Abstract
Since short cycles are (empirically) detrimental to message passing, determining the girth of a given code is of interest in coding theory. Halford et al. studied codes which do not have a 4-cycle-free Tanner graph representation. It is natural to then ask which codes must have girth 8. In this paper, a new necessary condition is derived for codes to have girth 8. Halford et al. made statements about the girth of high rate well known codes but the girth of lower rate codes remain open. In this work, we investigate girth of low rate Reed–Muller, BCH and Reed–Solomon codes.
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Augot D., vit Vehel F.L.: Bounds on the minimum distance of the duals of BCH codes. IEEE Trans. Inform. Theory 42, 1257–1260 (1996)
Etzion T., Trachtenberg A., Vardy A.: Which codes have cycle-free Tanner graphs?. IEEE Trans. Inform. Theory 45, 2173–2181 (1999)
Győri E.: C 6-free bipartite graphs and product representation of squares. Discrete Math. 165/166, 371–375 (1997)
Győri E.: Triangle-free hypergraphs. Comb. Probab. Comput. 15, 185–191 (2006)
Halford T.R., Chugg K.M.: The tradeoff between cyclic topology and complexity in graphical models of linear codes. Forty-Fourth Allerton Conf. on Commun., Control, and Computing, Monticello, IL, September (2006).
Halford T.R., Grant A.J., Chugg K.M.: Which codes have 4-cycle free tanner graph?. IEEE Trans. Inform. Theory 52, 4219–4223 (2006)
Hoory Sh.: The size of bipartite graphs with a given girth. J. Comb. Theory, Ser. B 86, 215–220 (2002)
Kim J.-L., Peled U.N., Perepelista I., Pless V., Friedland S.: Explicit construction of families of LDPC codes with no 4-cycles. IEEE Trans. Inform. Theory 50, 2378–2388 (2004)
Liu Z., Pados D.A.: LDPC codes from generalized polygons. IEEE Trans. Inform. Theory 51, 3890–3898 (2005)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam, New York, Oxford (1978)
Neuwirth S.: The size of bipartite graphs with girth eight, Nov. 2008, arXiv:math/0102210.
Sankaranarayanan S., Vasic B.: Iterative decoding of linear block codes: a parity-check orthogonalization approach. IEEE Trans. Inform. Theory 51, 3347–3353 (2005)
Tanner R.M.: A recursive approach to low-complexity codes. IEEE Trans. Inform. Theory 27, 533–547 (1981)
van Maldeghem H.: Generalized Polygons. Birkhäuser Verlag (1998).
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Communicated by T. Helleseth.
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Sakzad, A., Sadeghi, MR. & Panario, D. Codes with girth 8 Tanner graph representation. Des. Codes Cryptogr. 57, 71–81 (2010). https://doi.org/10.1007/s10623-009-9349-0
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DOI: https://doi.org/10.1007/s10623-009-9349-0