You must submit your assignments by next Monday at the latest.
Is "next" optional in the above sentence? I think by "by Monday" should be clear enough and "next" is redundant.
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It is impossible to say whether you can remove next from next Monday without additional context. It may make no difference, but it may make no difference because both are equally weak.
Next Monday itself can be ambiguous, as to whether it is referring to Monday next week or the week after that. Monday alone would pose the same problem if you are at a point in the week where next Monday and this Monday would differ, and adds the possibility that Monday or this Monday refers to this past Monday rather than this coming Monday.
Even further, if you're referring to something occurring over multiple weeks, you must also distinguish between references to fixed dates and references to points in the cycle, in which case it might not just be this Monday or next Monday, but the next Monday.
You don't indicate why you want the shorter form, but do remember that language is not software, or logic. Redundancy is neither inherently erroneous nor inefficient. The language is full of idiomatic pleonasms from advance warning to water hydrant that some people may dislike, but which most others accept without any thought.
This is something that confuses native English speakers as it is often used in two different ways. Whenever someone tells me something like "next Monday" I always have to ask if they are referring to this-coming Monday (i.e. the closest Monday) or the Monday following the closest Monday. This is because some people think the closest Monday is the so-called "this Monday" and the following is the so-called "next Monday".
Anyhow, to answer your question, yes, It's optional.
As noted in other answers, there is potential for ambiguity in the meaning of next. Hence, it is usually best when dealing with important matters, and especially in written communications, to use a specific date.
When speaking I might say
Your assignments must be submitted by next Monday, the 23rd July.
making my meaning clear. In writing I would use the slightly more formal:
Your assignments must be submitted by Monday 23rd July.