Soft and Hard Decision Decoding Performance

Single parity check node adapted to polar codes with dynamic frozen bit equivalent to binary linear block codes

“…In practice, a linear code is thus represented by a matrix 𝐺, called a generator matrix, of size 𝑘 * 𝑛 with 𝑘 < 𝑛, which takes words from 𝔽 𝑞 𝑘 and returns codewords from 𝔽 𝑞 𝑛 . The coding is done by partitioning the original message into blocks of 𝑘 bits, then multiplying each block by the matrix 𝐺, resulting in codewords of 𝑛 bits as (1).…”

“…The coding process consists in constructing a vector 𝑢 containing 𝑛 − 𝑘 frozen bits set to 0 and 𝑘 information bits, it is constructed in such a way that the information bits are situated on the positions {𝑖 ∉ ℱ} of the most reliable channels, and the frozen bits in the bad channels indices {𝑖 ∈ ℱ}. The corresponding codeword can then be computed similarly to (1) as 𝑐 𝑝 = 𝑢 * 𝐺 𝑝 .…”

“…It consists of encoding the information by adding redundancy by the encoder in order to obtain a codeword, and the decoder uses this redundancy to rectify the transmission errors due to the channel noise. Generally, two types of decoding algorithms exist [1]. The first is hard decision decoding, which manipulates the received codeword in binary form.…”