This paper gives a unified algebraic solution to seven open problems of Wang, Tang and Ding on cyclic, negacyclic and constacyclic codes supporting designs. For the cyclic code \[ C\left(\frac{p^s-1}{2},\frac{p^s+1}{2}\right), \] a Cayley parametrization of the unit circle reduces the trace-zero condition to a semilinear equation on \(\PG(1,q)\). Its large root sets are exactly the \(\F_{p^{\gcd(m,s)}}\)-sublines, yielding the complementary design \[ \overline{S(3,q_0+1,q+1)}. \] For the length \(q^2+1\) negacyclic code, a quotient transport from \(\U_{2(q^2+1)}\) to \(\U_{q^2+1}\) and a unit-circle parametrization show that the minimum zero sets are precisely the Baer sublines of \(\PG(1,q^2)\). Equivalently, the corresponding support design is the complement of the non-tangent plane sections of an elliptic quadric \(\Q^-(3,q)\). For constacyclic ovoid codes of length \(q^2+1\) over \(\F_q\), the exact existence criterion is \[ λ\in\F_q^*,\qquad \exists\ λ\text{-constacyclic ovoid code} \Longleftrightarrow λ\notin(\F_q^*)^2. \] In particular, negacyclic ovoid codes exist exactly when \(q\equiv3\pmod4\). The proof uses the corrected projective-order congruence \[ a=(q+1)c,\qquad c\equiv b\pmod{q-1},\qquad \operatorname{ord}(θ\F_q^*)=\frac{q^2+1}{\gcd(q^2+1,c)}. \] The paper also derives a universal weight enumerator for lifted ovoid codes over extension fields, independent of the chosen ovoid. Finally, consecutive-root negacyclic MDS codes are constructed to give complete simple \(5\)-designs, including a proper negacyclic \([11,5,7]_{23}\) code whose minimum supports form the complete \(5-(11,7,15)\) design.

翻译：本文给出了王、唐与丁提出的关于循环码、负循环码及常循环码支持设计的七个未解决问题的统一代数解法。对于循环码 \[ C\left(\frac{p^s-1}{2},\frac{p^s+1}{2}\right), \] 通过单位圆的凯莱参数化，将迹零条件归结为 \(\PG(1,q)\) 上的半线性方程。其大根集恰为 \(\F_{p^{\gcd(m,s)}}\) 子线，从而得到互补设计 \[ \overline{S(3,q_0+1,q+1)}。\] 对于长度为 \(q^2+1\) 的负循环码，通过从 \(\U_{2(q^2+1)}\) 到 \(\U_{q^2+1}\) 的商传输与单位圆参数化，证明最小零集恰为 \(\PG(1,q^2)\) 中的贝尔子线。等价地，相应支撑设计是椭圆二次曲面 \(\Q^-(3,q)\) 的非切平面截面的补集。对于 \(\F_q\) 上长度为 \(q^2+1\) 的常循环卵形线码，其存在性的精确判据为 \[ λ\in\F_q^*,\qquad \exists\ λ\text{-常循环卵形线码} \Longleftrightarrow λ\notin(\F_q^*)^2。\] 特别地，当且仅当 \(q\equiv3\pmod4\) 时存在负循环卵形线码。证明采用了修正的射影阶同余式 \[ a=(q+1)c,\qquad c\equiv b\pmod{q-1},\qquad \operatorname{ord}(θ\F_q^*)=\frac{q^2+1}{\gcd(q^2+1,c)}。\] 本文还导出了扩域上提升卵形线码的通用重量枚举器，该枚举器独立于所选卵形线。最后，构造了连续根负循环MDS码，以给出完全简单 \(5\)-设计，其中包括一个本征负循环 \([11,5,7]_{23}\) 码，其最小支撑构成完全 \(5-(11,7,15)\) 设计。