Convolutional cylinder-type block-circulant cycle codes

Document Type: Research Paper

Authors

1 Shahrekord University

2 Isfahan Mathematics House

Abstract

In this paper‎, ‎we consider a class of column-weight two‎ ‎quasi-cyclic low-density parity check codes in which the girth can be large enough‎, ‎as an‎ ‎arbitrary multiple of 8‎. ‎Then we devote a convolutional form to these codes‎, ‎such that‎ ‎their generator matrix can be obtained by elementary row and‎ ‎column operations on the parity-check matrix‎. ‎Finally‎, ‎we show that the free distance of the convolutional codes is equal to the minimum distance of‎
‎their block counterparts‎.

Keywords

Main Subjects


R. G. Gallager (1962). Low-density parity-check codes. IRE Trans.. IT-8 (1), 21-28
D. J. C. MacKey (1999). Good error-correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory. 45 (2), 399-432
S. Y. Chung, G. D. Forney, T. J. Richardson and R. L. Urbanke (2001). On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit. IEEE Commun. Lett.. 5 (2), 58-60
F. R. Kschischang, B. J. Frey and H. -A. Loeliger (2001). Factor graphs and the sum-product algorithm. IEEE Trans. on Inf. Theory. 47 (2), 498-519
T. D. Souza (2005). Cycle Codes. EPFL / ALGO-LMA, SSC 6th semester.
Y. Levy and D. J. Costello (1993). An algebraic approach to constructing convolutional codes from quasi-cyclic codes. DIMACS Series in Discr. Math. and Theor. Comp. Sci.. 14, 189-198
R. M. Tanner, D. Sridhara, A. Sridharan, T. E. Fuja and D. J. Costello (2004). LDPC block and convolutional codes based on circulant matrices. IEEE Trans. Inform. Theory. 50 (12), 2966-2984
R. M. Tanner (1987). Convolutional codes from quasi-cyclic codes: A link between the theories of block and convolutional codes. Computer Research Laboratory, Technical Report, USC-CRL-87-21.
H. Song, J. Liu and B. V. K. V. Kumar (2002). Low complexity LDPC codes for magnetic recording. in IEEE Globecom 2002, Taipei, Taiwan, R.O.C..
H. Song, J. Liu and B. V. K. V. Kumar (2002). Low complexity LDPC codes for partial response channels. in IEEE Globecom 2002, Taipei, Taiwan. 2, 1294-1299
H. Song, J. Liu and B. V. K. V. Kumar (2004). Large girth cycle codes for partial response channels. IEEE Trans. Magn.. 40 part 2 (4), 3084-3086
X.-Y. Hu and E. Eleftheriou (2004). Binary representation of cycle Tannergraph codes. in Proc. IEEE Intern. Conf. on Commun., Paris, France. , 528-532
C. Poulliat, M. Fossorier and D. Declercq (2008). Design of regular $(2, d_c)$ LDPC codes over GF(q) using their binary images. IEEE Trans. Commun.. 56 (10), 1626-1635
M. Esmaeili and M. Gholami (2009). Geometrically-structured maximum-girth LDPC block and convolutional codes. IEEE journal of selected area of commun.. 27 (6), 831-845
M. Esmaeili and M. Gholami (2008). Maximum-girth Slope-based Quasi-cyclic $(2,kgeq 5)$-LDPC Codes. IET Commun.. 2 (10), 1251-1262
M. Gholami and M. Esmaeili (2012). Maximum-girth Cylinder-type Block-circulant LDPC Codes. IEEE Trans. on Commun.. 60 (4), 952-962
M. Esmaeili and M. Gholami (2009). Type-III LDPC Convolutional Codes. Contemporary Engineering Sciences. 2 (2), 85-93
D. J. Costello (1969). A construction technique for random-error correcting convolutional codes. IEEE Trans. Information Theory. IT-15 (5), 631-636
A. J. Viterbi (1967). Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Information Theory. 12 (2), 260-269
S. Lin and D. J. Costello (2004). Error Control Coding. 2nd Edition, Prentice Hall.