'under the designation' is correct and usable in written English.
You can use it when you want to indicate that something has been assigned a certain label or classification. For example, "All materials approved by the organization must be stored under the designation 'Organization Materials'".
The Enfield-designed pistol was quickly accepted under the designation Revolver, No. 2 Mk I, and was adopted in 1932, followed in 1938 by the Mk I* (spurless hammer, double action only), and finally the Mk I** (simplified for wartime production) in 1942.
In 1959, Colt, a much larger firearms company, bought the design from Stoner's struggling Armalite, and produced it for the military under the designation M-16.
The committee accepted the proposal, calling it the "Type IV" (of five designs), and awarded a production contract to de Havilland under the designation Type 106 in February 1945.
Throughout western Europe, towns acquired various kinds of municipal institutions loosely grouped under the designation commune.
Some forms of threat are occasionally singled out for separate statutory treatment under the designation "blackmail".
A variety of Gnostic sects, such as the Naassenes and Cainites, are included under the designation Ophites.
Construction of the controlled-access highway was scheduled in late 1957 under the designations Interstate 82S and Interstate 80N.
This has become the standard modeling paradigm in computer simulation experiments (cf. 2– 5), known under the designations of Gaussian process (GP) modelling and kriging analysis.
Even under Dodd-Frank, the designation process is a lengthy one involving extensive study and numerous opportunities for a company to question and challenge a designation.
Local officials selected the zones from a group of areas deemed eligible for the designation under a federal formula that factors in income and poverty levels.
Under the Superfund designation, the E.P.A. can compel polluters to pay.
