New ring-linear codes from dualization in projective Hjelmslev geometries
详细信息 查看全文
- 作者：Michael Kiermaier (1)
Johannes Zwanzger (2) - 关键词：Ring ; linear code ; Kerdock code ; Lee weight ; Homogeneous weight ; Galois ring ; Gray map ; Hjelmslev geometry ; 94B05 ; 94B27 ; 51C05 ; 51E20 ; 05B25
- 刊名：Designs, Codes and Cryptography
- 出版年：2013
- 出版时间：3 - January 2013
- 年：2013
- 卷：66
- 期：1
- 页码：39-55
- 全文大小：245KB
- 参考文献：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), 2065-069 (2022) CrossRef
2. Bonnecaze A., Solé P., Calderbank A.R.: Quaternary quadratic residue codes and unimodular lattices. IEEE Trans. Inf. Theory 41(2), 366-77 (1995) CrossRef
3. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. the user language. J. Symb. Comput. 24(3-), 235-65 (1997) CrossRef
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, 95-8 (1993) CrossRef
5. Byrne E., Greferath M., Honold T.: Ring geometries, two-weight codes, and strongly regular graphs. Des. Codes Cryptogr. 48(1), 1-6 (2008) CrossRef
6. Byrne E., Greferath M., Kohnert A., Skachek V.: New bounds for codes over finite frobenius rings. Des. Codes Cryptogr. 57(2), 169-79 (2010) CrossRef
7. Calderbank A.R., McGuire G.: Construction of a (64,237,12) code via Galois rings. Des. Codes Cryptogr. 10, 157-65 (1997) CrossRef
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), 217-26 (1996) CrossRef
9. Constantinescu I., Heise W.: A metric for codes over residue class rings. Probl. Inf. Transm. 33, 208-13 (1997)
10. Delsarte P., Goethals J.M.: Alternating bilinear forms over GF / q . J. Comb. Theory Ser. A 19(1), 26-0 (1975) CrossRef
11. Feulner T.: Canonization of linear codes over ${\mathbb{Z}_4}$ . Adv. Math. Commun. 5(2), 245-66 (2011) CrossRef
12. Goethals J.M.: Nonlinear codes defined by quadratic forms over GF(2). Inf. Control 31(1), 43-4 (1976) CrossRef
13. Grassl M.: Code Tables: Bounds on the parameters of various types of codes. www.codetables.de.
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), 2522-524 (1999) CrossRef
15. Greferath M., Schmidt S.E.: Finite-ring combinatorics and MacWilliams-equivalence theorem. J. Comb. Theory Ser. A 92(1), 17-8 (2000) CrossRef
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), 301-19 (1994) CrossRef
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. 4-2 (2000).
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. 1807-813 (2010).
19. Honold T., Landjev I.: Linear codes over finite chain rings. Electron. J. Comb. 7,–R11 (2000)
20. Honold T., Landjev I.: On arcs in projective Hjelmslev planes. Discret. Math. 231(1-), 265-78 (2001) CrossRef
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, 18-9 August 2008, pp. 60-23. World Scientific, Singapore (2009).
22. Honold T., Landjev I.: The dual construction for arcs in projective Hjelmslev spaces. Adv. Math. Commun. 5(1), 11-1 (2011) CrossRef
23. Honold T., Nechaev A.A.: Weighted modules and representations of codes. Probl. Inf. Transm. 35(3), 205-23 (1999)
24. Huffman W.C., Pless V.: Fundamentals of error-correcting codes. Cambridge University Press, Cambridge (2003) CrossRef
25. Kerdock A.M.: A class of low-rate nonlinear binary codes. Inf. Control 20, 182-87 (1972) CrossRef
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. 112-19 (2007).
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. 2617-619 (2008).
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. Kiermaier M., Zwanzger J.: Online tables of linear codes over finite chain rings. http://codes.uni-bayreuth.de.
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. 929-32 (2010).
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), 275-86 (2011) CrossRef
32. Kiermaier M., Zwanzger J.: New ring-linear codes from geometric dualization. In: Proceedings of the Seventh International Workshop on Coding and Cryptography, pp. 111-20 (2011).
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), 183-84 (1994) CrossRef
34. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
35. McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974)
36. Nechaev A.A.: Kerdock code in a cyclic form. Discret. Math. Appl. 1(4), 365-84 (1991) CrossRef
37. Nechaev A.A.: Finite rings with applications. In: Hazewinkel, M. (eds) Handbook of Algebra, vol 5, chap 5., pp. 213-20. North-Holland, Amsterdam (2008) CrossRef
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. 31-4 (1996).
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 23-7, Lecture Notes in Computer Science, vol. 1255, pp. 263-76. Springer, Berlin (1997).
40. Nordstrom A.W., Robinson J.P.: An optimum nonlinear code. Inf. Control 11(5-), 613-16 (1967) CrossRef
41. Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over Z4. IEEE Trans. Inf. Theory 42(5), 1594-600 (1996) CrossRef
42. Preparata F.P.: A class of optimum nonlinear double-error-correcting codes. Inf. Control 13(4), 378-00 (1968) CrossRef - 作者单位：Michael Kiermaier (1)
Johannes Zwanzger (2)
1. Mathematisches Institut, Universit?t Bayreuth, 95440, Bayreuth, Germany
2. Siemens AG, CT T DE IT1, Otto-Hahn-Ring 6, 81739, München, Deutschland - ISSN：1573-7586
文摘
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).