05AB1E, score = 1944 (9 bytes · 216)

žBœε3yJ}J

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How it works

žBœε3yJ}J
žB          push 1024
  œ         permutations: ["1024", "1042", "1204", …, "4201"]
   ε   }    map over elements y:
    3       push 3
     y      push y again
      J     concatenate the stack
        J   concatenate

Pyth, score = 2160 (10 bytes · 216)

sm+d+4d.p4

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How it works

 m     .p4   map over all permutations d of [0, 1, 2, 3]:
  +d+4d        d + [4] + d
s            concatenate