Compact coding of integer into bitstring
Python, 171
def F(x,m):
 a=0;b=[9]
 while len(b)&lt;m:
    if 4**len(bin(len(b)))*2&gt;64*a&lt;4**len(bin(m)):
     if(a&amp;a+1)**a:b+=[a];a+=1
     else:a*=2
    else:a=b.pop()*2
 return bin((b+[a])[x])[2:]

Note that lines that appear to begin with 4 spaces actually begin with a tab.
Testing, with bitpwner's test script:
         1        2        3        4        5        6        7        8        9        10       11       12       13       14       15      
F(x,1)   0       
F(x,2)   0        1       
F(x,3)   0        10       11      
F(x,4)   0        10       110      111     
F(x,5)   0        10       110      1110     1111    
F(x,6)   0        100      101      110      1110     1111    
F(x,7)   0        100      101      1100     1101     1110     1111    
F(x,8)   0        100      101      110      11100    11101    11110    11111   
F(x,9)   0        100      101      110      11100    11101    11110    111110   111111  
F(x,10)  0        100      101      1100     1101     11100    11101    11110    111110   111111  
F(x,11)  0        100      101      1100     1101     11100    11101    111100   111101   111110   111111  
F(x,12)  0        100      101      1100     11010    11011    11100    11101    111100   111101   111110   111111  
F(x,13)  0        100      101      1100     11010    11011    11100    111010   111011   111100   111101   111110   111111  
F(x,14)  0        100      101      11000    11001    11010    11011    11100    111010   111011   111100   111101   111110   111111  
F(x,15)  0        100      101      11000    11001    11010    11011    111000   111001   111010   111011   111100   111101   111110   111111  

Explanation:
This is all based on the observation that between two consecutive elements of the code, F(x,m) and F(x+1,m), we always add one to the binary number, then multiply by two some number of times. If these steps are followed, then it is a valid code. The rest simply tests to make sure it is short enough.
Golfs:
175 -&gt; 171: changed 2**(2*... to 4**