Generalized Binary Representation for the Nonbinary LDPC Code With Decoder Design

Abstract

In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e.,C. A new methodology for constructing the binary representation [generalized binary representation (GBR)] ofC is proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother codeC under nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.Index Terms-Non-binary LDPC code, binary image, binary Gaussian channel, binary symmetric channel.

Decoding of NB-LDPC codes over Subfields

“…Note that the sixteen vectors in Table II 11 α, α 9 , α 10 , α 12 0 : 00000 α 3 : to derive the remaining three. The dependancies could easily be captured through the parity-check equations of the code.…”

Decoding of NB-LDPC codes over Subfields

“…In channel coding terms, they are the 16 codewords of a (2, 5) linear code over F 2 2 . Thus, values of some γ ∈ F 2 4 with respect to 2 11 1, α 4 , α 12 , α 13 α 2 , α 3 , α 5 , α 9 α 7 , α 8 , α 10 , α 14 Q1 0, 1, α 5 , α 10 α, α 2 , α 4 , α 8 α 6 , α 7 , α 9 , α 13 α 3 , α 11 , α 12 , α 14 Q2 0, α 4 , α 9 , α 14 1, α, α 3 , α 7 α 5 , α 6 , α 8 , α 12 α 2 , α 10 , α 11 , α 13 Q3 0, α 2 , α 7 , α 12 1, α 8 , α 9 , α 11 α, α 5 , α 13 , α 14 α 3 , α 4 , α 6 , α 10 Q4 0, α 3 , α 8 , α 13 1, α 2 , α 6 , α 14 α 4 , α 5 , α 7 , α 11 α, α 9 , α 10 , α 12 0 : 00000 α 3 : ωω 2 1ω 2 0 α 7 : ω 2 ω10ω α 11 : 0ω 2 ω 2 1ω 1 : 10111 α 4 : 110ω 2 ω α 8 : ω 2 1ω10 α 12 : 1ω 2 ω0ω 2 α : 011ωω 2 α 5 : ω0ωωω α 9 : ωω01ω 2 α 13 : 1ωω 2 ω0 α 2 : ω1ω 2 01 α 6 : 0ωωω 2 1 α 10 : ω 2 0ω 2 ω 2 ω 2 α 14 : ω 2 ω 2 0ω1 to derive the remaining three. The dependancies could easily be captured through the parity-check equations of the code.…”

“…Additionally, a decoding algorithm for NB-LDPC codes over the binary erasure channel was introduced in [10]. This strategy is adapted to general channels, as discussed in [11]. In [12], the authors used the binary image of the non-binary PCM to decode NB-LDPC codes.…”

Decoding of NB-LDPC codes over Subfields

“…Note that a position i in these vectors map to quotient group Q i as given in Table I. 11 1, α 4 , α 12 , α 13 α 2 , α 3 , α 5 , α 9 α 7 , α 8 , α 10 , α 14 Q 1 0, 1, α 5 , α 10 α, α 2 , α 4 , α 8 α 6 , α 7 , α 9 , α 13 α 3 , α 11 , α 12 , α 14 Q 2 0, α 4 , α 9 , α 14 1, α, α 3 , α 7 α 5 , α 6 , α 8 , α 12 α 2 , α 10 , α 11 , α 13 Q 3 0, α 2 , α 7 , α 12 1, α 8 , α 9 , α 11 α, α 5 , α 13 , α 14 α 3 , α 4 , α 6 , α 10 Q 4 0, α 3 , α 8 , α 13 1, α 2 , α 6 , α 14 α 4 , α 5 , α 7 , α 11 α, α 9 , α 10 , α 12 Note that the sixteen vectors in Table II form a 2dimensional space over F 2 2 . In channel coding terms, they are the 16 codewords of a (2, 5) linear code over F 2 2 .…”

“…Additionally, a decoding algorithm for NB-LDPC codes over the binary erasure channel was introduced in [10]. This strategy is adapted to general channels, as discussed in [11]. In [12], the authors used the binary image of the nonbinary PCM to decode NB-LDPC codes.…”