Polynomial Constructions and Deletion-Ball Geometry for Multiset Deletion Codes

We study error-correcting codes in the space $\mathcal{S}_{n,q}$ of length-$n$ multisets over a $q$-ary alphabet under the deletion metric, motivated by permutation channels in which ordering is completely lost and errors act only on symbol multiplicities. We develop two complementary directions. First, we present polynomial Sidon-type constructions over finite fields, in both projective and affine forms, yielding multiset $t$-deletion-correcting codes in the regime $t<q$ with redundancy $t+O(1)$, independent of the blocklength $n$. Second, we develop a geometric analysis of deletion balls in $\mathcal{S}_{n,q}$. Using difference-vector representations together with a diagonal reduction of the relevant generating functions, we derive exact generating-function expressions for individual deletion-ball sizes, exact formulas for the number of ordered pairs of multisets at a fixed distance $m$, and consequently for the average ball size. We prove that radius-$r$ deletion balls are minimized at extreme multisets and maximized at the most balanced multisets, giving a formal global characterization of extremal centers in $\mathcal{S}_{n,q}$. We further relate the maximal-ball value to the ideal difference set $S_{q-1}(r,r)$ through boundary truncation, obtaining explicit closed forms for $q=2$ and $q=3$. These geometric results lead to volume-based bounds on code size, including sphere-packing upper bounds, a boundary-aware analysis of code--anticode arguments, and Gilbert--Varshamov-type lower bounds governed by exact average ball sizes. For fixed $q$ and $t$, the resulting average-ball lower bound matches the interior-difference-set scale asymptotically.