Can uniqueness of strings (each with an equal number of 1's and 0's) be decided in logspace? Let k be a positive even integer.

Given a list of (k choose k/2) k-sequences, each with an equal number of 1's and 0's, is distinctness decidable in logspace if we're only allowed to read the input once?

In more detail:

Let a sequence have k bits.

Let each sequence have an equal number of 1's and 0's.

There are (k choose k/2) such sequences.

Suppose I have a list of sequences.

Suppose there are (k choose k/2) sequences in the list.

Can we decide each sequence is distinct from the others in logspace?

If we can read the input multiple times, we can use two logspace counters - and ensure each sequence is not equal to the sequences that follow.

What if we're only allowed to read the input once?
Hint: let $n={k \choose k/2}$.  Consider the state of the algorithm after reading $n/2$ of the sequences.  How many possible states are there, that it could be in at that point?