The comments point to a question about creating sub-types, which is basically a domain within another domain. This would allow you to restrict the code field so that only one value was valid for a given value in the name field, but at best that means you can pick it from a dropdown - it will not fill it in automatically. As discussed at Coded Geodatabase Domain Values behaving oddly with special characters in the description field domains are only enforced at time of validation. It also means that with 700 values, you'd need 700 sub-types. That is not an efficient way to do this.
Basically you want two fields to represent the same conceptual value. There are only two ways I know of to do this automatically. One is create a Feature Template, which stores what all or some of the attributes for a new feature should be. But again, you're looking at 700 feature templates, so not really practical. The other way involves what are known as triggers and may or may not be available depending on your underlying database. Another question here, Automatically update field value in one column based on change to another column in same table? talks about their use.
You could also create a lookup table of name/code, and then use a script to Join Field (or update) the code based on name. However that doesn't happen as you edit - it's something that must be run on the table after it's been edited, either on some regular basis or manually. (The script could also be run as the result of a trigger.)