N.J. Stat. § 19:5-1
A political party may nominate candidates for public office at primary elections provided for in this Title, elect committees for the party within the State, county or municipality, as the case may be, and in every other respect may exercise the rights and shall be subject to the restrictions herein provided for political parties; except that no political party which fails to poll at any primary election for a general election at least ten per centum (10%) of the votes cast in the State for members of the General Assembly at the next preceding general election, held for the election of all of the members of the General Assembly, shall be entitled to have a party column on the official ballot at the general election for which the primary election has been held. In such case the names of the candidates so nominated at the primary election shall be printed in the column or columns designated "Nomination by Petition" on the official ballot under the respective titles of office for which the nominations have been made, followed by the designation of the political party of which the candidates are members.
N.J.S. § 19:5-1