Possible Cause

ABS/TCS control module malfunction.
Open circuit between ABS/TCS control module terminal "J" and PCM terminal "3W". See Fig 1 .
Short to ground circuit between ABS/TCS control module terminal "J" and PCM terminal "3W".
Short to power circuit between ABS/TCS control module terminal "J" and PCM terminal "3W".
Poor connection at ABS/TCS control module or PCM connector.
PCM malfunction.

Fig 1: Identifying DTC P1628: PCM - ABS/TCS Control Module Line Communication Error, Circuit & Connector Terminals

Courtesy of MAZDA MOTORS CORP.