## Recursive constructions I

In the case of puncturing the situation is analogous. The inverse of puncturing is called lengthening and is clearly not always possible. Append a new coordinate, where the entry is chosen such that the new lengthened codeword has even weight. This means: the entry in the last coordinate is 0 if the weight of the old codeword is even, the entry in the last coordinate is 1 if the old weight was odd. Clearly the new code is linear and has the same dimension as the old one. Append a new coordinate, where the entry is chosen such that the new lengthened codeword has even weight. This means: the entry in the last coordinate is 0 if the weight of the old codeword is even, the entry in the last coordinate is 1 if the old weight was odd. Clearly the new code is linear and has the same dimension as the old one.