the entire stud season within the state.