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The GPS position displayed on a map with NAD27 State Plane coordinates is incorrect by 40 feet or more in both the north/south and east/west directions.
You have already confirmed that the GPS Datum setting in the GPS page of the ArcPad Options dialog box is correct. Note that this setting refers to the datum of the coordinates being output by the GPS receiver, not the datum of the map. This is usually WGS84.
The NAD27 State Plane map was probably created from WGS84 or NAD83 data and transformed using the NADCON grid-based transformation method.
ArcPad does not support grid-based datum transformation methods. Only equation-based datum transformation methods are supported. As a result, the incoming GPS coordinates are being transformed to NAD27 using a different method than was used to transform the original data.
Transform the NAD27 shapefiles to WGS84 or NAD83 using the projection tools for ArcView GIS 3.x or ArcGIS. Use these transformed shapefiles in ArcPad with GPS to collect new data or modify existing data. Then transform the shapefiles back to NAD27 using the projection tools again.