Generate a Universal-binary-function Lookup Table This is tangentially related to my quest to invent an esoteric programming language.

A table of the binary numbers 0 .. 15 can be used to implement a Universal Binary Function using indexing operations. Given two 1-bit inputs X and Y, all 16 possible functions can be encoded in a 4-bit opcode.

X Y  F|0 1 2 3 4 5 6 7 8 9 A B C D E F
- -    - - - - - - - - - - - - - - - -  
0 0    0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1    0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 0    0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
1 1    0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
       -     -   - -   -   -   -   - -
       0    ~X  ~Y ^   &amp;   Y   X   | 1
      ZERO    NOT-Y   AND         OR
          NOT-X   XOR              ONE


So this set of 16 functions can be applied to binary inputs as the function  


  U(f,x,y): (f >> ((x&lt;&lt;1) | y)) &amp; 1,   


or  


  U(f,x,y): (f / 2^(x&times;2 + y)) % 2,  


or with indexing or matrix partitioning.

It will be useful to know the most compact way to represent or generate such a table of values for any possible languages to be built upon this type of binary operation.

The goal:

Generate this exact text output:

0101010101010101
0011001100110011
0000111100001111
0000000011111111


That's it! Shortest-code wins.
J, 10 (13?) characters

|.|:#:i.16


Number list:

   i.16
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15`


to binary:

   #:i.16
0 0 0 0
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


Transpose:

   |:#:i.16
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1


Reverse:

   |.|:#:i.16
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Do we need to remove the spaces? Looking at the other J answer it seems we do so we'll need to add 3 characters and borrow the 1": from Jan's answer.