Matices

NB-LDPC matrices and simulation results

Several NB-LDPC matrices are proposed in order to evaluate and compare the performance of NB-LDPC decoder. Matrices are described using the 'alist' format . The 'alist' format is modified for NB matrices using the q-ary 'alist' format .

The matrix element are given as the power of the primitive element. GF(64) constructed using the irreductive polynomial Q[X] = X⁶ + X + 1. Similarly, GF(256) is defined thanks to the irreducible polynomial Q[X] = X⁸ + X⁴ + X³ + X² + 1. The binary representation of the finite field can be obtained using the following matlab function build_GF.m

NB-LDPC matrices on GF(64)

NB-LDPC matrices on GF(64)
N_bin	K_bin	Rate	GF	N_symb	K_symb	d_v	d_c	Girth	#g*	#g+2	#g+4	d_min	alist	Author	Simulation results
96	48	1/2	64	16	8	2	4	8	36	0	0	11	alist1	[KL]	[ML]	4 Bubbles
576	288	1/2	64	96	48	2	4	8	4	24	20	?	alist1	[KL]	4 Bubbles
576	480	0.83	64	96	80	2	12	6	256	2136	16128	6	alist1	[KL]	[ML]	4 Bubbles	8 Bubbles	min-max
864	720	0.83	64	144	120	2	12	?	?	?	?	?	alist1	[UBS]	4 Bubbles
2304	1152	1/2	64	384	192	2	4	16	1164	5856	0	?	alist1	[KL]	4 Bubbles
9216	8064	0.83	64	1536	1344	2	16	?	?	?	?	?	alist1	[UBS]	4 Bubbles

NB-LDPC matrices on GF(256)

NB-LDPC matrices on GF(256)
N_bin	K_bin	Rate	GF	N_symb	K_symb	d_v	d_c	Girth*, #g, #g+2, #g+4	d_min	alist	Author	Simulation results
96	48	1/2	256	12	6	2	4	6, 8, 15, 24	?	alist1	[KL]	4 Bubbles
128	64	1/2	256	16	8	2	4	8, 36, 0, 0	13	alist1	[KL]	[ML]	4 Bubbles
128	64	1/2	256	16	8	2	4	8, 36, 0, 0	14	alist1	[U-NBPB]	4 Bubbles	8 Bubbles
256	128	1/2	256	32	16	2	4	?	?	alist1	[U-NBPB]	8 Bubbles
512	256	1/2	256	64	32	2	4	?	?	alist1	[U-NBPB]	8 Bubbles
512	256	1/2	256	64	32	2	4	8, 16, 0, 0	?	alist1	[KL]	4 Bubbles
576	480	0.83	256	72	60	2	12	4, 18, 255, 1809	?	alist1	[KL]	4 Bubbles	8 Bubbles	min-max
1152	960	0.83	64	144	120	2	12	?	?	?	?	?	alist1	[UBS]	4 Bubbles

* #g is the number of cycles of length g and #g+2 is the number of cycles of length g+2. These values are obtained using [1].
[KL] These LDPC codes have been designed at the university of Kaiserslautern using optimized rows and optimized short loops according to [2] and [3].
[ML] Helmling, Michael and Scholl, Stefan: Database of Channel Codes and ML Simulation Results. University of Kaiserslautern, 2015. URL: www.uni-kl.de/channel-codes
[UBS] These LDPC codes have been designed At the University of South Brittany.
[U-NBPB] Unconstrained Non-binary Protograph-Based LDPC Codes [4].
[1] Halford, T.R.; Chugg, K.M., "An algorithm for counting short cycles in bipartite graphs," in Information Theory, IEEE Transactions on , vol.52, no.1, pp.287-292, Jan. 2006 .
[2] Stefan Scholl: Entwurf und Decodierung nicht-binärer Low-Density-Parity-Check-Codes. Diploma thesis, University of Kaiserslautern, Mar. 2010.
[3] 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.
[4] L. Dolecek, D. Divsalar, Y. Sun and B. Amiri, "Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs," in IEEE Transactions on Information Theory, July 2014.