A-6.3 Approximate Decomposition

If no level $\ell$ satisfies ${\beta }^{\ell }\mid q$, then some lower bits of $q$ must be discarded during decomposition, as shown in Figure 2. Such lower bits can be rounded to the nearest multiple of ${\displaystyle \frac{q}{{\beta }^{\ell }}}$ during decomposition. In such a case, the decomposition is an approximate decomposition. Formally, when ${\beta }^{\ell }\nmid q$ we can write

(using nearest-integer rounding and identifying $\gamma$ with its integer representative). The polynomial case is analogous, coefficient-wise.