Parity check matrices and product representations of squares

N. Alon, L. Babai and A. Itai: A fast and simple randomized parallel algorithm for the maximal independent set problem, J. Algorithms 7(4) (1986), 567–583.

N. Alon, O. Goldreich, J. Håstad and R. Peralta: Simple constructions of almost k-wise independent random variables, Random Structures Algorithms 3(3) (1992), 289–304.

N. Alon, S. Hoory and N. Linial: The Moore bound for irregular graphs, Graphs Combin. 18(1) (2002), 53–57.

N. Alon and J. H. Spencer: The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York, second edition, 2000. With an appendix on the life and work of Paul Erdős.

N. Alon, R. Yuster and U. Zwick: Color-coding, J. Assoc. Comput. Mach. 42(4) (1995), 844–856.

N. Alon, R. Yuster and U. Zwick: Finding and counting given length cycles, Algorithmica 17(3) (1997), 209–223.

L. Bazzi, M. Mahdian and D. A. Spielman: The minimum distance of Turbo-like codes, preprint, 2003.

C. Berrou, A. Glavieux and P. Thitimajshima: Near Shannon Limit Error Correcting Codes and Decoding: Turbo Codes; in Proceedings of IEEE International Communications Conference, pages 1064–1070, 1993.

C. Bertram-Kretzberg, T. Hofmeister and H. Lefmann: Sparse 0–1 matrices and forbidden hypergraphs, Combin. Probab. Comput. 8(5) (1999), 417–427.

C. Bertram-Kretzberg and H. Lefmann: MODp-tests, almost independence and small probability spaces; Random Structures Algorithms 16(4) (2000), 293–313.

A. Beutelspacher and U. Rosenbaum: Projective geometry: from foundations to applications; Cambridge University Press, Cambridge, 1998.

B. Bollobás: Extremal graph theory, Dover Publications Inc., Mineola, NY, 2004. Reprint of the 1978 original.

J. Bondy and M. Simonovits: Cycles of even length in graphs, J. Combinatorial Theory Ser. B 16 (1974), 97–105.

M. Breiling: A logarithmic upper bound on the minimum distance of Turbo codes, IEEE Transactions on Information Theory 50(8) (2004), 1692–1710.

A. E. Brouwer: Block designs, in: Handbook of combinatorics, Vol. 1, 2, pages 693–745, Elsevier, Amsterdam, 1995.

W. G. Brown, P. Erdős and V. T. Sós: Some extremal problems on r-graphs, in: New directions in the theory of graphs, Proc 3rd Ann. Arbor Conference on Graph Theory, Academic Press, New York, 55–63, 1973.

M. C. Davey and D. J. C. MacKay: Low-density parity check codes over GF(q), IEEE Communications Letters 2(6) (1998), 165–167.

J. D. Dixon: Asymptotically fast factorization of integers, Math. Comp. 36(153) (1981), 255–260.

P. Erdős: On some applications of graph theory to number theoretic problems, Publ. Ramanujan Inst. 1 (1969), 131–136.

P. Erdős, A. Sárközy and V. T. Sós: On product representations of powers, I; European J. Combin. 16(6) (1995), 567–588.

R. G. Gallager: Low Density Parity Check Codes, MIT Press, Cambridge MA, 1963. Research Monograph Series, no. 21.

E. Győri: C 6-free bipartite graphs and product representation of squares, Discrete Math. 165/166 (1997), 371–375. Graphs and combinatorics (Marseille, 1995).

S. Hoory: The size of bipartite graphs with a given girth, J. Combin. Theory Ser. B 86(2) (2002), 215–220.

N. Kahale and R. Urbanke: On the minimum distance of parallel and serially concatenated codes, IEEE Trans. Inform. Theory, to appear.

H. Lefmann: Sparse parity-check matrices over finite fields (extended abstract), in: Computing and combinatorics, volume 2697 of Lecture Notes in Comput. Sci., pages 112–121, Springer, Berlin, 2003.

H. Lefmann, P. Pudlák and P. Savický: On sparse parity check matrices, Des. Codes Cryptogr. 12(2) (1997), 107–130.

D. J. C. MacKay: Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory 45(2) (1999), 399–431.

F. J. Mac Williams and N. J. A. Sloane: The theory of error-correcting codes, I; North-Holland Publishing Co., Amsterdam, 1977, North-Holland Mathematical Library, Vol. 16.

F. J. MacWilliams and N. J. A. Sloane: The theory of error-correcting codes, II; North-Holland Publishing Co., Amsterdam, 1977, North-Holland Mathematical Library, Vol. 16.

A. Naor and J. Verstraëte: A note on bipartite graphs without 2k-cycles, Combin. Probab. Comput. 14(5–6) (2005), 845–849.

C. Pomerance: A tale of two sieves, Notices Amer. Math. Soc. 43(12) (1996), 1473–1485.

C. Pomerance and A. Sárközy: Combinatorial number theory, in: Handbook of combinatorics, Vol. 1, 2, pages 967–1018, Elsevier, Amsterdam, 1995.

G. N. Sárközy: Cycles in bipartite graphs and an application in number theory, J. Graph Theory 19(3) (1995), 323–331.

M. Sipser and D. A. Spielman: Expander codes, IEEE Trans. Inform. Theory 42(6/1) (1996), 1710–1722. Codes and complexity.

D. A. Spielman: Linear-time encodable and decodable error-correcting codes, IEEE Trans. Inform. Theory 42(6/1) (1996), 1723–1731. Codes and complexity.

J. Verstraëte: On arithmetic progressions of cycle lengths in graphs, Combin. Probab. Comput. 9(4) (2000), 369–373.

H. C. Williams and J. O. Shallit: Factoring integers before computers, in: Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993), volume 48 of Proc. Sympos. Appl. Math., pages 481–531, Amer. Math. Soc., Providence, RI, 1994.