# 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:
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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.  Google Scholar [2] T. Feulner, Canonization of linear codes over $\mathbbZ_4$, Adv. Math. Commun., 5 (2011), 245-266. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [7] T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Comb., 7 (2000), #R11.  Google Scholar [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.  Google Scholar [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. Google Scholar [10] T. Honold and A. A. Nechaev, Weighted modules and representations of codes, Prob. Inform. Trans., 35 (1999), 205-223.  Google Scholar [11] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003.  Google Scholar [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.  Google Scholar [13] S. Litsyn, E. Rains and N. Sloane, Table of nonlinear binary codes,, \url{http://www.eng.tau.ac.il/~litsyn/tableand/}, ().   Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [18] R. Raghavendran, Finite associative rings, Composito Math., 21 (1969), 195-229.  Google Scholar [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.  Google Scholar [20] V. Zinoviev and S. Litsyn, On the general construction of codes shortening, Prob. Inform. Trans., 23 (1987), 111-116.  Google Scholar [21] J. Zwanzger, Linear codes over finite chain rings,, \url{http://www.mathe2.uni-bayreuth.de/20er/codedb/}, ().   Google Scholar [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.  Google Scholar
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