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.

NbinKbinRateGFNsymKsymnon-binary matrixbinary image
96480.5GF(64)168alist or matrixalist or matrix
96480.5GF(256)126alist or matrixalist or matrix
128640.5GF(256)168alist or matrixalist or matrix
128*640.5GF(256)168alist or matrixalist or matrix
5122560.5GF(256)6432alist or matrixalist or matrix
5762880.5GF(64)9648alist or matrixalist or matrix
5762880.5GF(256)7236alist or matrixalist or matrix
5764800.83GF(64)9680alist or matrixalist or matrix
5764800.83GF(256)7260alist or matrixalist or matrix
230411520.5GF(64)384192alist or matrixalist or matrix


* 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.


[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