A Homomorphic Secret Sharing (HSS) scheme is a secret-sharing scheme that shares a secret $x$ among $s$ servers, and additionally allows an output client to reconstruct some function $f(x)$, using information that can be locally computed by each server. A key parameter in HSS schemes is download rate, which quantifies how much information the output client needs to download from each server. Recent work (Fosli, Ishai, Kolobov, and Wootters, ITCS 2022) established a fundamental limitation on the download rate of linear HSS schemes for computing low-degree polynomials, and gave an example of HSS schemes that meet this limit. In this paper, we further explore optimal-rate linear HSS schemes for polynomials. Our main result is a complete characterization of such schemes, in terms of a coding-theoretic notion that we introduce, termed optimal labelweight codes. We use this characterization to answer open questions about the amortization required by HSS schemes that achieve optimal download rate. In more detail, the construction of Fosli et al. required amortization over $\ell$ instances of the problem, and only worked for particular values of $\ell$. We show that -- perhaps surprisingly -- the set of $\ell$'s for which their construction works is in fact nearly optimal, possibly leaving out only one additional value of $\ell$. We show this by using our coding-theoretic characterization to prove a necessary condition on the $\ell$'s admitting optimal-rate linear HSS schemes. We then provide a slightly improved construction of optimal-rate linear HSS schemes, where the set of allowable $\ell$'s is optimal in even more parameter settings. Moreover, based on a connection to the MDS conjecture, we conjecture that our construction is optimal for all parameter regimes.