How to enumerate constants of a certain type

I'd like to ensure with a test, that for each APIErrorCode constant defined as below, the map APIErrorCodeMessages contains an entry. How can I enumerate all constants of a certain type in Go?

// APIErrorCode represents the API error code
type APIErrorCode int

const (
    // APIErrorCodeAuthentication represents an authentication error and corresponds with HTTP 401
    APIErrorCodeAuthentication APIErrorCode = 1000
    // APIErrorCodeInternalError represents an unknown internal error and corresponds with HTTP 500
    APIErrorCodeInternalError APIErrorCode = 1001
)

// APIErrorCodeMessages holds all error messages for APIErrorCodes
var APIErrorCodeMessages = map[APIErrorCode]string{
    APIErrorCodeInternalError: "Internal Error",
}

I've looked into reflect and go/importer and tried tools/cmd/stringer without success.