Advances in Mathematics of Communications (AMC)

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

Pages: 275 - 286, Volume 5, Issue 2, May 2011      doi:10.3934/amc.2011.5.275

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Michael Kiermaier - Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany (email)
Johannes Zwanzger - Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany (email)

Abstract: 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.

Keywords:  Coding theory, Gray image, minimum distance, Hjelmslev geometry, hyperoval.
Mathematics Subject Classification:  Primary: 94B05; Secondary: 51C05, 51E21.

Received: April 2010;      Revised: August 2010;      Available Online: May 2011.