Study of proper circulant weighing matrices with weight 9

Circulant weighing matrices of weight 22t

“…For any i, suppose A (2) i = γ −1 A i for some γ ∈ H. Thus we have (C 00 + C 01 ) (2) ≡ γ −1 (C 00 + C 01 ) mod 2 and hence X (2) = γ −1 X. This implies β −1 X = γ −1 X and so X is a union of cosets of βγ −1 .…”

“…Then (βA 0 ) (2) = βA 0 and since the supports of C 00 , C 01 , C 02 are disjoint, we have [β(C 00 − C 01 )] (2) = β(C 00 − C 01 ) and (βC 02 ) (2) = βC 02 .…”

“…Since the coefficients of s−1 i=0 2 i (C i0 +C i1 ) lie between − s−1 i=0 2 i and s−1 i=0 2 i and the supports of s−1 i=0 2 i (C i0 + C i1 ) and D are disjoint, we have (βD) (2) …”

“…There exists β ∈ H such that [β(C 00 − C 01 )] (2) = β(C 00 − C 01 ) and (βC 02 ) (2) = βC 02 . Furthermore, if support(C 00 ) ∪ support(C 01 ) is not a union of cosets of g for any g ∈ H\{1}, then (βD) (2) = βD and (βC ij ) (2) = βC ij for all i, j.…”

“…Furthermore, if support(C 00 ) ∪ support(C 01 ) is not a union of cosets of g for any g ∈ H\{1}, then (βD) (2) = βD and (βC ij ) (2) = βC ij for all i, j.…”