The Ternary and Quaternary Cases of the $\{2,4\}$-Distance Code Conjecture

For an integer $q\ge2$, let $A_q(n,\{2,4\})$ be the largest cardinality of an unrestricted $q$-ary code of length $n$ in which any two distinct words have Hamming distance $2$ or $4$. We prove that \[ A_q(n,\{2,4\})=1+\binom n2 \qquad(q\in\{3,4\},\ n\ge6), \] thereby settling the two remaining a