Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 533–557 | Cite as

On the bounds and achievability about the ODPC of \(\mathcal{GRM }(2,m)^*\) over prime fields for increasing message length

  • Xiaogang Liu
  • Yuan LuoEmail author


The optimum distance profiles of linear block codes were studied for increasing or decreasing message length while keeping the minimum distances as large as possible, especially for Golay codes and the second-order Reed–Muller codes, etc. Cyclic codes have more efficient encoding and decoding algorithms. In this paper, we investigate the optimum distance profiles with respect to the cyclic subcode chains (ODPCs) of the punctured generalized second-order Reed–Muller codes \(\mathcal{GRM }(2,m)^*\) which were applied in Power Control in OFDM Modulations, in channels with synchronization, and so on. For this, two standards are considered in the inverse dictionary order, i.e., for increasing message length. Four lower bounds and upper bounds on ODPC are presented, where the lower bounds almost achieve the corresponding upper bounds in some sense. The discussions are over nonbinary prime fields.


Cyclic code Exponential sum Generalized Reed–Muller code  Optimum distance profile Quadratic form 

Mathematics Subject Classification

94B05 94B65 



This work was supported in part by the National Basic Research Program of China under Grants 2012CB316100, 2013CB338004, and the National Natural Science Foundation of China under Grant 61271222. We would also like to acknowledge the foundation TS0520103001 of Shanghai Jiao Tong University and University of Leuven.


