The USS Kobayashi Maru (ECS-1022) was a Federation starship that served as a component of the Kobayashi Maru simulation, a no-win-scenario at the Starfleet Academy. In the scenario, the vessel was under attack by several Klingon warbirds. (Star Trek)
John Eaves and Alex Jaeger designed the Kobayashi Maru for Star Trek. They based the design on Roger Sorensen's blueprints from 1983 of the prime universe class 3 neutronic fuel carrier Kobayashi Maru  and its image on the cover of the novel The Kobayashi Maru,  which in turn, was a depiction of a Tritium-class starship (β), originally designed by Rick Sternbach for the Spaceflight Chronology. (p. 130)
In the opinion of Industrial Light & Magic art director Alex Jaeger, the starship Kobayashi Maru had a similar design to the USS Excelsior (β) that was planned to feature in the film Star Trek but did not make the transition to the screen (although it was essentially a dark-colored variant of the Armstrong-type). Jaeger specifically commented about the Kobayashi Maru, "It resembled the Excelsior, but was more of a cargo version." (Cinefex, No. 118, p. 60; Star Trek - The Art of the Film, p. 59)
According to the virtual collectible card battle game Star Trek: Rivals, card #41, the USS Kobayashi Maru has the registration NCC-S3700. The CGI model of this ship made an appearance in the Star Trek Encyclopedia (4th ed., vol. 1, p. 434) where the registry and name could barely be made out. The registry appears in the entry for this ship on the same page and in the starship list on p. 331 of volume 2.