Character and Multiplier Obstructions for Circulant Weighing Matrices

Abstract

We prove the nonexistence of eight circulant weighing matrices from the remaining table of orders at most $200$ and weights at most $100$. The proofs combine contraction, character evaluation on the kernel of a contraction, multiplier methods, and exact finite computations. For $CW(105,36)$, the contracted matrix is unique up to equivalence. Applying a nonprincipal character of the $C_3$ kernel gives an element over the Eisenstein integers; reduction modulo $1-\omega$ gives a word in a ternary cyclic code of length $35$, and exact enumeration rules out every required Eisenstein-unit lift. For $CW(140,36)$, the real-valued character $Y\mapsto-1$ of the $C_4$ kernel is incompatible with the same contracted class. For weight $64$, the faithful character $Y\mapsto i$ of a $C_4$ kernel first gives an element of $\mathbb{Z}[i][C_m]$; a generalized multiplier then forces constancy on multiplication-by-$2$ orbits, and exact correlation calculations eliminate orders $140$, $180$, and $196$. The three weight-$49$ cases are settled by the ordinary prime-power multiplier, with contraction where needed. Consequently none of $CW(105,36)$, $CW(140,36)$, $CW(116,49)$, $CW(120,49)$, $CW(192,49)$, $CW(140,64)$, $CW(180,64)$, and $CW(196,64)$ exists.