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Kibler, Baumert, Lander, and Kopilovich (cf. [7], [1], [10], and [8] respectively), studied the existence of \( (v, k, \lambda) \)-abelian difference sets with \( k \leq 100 \). In Lander and Kopilovich’s works, there were \( 13 \) and \( 8 \) \( (v, k, \lambda) \) tuples, respectively, in which the problem was open. Later, several authors have completed these studies and nowadays the problem is open for \( 6 \) and \( 7 \) tuples,
respectively. Jungnickel (cf. [9]) lists some unsolved problems on difference sets. One of them is to extend Lander’s table somewhat. By following this idea, this paper deals with the existence or non-existence of \( (v, k, \lambda) \)-abelian difference sets with \( 100 < k \leq 150 \). There exist \( 277 \) tuples that satisfy the basic relationship between the parameters \( v \), \( k \), and \( \lambda \), \( k \leq v/2 \),
Schutzenberger and Bruck-Chowla-Ryser's necessary conditions, and \( 100 < k \leq 150 \). In order to reduce this number, we have written in C several programs which implement some known criteria on non-existence of difference sets. We conclude that the only \( (v, k, \lambda) \) tuples, with \( 100 < k \leq 150 \), for which a difference set in some abelian group of order \( v \) can exist are
\begin{align*}
&(10303, 102, 1), (10713, 104, 1), (211, 105, 52), (11557, 108, 1), \\
&(223, 111, 55), (11991, 110, 1), (227, 113, 56), (12883, 114, 1), \\
&(378, 117, 386), (239, 119, 59), (256, 120, 56), (364, 121, 40), \\
&(243, 121, 60), (14763, 122, 1), (251, 125, 62), (15751, 126, 1), \\
&(351, 126, 45), (255, 127, 63), (16257, 128, 1), (16513, 129, 1), \\
&(263, 131, 65), (17293, 132, 1), (1573, 132, 11), (1464, 133, 12), \\
&(271, 135, 67), (18907, 138, 1), (19461, 140, 1), (283, 141, 70), \\
&(22351, 150, 1), (261, 105, 42), (429, 198, 27), (1200, 110, 10), \\
&(768, 118, 18), (841, 120, 17), (715, 120, 20), (5085, 124, 3), \\
&(837, 133, 21), (419, 133, 42), (1225, 136, 15), (361, 136, 51), \\
&(1975, 141, 10), (1161, 145, 18), (465, 145, 45), (5440, 148, 4), \\
&(448, 150, 50).
\end{align*}
It is known that there exist difference sets for the first \( 29 \) tuples and the problem is open for the remaining \( 16 \). Besides, in Table 1, we give the criterion that we have applied to determine the non-existence of \( (v, k, \lambda) \)-difference
sets for the rest of the tuples.