Mary,

I think the confusion is that Mathematica "strings" involve text strings as Sander demonstrated. I believe that your "strings" are lists of binary digits (and not Mathematica strings). If this is correct, This should work for you:

In[2]:= list = Range[15]
Out[2]= {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}

IntegerDigits converts my list of numbers to 4 digit binary "Strings" in Mathematica -- these are lists of digits

In[3]:= mylist = Map[IntegerDigits[#,2,4]&,list]
Out[3]= {{0,0,0,1},{0,0,1,0},{0,0,1,1},{0,1,0,0},{0,1,0,1},{0,1,1,0},{0,1,1,1},{1,0,0,0},{1,0,0,1},{1,0,1,0},{1,0,1,1},{1,1,0,0},{1,1,0,1},{1,1,1,0},{1,1,1,1}}

Make a function that picks out the lists with a certain number of 1's

In[5]:= SelectHamming[str_List,n_Integer]:=Select[str,Count[#,1]==n&]
In[6]:= SelectHamming[mylist,2]
Out[6]= {{0,0,1,1},{0,1,0,1},{0,1,1,0},{1,0,0,1},{1,0,1,0},{1,1,0,0}}

I hope this helps

The correct interpretation was

SelectHamming

I am very grateful for your help, I didn't think of simply counting the 1's, I was stuck trying to make it work with the HammingDistance since that is what I found when looking for "hamming weight". The bulk of the workaround (generating a list of 0's of length 2^(2^n) and calculating the hamming distance to that) meant that I ran out of memory already with functions on 6 variables (strings of length 32) Beginners mistake!

I am working on bent functions, (maximally nonlinear boolean functions) and it gets unwieldy pretty fast. I narrowed down the list where interesting functions lived with a little program called spaceF, and I needed to feed subsets of that space with hamming weight n into another program.

Mary

Thanks, but I still can't tweak it to work with my input.

SpaceF takes an integer and returns a list of binary strings like say [{0,0,1,1}, {0,1,1,1},{1,0,0,0}]. What I really want is a program which works with SpaceF so that I can generate a sublist of all entries with hamming weight n.

Mary

Or, may be this is the solution:

listofstrings = {"hello", "test", "apple", "banana", "world", "soap", "man", "an", "Hamming", "dodecahedron", "Wolfram Community",  "Wolfram Language", "Paintings"};
SelectHammingWeight[x_List, z_String, n_Integer] := Select[StringJoin /@ StringSplit[x, z], StringLength[#] == n &]
Manipulate[SelectHammingWeight[listofstrings, "a", n], {n, 1, 16, 1}]