DFA for a set {1,2,4,8,..,2^n} written in binary

Question

I have to construct a minimal Deterministic Finite State Automaton(DFA) for a set :-

L = {1,2,2 ² ,2 ³ ,2 ⁴ ....,2 ⁿ } where numbers 1,2,2 ² ,2 ³ ,2 ⁴ ....,2 ⁿ written in binary and Input alphabet = {0,1} and 'n' is a non-negative integer.

I have constructed a minimal DFA as :-

Regular Expression(RegEx) for this DFA should be :- 10* because Language contains strings {1,10,100,1000,....}

Now, I am confused whether should I have to consider a unique representation for every string of the given set(Language) or not.
Here , string '1' has infinitely many representations like {1,01,001,0001,.....} Similarly, I can get many representations for every string of the set by appending zeros initially.
So, If I consider it then I am getting RegEx as 0*10*.
According to this RegEx, my equivalent minimal DFA will be :-

So, My question is : which DFA is correct according to the given description of the set which I have written initially. Should I have to mention that all strings should be started with '1' or every string has unique representation for the precise description of the language so that I can get the unique minimal DFA for the given set.
Please help!

Answer 1

我将使用第二个数字，因为它更灵活，如果您需要将数字存储在内存字段大于您的数字（位数）的情况下，则需要添加初始零以进行存储它。