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Designs, Codes and Cryptography

, Volume 66, Issue 1–3, pp 39–55 | Cite as

New ring-linear codes from dualization in projective Hjelmslev geometries

  • Michael Kiermaier
  • Johannes Zwanzger
Article

Abstract

In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include \({\mathbb {Z}_4}\) as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. Applying it to the \({\mathbb {Z}_4}\) -preimages of the Kerdock codes and a related family of codes we will call Teichmüller codes, we get two new infinite series of codes and compute their symmetrized weight enumerators. In some cases, residuals of the original code give further interesting codes. The generalized Gray map translates our codes into ordinary, generally non-linear codes in the Hamming space. The obtained parameters include (58, 27, 28)2, (60, 28, 28)2, (114, 28, 56)2, (372, 210, 184)2 and (1988, 212, 992)2 which provably have higher minimum distance than any linear code of equal length and cardinality over an alphabet of the same size (better-than-linear, BTL), as well as (180, 29, 88)2, (244, 29, 120)2, (484, 210, 240)2 and (504, 46, 376)4 where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).

Keywords

Ring-linear code Kerdock code Lee weight Homogeneous weight Galois ring Gray map Hjelmslev geometry 

Mathematics Subject Classification

94B05 94B27 51C05 51E20 05B25 

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References

  1. 1.
    Aydin N., Ray-Chaudhuri D.K.: Quasi-cyclic codes over \({\mathbb{Z}_4}\) and some new binary codes. IEEE Trans. Inf. Theory 48(7), 2065–2069 (2022)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonnecaze A., Solé P., Calderbank A.R.: Quaternary quadratic residue codes and unimodular lattices. IEEE Trans. Inf. Theory 41(2), 366–377 (1995)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brouwer A.E., Tolhuizen L.M.G.M.: A sharpening of the Johnson bound for binary linear codes and the nonexistence of linear codes with Preparata parameters. Des. Codes Cryptogr. 3, 95–98 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Byrne E., Greferath M., Honold T.: Ring geometries, two-weight codes, and strongly regular graphs. Des. Codes Cryptogr. 48(1), 1–16 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Byrne E., Greferath M., Kohnert A., Skachek V.: New bounds for codes over finite frobenius rings. Des. Codes Cryptogr. 57(2), 169–179 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Calderbank A.R., McGuire G.: Construction of a (64,237,12) code via Galois rings. Des. Codes Cryptogr. 10, 157–165 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Calderbank A.R., McGuire G., Kumar V., Helleseth T.: Cyclic codes over \({\mathbb{Z}_4}\) , locator polynomials, and Newton’s identities. IEEE Trans. Inf. Theory 42(1), 217–226 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Constantinescu I., Heise W.: A metric for codes over residue class rings. Probl. Inf. Transm. 33, 208–213 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Delsarte P., Goethals J.M.: Alternating bilinear forms over GFq. J. Comb. Theory Ser. A 19(1), 26–50 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Feulner T.: Canonization of linear codes over \({\mathbb{Z}_4}\) . Adv. Math. Commun. 5(2), 245–266 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Goethals J.M.: Nonlinear codes defined by quadratic forms over GF(2). Inf. Control 31(1), 43–74 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Grassl M.: Code Tables: Bounds on the parameters of various types of codes. www.codetables.de.
  14. 14.
    Greferath M., Schmidt S.E.: Gray isometries for finite chain rings and a nonlinear ternary (36, 312, 15) code. IEEE Trans. Inf. Theory 45(7), 2522–2524 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Greferath M., Schmidt S.E.: Finite-ring combinatorics and MacWilliams’ equivalence theorem. J. Comb. Theory Ser. A 92(1), 17–28 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{Z}_4}\) -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)zbMATHCrossRefGoogle Scholar
  17. 17.
    Hemme L., Honold T., Landjev I.: Arcs in projective Hjelmslev spaces obtained from Teichmüller sets. In: Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory 2000, pp. 4–12 (2000).Google Scholar
  18. 18.
    Honold T.: Two-intersection sets in projective Hjelmslev spaces. In: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pp. 1807–1813 (2010).Google Scholar
