Algebraic Resolutions of Seven Open Problems on Cyclic and Negacyclic Codes Supporting Designs

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 $$\lambda\in\F_q^*,\qquad \exists\ \lambda\text{-constacyclic ovoid code} \Longleftrightarrow \lambda\notin(\F_q^*)^2.$$ In particular, negacyclic ovoid codes exist exactly when $q\equiv3\pmod4$. The proof uses the corrected projective-order congruence $$a=(q+1)c,\qquad c\equiv b\pmod{q-1},\qquad \operatorname{ord}(\theta\F_q^*)=\frac{q^2+1}{\gcd(q^2+1,c)}.$$ 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.