  1. 1.
    Arıkan E.: Channel polarization: a method for constructing capacity-achieving codes for symmetric binaryinput memoryless channels. IEEE Trans. Inf. Theory 55(7), 3051–3073 (2009).Google Scholar
  2. 2.
    Assmus Jr E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).Google Scholar
  3. 3.
    Bruen A.: Blocking sets and low-weight codewords in the generalized reed-muller codes. In: Bruen A., Wehlau D., Society C.M. (eds.) Error-Correcting Codes, Finite Geometries, and Cryptography. Contemporary Mathematics, vol. 525. American Mathematical Society, Southport, pp. 161–164 (2010).Google Scholar
  4. 4.
    Burnashev M., Dumer I.: Error exponents for recursive decoding of Reed–Muller codes on a binary-symmetric channel. IEEE Trans. Inf. Theory 52(11), 4880–4891 (2006).Google Scholar
  5. 5.
    Cameron P.J., Von Lint J.H.: Graph Theory, Coding Theory and Block Designs. London Mathematical Society. Lecture Note, vol. 19. Cambridge University Press, London (1975).Google Scholar
  6. 6.
    Chen Y., Han Vinck A.J.: A lower bound on the optimum distance profiles of the second-order Reed–Muller codes. IEEE Trans. Inf. Theory 56(9), 4309–4320 (2010).Google Scholar
  7. 7.
    Davis J.A., Jedwab J.: Peak-to-mean power control in OFDM, Golay complementary sequences and Reed–Muller codes. IEEE Trans. Inf. Theory 45(7), 2397–2417 (1999).Google Scholar
  8. 8.
    Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, (1973).Google Scholar
  9. 9.
    Delsarte P., Goethals J.M.: Alternating bilinear forms over GF(q). J. Comb. Theory. 19(A), 26–50 (1975).Google Scholar
  10. 10.
    Delsarte P., Goethals J.M., MacWilliams F.J.: On GRM and their relatives. Inf. Contract. 16(5), 403–442 (1970).Google Scholar
  11. 11.
    Ding C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55(3), 955–960 (2009).Google Scholar
  12. 12.
    Ding C., Liu Y., Ma C., Zeng L.: The weight distributions of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 57(12), 8000–8006 (2011).Google Scholar
  13. 13.
    Draper S., Hou X.D.: Explicit evaluation of certain exponential sums of quadratic functions over \({\mathbb{F}}_{p^n}\), p odd. arXiv:0708.3619v1 (2007).Google Scholar
  14. 14.
    Egawa Y.: Association schemes of quadratic forms. J. Comb. Theory. 38(A), 1–14 (1985).Google Scholar
  15. 15.
    Elias P.: List decoding for noisy channels. Technical Report 335, Research Laboratory of Electronics, MIT (1957).Google Scholar
  16. 16.
    Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).Google Scholar
  17. 17.
    Geil O.: On the second weight of generalized Reed–Muller codes. Des. Codes Cryptogr. 48, 323–330 (2008).Google Scholar
  18. 18.
    Han Vinck A.J., Luo Y.: Optimum distance profiles of linear block codes. In: Proceedings of IEEE International Symposium on Information Theory, pp. 1958–1962 (2008).Google Scholar
  19. 19.
    Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths \(n_1((q^l-1)/N)\). Discret. Math. 18, 179–211 (1977).Google Scholar
  20. 20.
    Holma H., Toskala A.: WCDMA for UMTSHSPA Evolution and LTE, 4th edn. Wiley, London (2007).Google Scholar
  21. 21.
    Kacewicz A., Wicker S.: Application of Reed–Muller codes for localization of malicious nodes. In: Proceedings of IEEE International Conference on Communications (ICC’10), Capetown (2010).Google Scholar
  22. 22.
    Kasami T., Lin S., Peterson W.: New generalization of Reed–Muller codes. Part I: primitive codes. IEEE Trans. Inf. Theory 14, 189–199 (1968).Google Scholar
  23. 23.
    Korada S.B., Şaşoğlu E., Urbanke R.: Polar codes: characterization of exponent, bounds, and constructions. IEEE Trans. Inf. Theory 56(12), 6253–6264 (2010).Google Scholar
  24. 24.
    Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).Google Scholar
  25. 25.
    Liu X., Luo Y.: The weight distributions of some cyclic codes with three or four nonzeros over \({\mathbb{F}}_3\). arXiv:1302.0394v1.Google Scholar
  26. 26.
    Liu X., Luo Y., Shum K.W.: On the optimum cyclic subcode chains of \({\cal RM}(2,m)^*\) for increasing message length. arXiv:1306.0710v1.Google Scholar
  27. 27.
    Luo Y., Han Vinck A.J.: On a classification of cyclic subcode chains. In: 2009 International Conference on Communications and Networking in China (ChinaCom-2009), Xi’an (2009).Google Scholar
  28. 28.
    Luo Y., Han Vinck A.J., Chen Y.: On the optimum distance profiles about linear block codes. IEEE Trans. Inf. Theory 56(3), 1007–1014 (2010).Google Scholar
  29. 29.
    Luo J., Tang Y., Wang H.: Cyclic codes and sequences: The generalized Kasami case. IEEE Trans. Inf. Theory 56(5), 2130–2142 (2010).Google Scholar
  30. 30.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1988).Google Scholar
  31. 31.
    McEliece R.J.: Irreducible cyclic codes and Gauss Sums. In: Combinatorics, Part I: Theory of Designs, Finite Geometry and Coding Theory, vol. 55, pp. 179–196. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam (1974).Google Scholar
  32. 32.
    Paterson K.G.: Generalized Reed–Muller codes and power control in OFDM modulation. IEEE Trans. Inf. Theory 46(1), 104–120 (2000).Google Scholar
  33. 33.
    Sloane N.J.A.: An introduction to association schemes and coding theory. In: Askey R. (ed.) Theory and Application of Special Functions, pp. 225–260. Academic Press, New York (1975).Google Scholar
  34. 34.
    Tanner R., Woodard J.: WCDMA—Requirements and Practical Design. Wiley, London (2004).Google Scholar
  35. 35.
    van Dijk M., Baggen S., Tolhuizen L.: Coding for informed decoders. In: Proceedings of IEEE International Symposium on Information Theory, p. 202. Washington, DC (2001).Google Scholar
  36. 36.
    Weldon Jr E.J.: New generalizations of the Reed–Muller codes. Part II: nonprimitive codes. IEEE Trans. Inf. Theory 14(3), 199–205 (1968).Google Scholar
  37. 37.
    Wozencraft J.: List Decoding. Technical Report 48, pp. 90–95. Quarterly Progress Report, Research Laboratory of Electronics, MIT (1958).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.National Mobile Communications Research LaboratorySoutheast UniversityNanjingChina

Personalised recommendations