Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Abstract

Abstract-Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of timeinvariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework.Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the -mostly moderate -decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.

The Construction and Viterbi Decoding of New (2k, k, l) Convolutional Codes

“…First, there are 2 k (l+1) branches in the trellis of (2k, k, l) convolutional codes. A branch has a codeword and so we can obtain a 2 k (l+1) ×1 matrix in i 0 i 1 …i l-1 i l ascending sort by 2 k -ary: C=[C 00...00 C 00...01 … C 00...0K C 00...10 … C KK...KK ] T (6) where C 00...00~C00...0K corresponds to the 2 k branches of state node S 00…0 , that can be derived from (3). Each element is 1×2k vector in (6), which can be converted into 2 k (l+1) ×2k matrix: 3 1 2 7 5 3 17 13 4 27 31 5 75 53 Since C is a constant matrix that can be obtained and converted to a bipolar code beforehand and can be stored in a "code word generator."…”

“…The first task is to verify the validity of the max module, and to determine a suitable memory depth. Take the (6,3,3) convolutional code as an example where 10 different memory depths are selected based on integer times of kl=9. When the Eb/No is respectively1.5db, 2db, and 2.5db, the BER is shown with or without the max module separately in Fig.…”

“…6. It can be seen that, except for (6,3,1), the other (2k, k, l) convolutional codes have different degrees of advantage over (2, 1, l) convolutional codes. …”

“…An RSC is a system code with the distance characteristics of a non-systematic code. Currently, convolutional LDPC codes [6,7] have become the new hot topic, as they can obtain a good cost performance when implementing the Belief Propagation (BP) algorithm. In addition, the quantum convolutional codes used in quantum communication have also become absorbing quantum error correcting codes [8].…”

“…However, even if k is not too large, as a result of the multiplier effect, the growth of memory length of the (2k, k, l) convolutional codes is more significant than those of the (2, 1, l) convolutional codes. So, we will pick out (6,3) and (8,4), which are two double loop cyclic codes, as embedded codes in [10] to construct (6, 3, l) and (8, 4, l …”