First of all it is clear that the samsung note 10 series will not get Android 13 but still get security patch updates in 2022. Samsumg provide 3 of Major updates for flagship devices and the Galaxy Note 10 series get Android 10,11,12. So, it is clear that there is no any chance that Samsung galaxy note 10 series will get Android 13.
Samsung Android 13 Supported Galaxy Devices List:
1) Galaxy S series:
Galaxy S21 Ultra (LTE/5G)
Galaxy S21+ (LTE/5G)
Galaxy S21 (LTE/5G)
Galaxy S20+ (LTE/5G)
Galaxy S20 (LTE/5G)
Galaxy S20 Ultra (LTE/5G)
2) Galaxy Note series:
Galaxy Note 20 Ultra (LTE/5G)
Galaxy Note 20 (LTE/5G)
Galaxy Note 10 Lite
3) Galaxy A series:
Galaxy A12 Nacho
Galaxy A03
4) Galaxy Z series:
Samsung Android 13 update devices list: 
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