循环称重矩阵的新不存在性结果
Character and Multiplier Obstructions for Circulant Weighing Matrices
中文总结 AI 辅助
该研究结合收缩、特征计算、乘子方法等，证明了8个不同阶数和权重的循环称重矩阵不存在。
中文摘要 AI 辅助
我们证明了8个循环称重矩阵的不存在性，这些矩阵来自阶数不超过200、权重不超过100的剩余表格。证明结合了收缩、收缩核上的特征值计算、乘子方法和精确有限计算。对于CW(105,36)，收缩后的矩阵在等价意义下唯一；应用C₃核的非主特征得到艾森斯坦整数上的一个元素，模1−ω约化后得到长度为35的三元循环码中的一个码字，精确枚举排除了所有所需的艾森斯坦单位提升。对于CW(140,36)，C₄核的实值特征Y↦−1与同一收缩类不兼容。对于权重64，C₄核的忠实特征Y↦i首先给出ℤ[i][Cₘ]的一个元素；广义乘子随后对乘2轨道强制常数性，精确相关计算排除了阶数140、180和196。三个权重49的情况通过普通素幂乘子解决，必要时结合收缩。因此，CW(105,36)、CW(140,36)、CW(116,49)、CW(120,49)、CW(192,49)、CW(140,64)、CW(180,64)和CW(196,64)均不存在。
英文摘要
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 $Y\mapsto-1$ 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.
