How to encrypt numeric data on a ticket
Given that we can add only 4 digits (13.3 bits) of redundancy, there is no way to use asymmetric cryptography. We'll have to use a secret key in the verification terminals.

This is an application for Format Preserving Encryption. It can act like a block cipher, except we can decide the block width, and deal with decimal rather than binary or hexadecimal digits. Methods for FPE are given in NIST&nbsp;SP&nbsp;800-38G. The previous link leads to references with other methods.

Take the 16-digit plaintext, append four zeroes, yielding 20 digits. Encipher that using FPE, yielding 20 digits, put that on the ticket.

On reading a ticket, check that it is 20 digits, decipher per FPE (yielding 20 digits), and consider the result valid when it ends with four zeroes; remove these.