Practical polar code construction using generalised generator matrices

Abstract

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, 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 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.

Simplified successive‐cancellation decoding using information set reselection for polar codes with arbitrary blocklength

“…However, most of the previous studies on SC decoder simplification were focused on the canonical N = 2 n case. As polar codes are in fact a general family of codes with a broad range of block‐length N = l n and polar codes with l > 2 are believed to have the potential to achieve better performance (see [14, 15]), it is meaningful to extend such simplification to the cases of l > 2. Such extension is by no means trivial since that, for the cases of l > 2, unlike the canonical case, there usually exist multiple generator matrices [16] for which no explicit or determined forms could be obtained so far.…”

Polar codes Bhattacharyya parameter generalisation

“…In this paper, a generalised technique for constructing polar codes for binary input channels is proposed. Our construction differs from previous works [11,12], since it depends on the Bhattacharyya parameter bounds. First, we deduce a generalised upper and lower bound of Bhattacharyya parameter.…”

Design of Rate-Compatible Polar Codes Based on Non-Uniform Channel Polarization

“…BCH kernel matrices proposed in [17] and the code decompositions proposed in [18] both have restrictions on the size of the kernel matrices. Square polarizing kernels larger than two have been proposed in [19], [20] and [21], while a PC construction with mixed kernel sizes has been proposed in [22] [23]. By considering different polarizing kernels of alternate dimensions, MK improves block length flexibility.…”