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 C3 kernel gives an element over the Eisenstein integers; reduction modulo 1−ω 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↦−1 of the C4 kernel is incompatible with the same contracted class. For weight $64$, the faithful character Y↦ i of a C4 kernel first gives an element of ℤ[i][Cm]; 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.
Related
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- Explicit weighing matrices W(n,k) for all n ≡ 0 pmod 4 with 4 ≤ n ≤ 108, and all 1 ≤ k ≤ n
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- Freeness for the G-circulant Decomposition of the Partial Transpose of Random Matrices
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BibTeX
@misc{tan2026character,
title = {{Character and Multiplier Obstructions for Circulant Weighing Matrices}},
author = {Ming Ming Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18468},
eprint = {2608.18468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18468},
}
RIS
TY - UNPB
TI - Character and Multiplier Obstructions for Circulant Weighing Matrices
AU - Ming Ming Tan
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.18468
UR - https://arxiv.org/abs/2608.18468
AB - 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−ω 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↦−1 of the C_4 kernel is incompatible with the same contracted class. For weight $64$, the faithful character Y↦ i of a C_4 kernel first gives an element of 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.
ER -