Non-Binary LDPC Codes
The following non-binary LDPC codes have been designed using optimized rows and optimized short loops according to [1] and [2]. For codes over GF(q) with q≥64 regular codes can be used with degree distribution of (dv,dc) = (2,4) for rate = 1/2 and (2,12) for rate = 5/6.
| Nbin | Kbin | Rate | GF | Nsym | Ksym | non-binary matrix | binary image |
|---|---|---|---|---|---|---|---|
| 96 | 48 | 0.5 | GF(64) | 16 | 8 | alist or matrix | alist or matrix |
| 96 | 48 | 0.5 | GF(256) | 12 | 6 | alist or matrix | alist or matrix |
| 128 | 64 | 0.5 | GF(256) | 16 | 8 | alist or matrix | alist or matrix |
| 128* | 64 | 0.5 | GF(256) | 16 | 8 | alist or matrix | alist or matrix |
| 512 | 256 | 0.5 | GF(256) | 64 | 32 | alist or matrix | alist or matrix |
| 576 | 288 | 0.5 | GF(64) | 96 | 48 | alist or matrix | alist or matrix |
| 576 | 288 | 0.5 | GF(256) | 72 | 36 | alist or matrix | alist or matrix |
| 576 | 480 | 0.83 | GF(64) | 96 | 80 | alist or matrix | alist or matrix |
| 576 | 480 | 0.83 | GF(256) | 72 | 60 | alist or matrix | alist or matrix |
| 2304 | 1152 | 0.5 | GF(64) | 384 | 192 | alist or matrix | alist or matrix |
| 64800 | 48600 | 0.75 | GF(256) | 8100 | 6075 | alist | alist |
| 64800 | 51840 | 0.8 | GF(256) | 8100 | 6480 | alist | alist |
* U-NBPB Code from [3]
GF Matrix Elements
The matrix elements are given as the power of the primitive element α (i.e. -inf or -1, 0, 1, ..., q-2). Note that the zero element has an exponent of -infinity. For better readability the exponent -inf is replaced by -1 in the format (not to be mixed up with the inverse of α). The primitive polynomials used to construct the fields and the binary images are:
GF(64) 1+x+x6
GF(256) 1+x2+x3+x4+x8
Binary Image Transformation
The binary image expansion can be used to transform the non binary matrix into a binary one. All non-binary elements are substituted by a qxq binary matrix. These q qxq matrices are constucted using a companion matrix, see [4] for further information.
Literature
[1] Stefan Scholl: Entwurf und Decodierung nicht-binärer Low-Density-Parity-Check-Codes. Diploma thesis, RPTU Kaiserslautern-Landau, Mar. 2010
[2] C. Poulliat, M. Fossorier and D. Declercq: Design of regular (2,dc)-LDPC codes over GF(q) using their binary images. IEEE Transactions on Communications, Oct. 2008
[3] B.-Y. Chang, D. Divsalar and L. Dolecek: Non-binary Protograph-Based LDPC Codes for Short Block-Lengths. IEEE Information Theory Workshop, 2012
[4] R. Horn and C. Johnson: Matrix Analysis. Cambridge University Press, Cambrigde, 1985