American Institute of Mathematical Sciences

May  2011, 5(2): 275-286. doi: 10.3934/amc.2011.5.275

A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval

 1 Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany, Germany

Received  April 2010 Revised  August 2010 Published  May 2011

In this paper we present a new non-free $\mathbb Z$4-linear code of length $29$ and size $128$ whose minimum Lee distance is $28$. Its Gray image is a nonlinear binary code with parameters $(58,2^7,28)$, having twice as many codewords as the biggest linear binary codes of equal length and minimum distance. The code also improves the known lower bound on the maximal size of binary block codes of length $58$ and minimum distance $28$.
Originally the code was found by a heuristic computer search. We give a geometric construction based on a hyperoval in the projective Hjelmslev plane over $\mathbb Z$4 which allows an easy computation of the symmetrized weight enumerator and the automorphism group. Furthermore, a generalization of this construction to all Galois rings of characteristic $4$ is discussed.
Citation: Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
References:
 [1] I. Constantinescu and W. Heise, A metric for codes over residue class rings, Prob. Inform. Trans., 33 (1997), 208-213. [2] T. Feulner, Canonization of linear codes over $\mathbbZ_4$, Adv. Math. Commun., 5 (2011), 245-266. [3] M. Greferath and S. E. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary $(36,3$12$,15)$ code, IEEE Trans. Inform. Theory, 45 (1999), 2522-2524. doi: 10.1109/18.796395. [4] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033. [5] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154. [6] T. Honold and M. Kiermaier, Classification of maximal arcs in small projective Hjelmslev geometries, in "Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-10),'' (2006), 112-117. [7] T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Comb., 7 (2000), #R11. [8] T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic, Finite Fields Appl., 11 (2005), 292-304. doi: 10.1016/j.ffa.2004.12.004. [9] T. Honold and I. Landjev, Linear codes over finite chain rings and projective Hjelmslev geometries, in "Codes over Rings. Proceedings of the CIMPA Summer School Ankara, Turkey, August 2008'' (ed. P. Solé), World Scientific, (2009), 60-123. [10] T. Honold and A. A. Nechaev, Weighted modules and representations of codes, Prob. Inform. Trans., 35 (1999), 205-223. [11] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. [12] A. M. Kerdock, A class of low-rate non-linear binary codes, Inform. Control, 20 (1972), 182-187. doi: 10.1016/S0019-9958(72)90376-2. [13] S. Litsyn, E. Rains and N. Sloane, Table of nonlinear binary codes, http://www.eng.tau.ac.il/~litsyn/tableand/ [14] A. A. Nechaev, Kerdock code in a cyclic form, Discrete Math. Appl., 1 (1991), 365-384. doi: 10.1515/dma.1991.1.4.365. [15] A. A. Nechaev and A. S. Kuzmin, Linearly presentable codes, in "Proceedings of the International Symposium on Information Theory and its Applications (ISITA) 1996,'' (1996), 31-34. [16] A. W. Nordstrom and J. P. Robinson, An optimum nonlinear code, Inform. Control, 11 (1967), 613-616. doi: 10.1016/S0019-9958(67)90835-2. [17] F. P. Preparata, A class of optimum nonlinear double-error-correcting codes, Inform. Control, 13 (1968), 378-400. doi: 10.1016/S0019-9958(68)90874-7. [18] R. Raghavendran, Finite associative rings, Composito Math., 21 (1969), 195-229. [19] H. C. A. van Tilborg, The smallest length of binary $7$-dimensional linear codes with prescribed minimum distance, Discrete Math., 33 (1981), 197-207. doi: 10.1016/0012-365X(81)90166-7. [20] V. Zinoviev and S. Litsyn, On the general construction of codes shortening, Prob. Inform. Trans., 23 (1987), 111-116. [21] J. Zwanzger, Linear codes over finite chain rings, http://www.mathe2.uni-bayreuth.de/20er/codedb/ [22] J. Zwanzger, A heuristic algorithm for the construction of good linear codes, IEEE Trans. Inform. Theory, 54 (2008), 2388-2392. doi: 10.1109/TIT.2008.920323.

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References:
 [1] I. Constantinescu and W. Heise, A metric for codes over residue class rings, Prob. Inform. Trans., 33 (1997), 208-213. [2] T. Feulner, Canonization of linear codes over $\mathbbZ_4$, Adv. Math. Commun., 5 (2011), 245-266. [3] M. Greferath and S. E. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary $(36,3$12$,15)$ code, IEEE Trans. Inform. Theory, 45 (1999), 2522-2524. doi: 10.1109/18.796395. [4] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033. [5] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbbZ_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154. [6] T. Honold and M. Kiermaier, Classification of maximal arcs in small projective Hjelmslev geometries, in "Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-10),'' (2006), 112-117. [7] T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Comb., 7 (2000), #R11. [8] T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic, Finite Fields Appl., 11 (2005), 292-304. doi: 10.1016/j.ffa.2004.12.004. [9] T. Honold and I. Landjev, Linear codes over finite chain rings and projective Hjelmslev geometries, in "Codes over Rings. Proceedings of the CIMPA Summer School Ankara, Turkey, August 2008'' (ed. P. Solé), World Scientific, (2009), 60-123. [10] T. Honold and A. A. Nechaev, Weighted modules and representations of codes, Prob. Inform. Trans., 35 (1999), 205-223. [11] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. [12] A. M. Kerdock, A class of low-rate non-linear binary codes, Inform. Control, 20 (1972), 182-187. doi: 10.1016/S0019-9958(72)90376-2. [13] S. Litsyn, E. Rains and N. Sloane, Table of nonlinear binary codes, http://www.eng.tau.ac.il/~litsyn/tableand/ [14] A. A. Nechaev, Kerdock code in a cyclic form, Discrete Math. Appl., 1 (1991), 365-384. doi: 10.1515/dma.1991.1.4.365. [15] A. A. Nechaev and A. S. Kuzmin, Linearly presentable codes, in "Proceedings of the International Symposium on Information Theory and its Applications (ISITA) 1996,'' (1996), 31-34. [16] A. W. Nordstrom and J. P. Robinson, An optimum nonlinear code, Inform. Control, 11 (1967), 613-616. doi: 10.1016/S0019-9958(67)90835-2. [17] F. P. Preparata, A class of optimum nonlinear double-error-correcting codes, Inform. Control, 13 (1968), 378-400. doi: 10.1016/S0019-9958(68)90874-7. [18] R. Raghavendran, Finite associative rings, Composito Math., 21 (1969), 195-229. [19] H. C. A. van Tilborg, The smallest length of binary $7$-dimensional linear codes with prescribed minimum distance, Discrete Math., 33 (1981), 197-207. doi: 10.1016/0012-365X(81)90166-7. [20] V. Zinoviev and S. Litsyn, On the general construction of codes shortening, Prob. Inform. Trans., 23 (1987), 111-116. [21] J. Zwanzger, Linear codes over finite chain rings, http://www.mathe2.uni-bayreuth.de/20er/codedb/ [22] J. Zwanzger, A heuristic algorithm for the construction of good linear codes, IEEE Trans. Inform. Theory, 54 (2008), 2388-2392. doi: 10.1109/TIT.2008.920323.

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