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

Golomb and Gong (1997 and 1999) considered binary sequences with the trinomial property. In this correspondence 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 2^m-1 with minimum distance exactly four