  Polar coding is a recently proposed coding technique that can provably
achieve the channel capacity. The polar code structure, which is based on the
original 2x2 generator matrix, polarises the channels, i.e., a portion of the
channel capacities approach 1, while the remaining channel capacities approach
0. Due to the specific size of this original generator matrix, polar codes can
only have code lengths equal to the powers of 2, resulting in inefficiency for
codes of practical lengths. In this paper, the performance of finite-length
polar codes over the binary erasure channel is analysed. A normalised
polarisation distance measure is defined and polar codes from different
generator matrices showing different amount of polarisation are compared using
this measure. Encoding structures for these generalised polar codes are
proposed and polarisation performances in both asymptotical and finite-length
cases are investigated for generator matrices of size 3x3 and 4x4. A
generalised decoder is also proposed for this generator matrix and its erasure
rate is compared with that of the original generator matrix. It is shown that
polar codes that have performance similar to the original construction can be
constructed and used for a variety of code lengths, not necessarily equal to
powers of 2, using generalised generator matrices.
