Postcodes in Nominatim
The blog post Nominatim and Postcodes describes the handling implemented since Nominatim 3.1.
Postcode centroids (aka 'calculated postcodes') are generated by looking at all postcodes of a country, grouping them and calculating the geometric centroid. There is currently no logic to deal with extreme outliers (typos or other mistakes in OSM data). There is also no check if a postcodes adheres to a country's format, e.g. if Swiss postcodes are 4 digits.
Regular updating calculated postcodes
The script to rerun the calculation is
nominatim refresh --postcodes
and runs once per night on nominatim.openstreetmap.org.
Finding places that share a specific postcode
In the Nominatim database run
SELECT address->'postcode' as pc,
osm_type, osm_id, class, type,
st_x(centroid) as lon, st_y(centroid) as lat
FROM placex
WHERE country_code='fr'
AND upper(trim (both ' ' from address->'postcode')) = '33210';
Alternatively on Overpass run the following query
[out:json][timeout:250];
area["name"="France"]->.boundaryarea;
(
nwr(area.boundaryarea)["addr:postcode"="33210"];
);
out body;
>;
out skel qt;