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Index the binary strings in lexicographic order, so that the empty string " " corresponds to 0, the string "0" corresponds to 1, the string "1" to 2, and so on.

I want a function that, given a natural number $n\geq0$, returns the binary string with lexicographic index $n$.

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up vote 2 down vote accepted

Try this function

bs[n_] := StringDrop[IntegerString[n + 1, 2], 1];

Table[{n, bs[n]}, {n, 0, 10}] // Grid
0      ""
1     "0"
2     "1"
3    "00"
4    "01"
5    "10"
6    "11"
7   "000"
8   "001"
9   "010"
10  "011"
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Here's what I came up with:

encode[x_Integer?NonNegative] := StringJoin @@ ToString /@ Rest@IntegerDigits[x + 1, 2]

Test it:

In[7]:= {encode[#], #} & /@ Range[0, 6] // InputForm

Out[7]//InputForm= {{"", 0}, {"0", 1}, {"1", 2}, {"00", 3}, {"01", 4}, {"10", 5}, {"11", 6}}
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