early next week; beginning of next week
Refers to the early part of the upcoming week, typically Monday or Tuesday. Used in scheduling and planning contexts.
Let's schedule the meeting for early next week.
The results are expected to come out at the beginning of next week.
来週 means 'next week' as a whole, whereas 来週明け narrows it down to the early part of next week.
Compound of 来週 (next week) and 明け (opening, beginning).