Practical polar code construction using generalised generator matrices

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 2 × 2 generator matrix, polarises the channels, that is, a portion of the channel capacities approach 1, whereas the remaining channel capacities approach 0. Owing 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 study, 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 3 × 3 and 4 × 4. 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.