A ℤ₄-linear code of high minimum Lee distance derived from a hyperoval

Titelangaben

Kiermaier, Michael ; Zwanzger, Johannes:
A ℤ₄-linear code of high minimum Lee distance derived from a hyperoval.
In: Advances in Mathematics of Communications. Bd. 5 (2011) Heft 2 . - S. 275-286.
ISSN 1930-5346
DOI: https://doi.org/10.3934/amc.2011.5.275

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Abstract

In this paper we present a new non-free ℤ₄-linear code of length 29 and size 128 whose minimum Lee distance is 28. Its Gray image is a nonlinear binary code with parameters (58,2⁷,28), 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 ℤ₄ 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.