Koi with a red head patch are called "Tancho." Most common are "Tancho Kohaku (all-white Koi with Tancho)," "Tancho Sanshoku (white Koi with Sumi similar to Shiro Bekko, and with Tancho)," and "Tancho Showa (Showa Sanshoku without red markings except for Tancho)," etc. However, "Tancho Goshiki (Koi of five colors with Tancho)," and "Tancho Hariwake" are rare.
Tancho do not form a single, independent kind of Nishikigoi; they all can be bred from Kohaku, Taisho Sankshoku or Showa Sanshoku. Their red patch happen to show up only in the head region. Tancho, therefore, can not be produced in bulk even if you so wish.
The essential point for appreciation is the red patch in the head region, of course. The red head patch sitting right at the center of the head region is the best. The white skin is also important as it is the milky white color that sets the red head patch off to advantage. The Sumi of Tancho Sanshoku and Tancho Showa are the same as Bekko and Shiro Utsuri respectively.