Nonexistence of CW(110, 100)

Abstract

Using character theoretic methods, we settle the existence status of a circulant weighing matrix of order 110 with weight 100. This fills a missing entry in recent tables, thereby answering the existence of previously open CW (110,100) with answer "NO".

Group invariant weighing matrices

“…It can be verified that there is no solution to (9) and (10) with |c i | ≤ 2 for all i. This gives a contradiction.…”

“…In the following, we present a partial generalization of Theorem 3.3 in [9]. Note that it contains McFarland's result as a special case.…”

“…It is straightforward to show that 9 is a multiplier and hence we can assume X (9) = X. Consequently, we can write…”

“…Arasu and Seberry [11] were the first to notice its applications to the studies of circulant weighing matrices. Arasu and Ma [9] extended the concept of multipliers from integer group rings to cyclotomic group rings, which helps to tackle cases where gcd(v, n) > 1. However, their result only covers cyclic groups.…”

“…The reader may refer to [39,Section 3.3.3] for the details. Then χ(S) is equivalent to θ 1 or θ 2 as defined in the following: with X, Y in Z[C 60 ] and ρ(Y ) is a CW (30,9). But CW (30,9) does not exist, see [3].…”