On the Existence of Abelian Difference Sets with $100<k\le 150$

Abstract

Kibler, Baumert, Lander, and Kopilovich (cf. [7], [1], [10], and [8] respectively), studied the existence of $(v,k,\lambda )$-abelian difference sets with $k\le 100$. 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 $100<k\le 150$. There exist $277$ tuples that satisfy the basic relationship between the parameters $v$, $k$, and $\lambda$, $k\le v/2$, Schutzenberger and Bruck-Chowla-Ryser's necessary conditions, and $100<k\le 150$. 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 $100<k\le 150$, for which a difference set in some abelian group of order $v$ can exist are $$\begin{array}{rl}& (10303,102,1),(10713,104,1),(211,105,52),(11557,108,1),\\ & (223,111,55),(11991,110,1),(227,113,56),(12883,114,1),\\ & (378,117,386),(239,119,59),(256,120,56),(364,121,40),\\ & (243,121,60),(14763,122,1),(251,125,62),(15751,126,1),\\ & (351,126,45),(255,127,63),(16257,128,1),(16513,129,1),\\ & (263,131,65),(17293,132,1),(1573,132,11),(1464,133,12),\\ & (271,135,67),(18907,138,1),(19461,140,1),(283,141,70),\\ & (22351,150,1),(261,105,42),(429,198,27),(1200,110,10),\\ & (768,118,18),(841,120,17),(715,120,20),(5085,124,3),\\ & (837,133,21),(419,133,42),(1225,136,15),(361,136,51),\\ & (1975,141,10),(1161,145,18),(465,145,45),(5440,148,4),\\ & (448,150,50).\end{array}$$ 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.