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：本文给出了关于支持设计的循环码、负循环码及常循环码的七个未解决问题的统一代数解法。对于循环码 \[ C\left(\frac{p^s-1}{2},\frac{p^s+1}{2}\right), \] 通过单位圆上的Cayley参数化，将迹零条件简化为\(\PG(1,q)\)上的半线性方程。其大根集恰好是\(\F_{p^{\gcd(m,s)}}\)-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\(\U_{q^2+1}\)的商传输及单位圆参数化，证明最小零点集正是\(\PG(1,q^2)\)中的Baer子线。等价地，相应的支撑设计是椭圆二次曲面\(\Q^-(3,q)\)非切平面截面的补集。对于\(\F_q\)上长度为\(q^2+1\)的常循环卵形码，其存在性的精确判据为：\[ λ\in\F_q^*,\qquad \exists\ λ\text{-常循环卵形码} \Longleftrightarrow λ\notin(\F_q^*)^2。 \] 特别地，负循环卵形码存在当且仅当\(q\equiv3\pmod4\)。证明采用了修正的投影阶同余式：\[ a=(q+1)c,\qquad c\equiv b\pmod{q-1},\qquad \operatorname{ord}(θ\F_q^*)=\frac{q^2+1}{\gcd(q^2+1,c)}。 \] 本文还推导了扩域上提升卵形码的通用重量枚举子，该枚举子独立于所选卵形。最后，构造了连续根负循环MDS码，以给出完全简单5-设计，包括一个本真的负循环\([11,5,7]_{23}\)码，其最小支撑构成完全\(5-(11,7,15)\)设计。