Transforming input text into code

This looks like a case where an Association[] is more appropriate:

as = AssociationThread @@ Transpose[{{"l", {1, 0}}, {"a", {1, 1}}, {"j", {0, 1, 0}},
                                     {"u", {0, 1, 1}}, {"e", {0, 0}}}]

Then you can do this:

StringJoin[Map[IntegerString, Lookup[as, Characters["lea"]], {2}]]
   "100011"

ClearAll[code] 
code = StringReplace[# -> StringRiffle[#2, ""] & @@@ #]&;

Example:

list = {{"l", {1, 0}}, {"a", {1, 1}}, {"j", {0, 1, 0}}, {"u", {0, 1, 1}}, {"e", {0, 0}}};

code[list]["lea"]

"100011"

I like J.M.'s solution. But since there is in Mathematica at least 10 ways to do the same thing, I thought to have a go at it

code = {{"l", {1, 0}}, {"a", {1, 1}}, {"j", {0, 1, 0}}, {"u", {0, 1, 1}}, {"e", {0, 0}}};
word = "ltea";
let = Characters[word];

StringJoin@Flatten[Cases[code, {x_, y__} /; 
       StringMatchQ[x, #] :> (ToString /@ y)] & /@ let]