What Are Secure Limits to Truncate the Authentication Tag (Mac) for Short Messages?
Cryptographic strength of an authentication tag strongly depends on the tag size and you shouldn't use a tag less than 64-bits for any application. 64-bits is really the minimum you should use. 

Such a short tag is acceptable for online protocols, where the intent is just to authenticate a packet that has been received. Since 64-bits will not be collusion resistant, previous tags will be reused before ~2^30 messages. To mitigate this, you might need to rotate key frequently. 

You can consider using SipHash function which is optimized for short inputs and has an output of 64-bits. 

https://eprint.iacr.org/2012/351