Asymptotically Tight Bounds for Generalized Covering Radii of Binary Primitive BCH Codes at All Higher Orders

We study how few parity-check columns are needed to span several prescribed syndromes of a binary primitive BCH code of length $2^m-1$, where $m$ is the extension degree. For the four-error-correcting family, the second generalized covering radius is exactly $11$ for $m\geq55$, and it is either $11$ or $12$ for $m\geq16$. For every fixed error parameter, we give an explicit stable two-value bound for the second radius together with an arithmetic criterion for exactness. For every generalized-covering order $t\geq2$, we also obtain explicit stable lower and upper bounds. Under an additional explicit field-size condition, their additive gap is bounded independently of $t$ for every fixed $e$, so the bounds are asymptotically tight as $t$ grows. The upper bound is the natural common-core count whenever $2\leq e\leq6$, and in general differs from it by a correction bounded independently of $t$. At order three this gives stable intervals for three through six errors. We further prove that, for three errors, every three-dimensional syndrome space confined to the highest coordinate has exact support size ten once $m\geq18$. A single self-contained completion-cover framework supplies both the exact second-order results and the asymptotically tight bounds at all higher orders.