  19. 19.
    Honold T., Landjev I.: Linear codes over finite chain rings. Electron. J. Comb. 7,–R11 (2000)MathSciNetGoogle Scholar
  20. 20.
    Honold T., Landjev I.: On arcs in projective Hjelmslev planes. Discret. Math. 231(1–3), 265–278 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Honold T., Landjev I.: Linear codes over finite chain rings and projective Hjelmslev geometries. In: P. Solé (ed.) Codes over Rings. Proceedings of the CIMPA Summer School Ankara, Turkey, 18–29 August 2008, pp. 60–123. World Scientific, Singapore (2009).Google Scholar
  22. 22.
    Honold T., Landjev I.: The dual construction for arcs in projective Hjelmslev spaces. Adv. Math. Commun. 5(1), 11–21 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Honold T., Nechaev A.A.: Weighted modules and representations of codes. Probl. Inf. Transm. 35(3), 205–223 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Huffman W.C., Pless V.: Fundamentals of error-correcting codes. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  25. 25.
    Kerdock A.M.: A class of low-rate nonlinear binary codes. Inf. Control 20, 182–187 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kiermaier M., Kohnert A.: New arcs in projective Hjelmslev planes over Galois rings. In: Proceedings of the Fifth International Workshop on Optimal Codes and Related Topics 2007, pp. 112–119 (2007).Google Scholar
  27. 27.
    Kiermaier M., Wassermann A.: On the minimum Lee distance of quadratic residue codes over \({\mathbb{Z}_4}\) . In: Proceedings of the International Symposium on Information Theory (ISIT) 2008, pp. 2617–2619 (2008).Google Scholar
  28. 28.
    Kiermaier M., Wassermann A.: Minimum weights and weight enumerators of \({\mathbb{Z}_4}\) -linear quadratic residue codes. IEEE Trans. Inf. Theory (2012). doi: 10.1109/TIT.2012.2191389.
  29. 29.
    Kiermaier M., Zwanzger J.: Online tables of linear codes over finite chain rings. http://codes.uni-bayreuth.de.
  30. 30.
    Kiermaier M., Zwanzger J.: A new series of \({\mathbb{Z}_4}\) -linear codes of high minimum Lee distance derived from the Kerdock codes. In: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pp. 929–932 (2010).Google Scholar
  31. 31.
    Kiermaier M., Zwanzger J.: A \({\mathbb{Z}_4}\) -linear code of high minimum Lee distance derived from a hyperoval. Adv. Math. Commun. 5(2), 275–286 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Kiermaier M., Zwanzger J.: New ring-linear codes from geometric dualization. In: Proceedings of the Seventh International Workshop on Coding and Cryptography, pp. 111–120 (2011).Google Scholar
  33. 33.
    Kuz’min A.S., Nechaev A.A.: Linearly representable codes and the Kerdock code over an arbitrary Galois field of characteristic 2. Russ. Math. Surv. 49(5), 183–184 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  35. 35.
    McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974)zbMATHGoogle Scholar
  36. 36.
    Nechaev A.A.: Kerdock code in a cyclic form. Discret. Math. Appl. 1(4), 365–384 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Nechaev A.A.: Finite rings with applications. In: Hazewinkel, M. (eds) Handbook of Algebra, vol 5, chap 5., pp. 213–320. North-Holland, Amsterdam (2008)CrossRefGoogle Scholar
  38. 38.
    Nechaev A.A., Kuzmin A.S.: Linearly presentable codes. In: Proceedings of the International Symposium on Information Theory and its Application (ISITA) 1996, pp. 31–34 (1996).Google Scholar
  39. 39.
    Nechaev A.A., Kuzmin A.S.: Trace-function on a Galois ring in coding theory. In: T. Mora, H. Mattson (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Proceedings of the 12th International Symposium AAECC-12, Toulouse, France, June 23–27, Lecture Notes in Computer Science, vol. 1255, pp. 263–276. Springer, Berlin (1997).Google Scholar
  40. 40.
    Nordstrom A.W., Robinson J.P.: An optimum nonlinear code. Inf. Control 11(5–6), 613–616 (1967)zbMATHCrossRefGoogle Scholar
  41. 41.
    Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over Z4. IEEE Trans. Inf. Theory 42(5), 1594–1600 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Preparata F.P.: A class of optimum nonlinear double-error-correcting codes. Inf. Control 13(4), 378–400 (1968)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Siemens AG, CT T DE IT1MünchenDeutschland

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