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If you're talking about Chinese, elementary or primary school should be 小学, not 小学校. 小学校 itself means a small or little school, nothing to do with grades.

Traditionally, Japanese colleges and universities are called 大学. (By contrast, they are called 大学校 in South Korea.)

There are special colleges and universities in Japan called 大学校、most of which are established by national government ministries or agencies or by local governments. Some of these 大学校 are degree granting, others are not.

Examples of 大学校 in Japan are:

防衛大学校　National Defense Academy

気象大学校 Meteorological College

水産大学校　National Fisheries University

税務大学校　National Tax College