On binary cyclic codes with codewords of weight three and binary sequences with the trinomial property


Pascale Charpin
INRIA, projet CODES
BP 105
78153 Le Chesnay Cedex, France
Pascale.Charpin@inria.fr

Aimo Tietäväinen
Department of Mathematics and TUCS
University of Turku
FIN-20014 Turku, Finland
tietavai@lena.utu.fi

Victor Zinoviev
Institute for Problems of Information Transmission
Bol'shoi Karetnyi 19
GSP-4 Moscow 101447, Russia

IEEE Transactions on Information Theory, To appear.

Abstract

In a recent paper, Golomb and Gong considered binary sequences with the trinomial property. In this paper we shall show that the sets of those sequences are (quite trivially) closely connected with binary cyclic codes with codewords of weight three. This approach gives us another way to deal with trinomial property problems. After disproving one conjecture formulated by Golomb and Gong, we exhibit an infinite class of sequences which do not have the trinomial property, corresponding to binary cyclic codes of length 2m-1 with minimum distance exactly four.