We study successor right-special strings over an alphabet $Σ$ of size $σ$, a minimal-branching analogue of de Bruijn strings, and ask how few Burrows--Wheeler transform (BWT) runs are possible. In a de Bruijn string of order $k$, every $(k-1)$-context has all $σ$ right-extensions; here, every context is still right-special but has exactly two right-extensions, chosen by a successor rule. For order $3$, we construct an explicit family $B_σ^{(3)}$, for every $σ\geq 2$, whose cyclic BWT has $r_c = σ^2 + 2$ runs. A suitable terminated linearization has the same run count, $r = r_c = σ^2 + 2$, while the smallest suffixient set has size $χ= 2σ^2 + 1$. The ratio $χ/r = 2 - 3/(σ^2 + 2)$ then quantifies how nearly this forced branching saturates the known bound $χ/r \leq 2$, which we have previously shown to be asymptotically tight. Compared with our earlier alphabet-growing construction, this improves the gap from $O(1/σ)$ to $O(1/σ^2)$. We also show that the order-$3$ pattern appears as a blockwise two-row projection of normalized linear-feedback shift register (LFSR) de Bruijn sequences over $\mathbb F_q$, when such primitive trinomials $x^3 - x + c$ exist. For higher orders, we analyze the natural boundary-merged candidate $L_{σ,k}$ using the last-to-first (LF) permutation: it fails for $k = 4$ and all $σ\geq 3$, while verified $k = 5$ instances for $σ\in {3,4}$ yield $χ/r$ ratios exceeding $1.96$.

翻译：本文研究字母表$Σ$（大小为$σ$）上的后继右特殊字符串——de Bruijn字符串在最小分支意义下的类比，并探究其Burrows-Wheeler变换（BWT）游程数的最小可能值。在$k$阶de Bruijn字符串中，每个$(k-1)$阶上下文具有全部$σ$种右扩展；而此处每个上下文虽仍为右特殊，但仅由后继规则选定恰好两种右扩展。对于$3$阶情形，我们为每个$σ\geq 2$构造了一个显式族$B_σ^{(3)}$，其循环BWT的游程数$r_c = σ^2 + 2$。合适的带终止线性化版本拥有相同游程数$r = r_c = σ^2 + 2$，而最小后缀集的大小为$χ= 2σ^2 + 1$。比值$χ/r = 2 - 3/(σ^2 + 2)$量化了这种强制分支在多大程度上趋近已知界$χ/r \leq 2$——我们此前已证明该界渐近紧确。与早期字母表增长构造相比，本方法将差距从$O(1/σ)$改进至$O(1/σ^2)$。我们还证明，当原始三项式$x^3 - x + c$存在时，$3$阶模式可表现为$\mathbb F_q$上归一化线性反馈移位寄存器（LFSR）de Bruijn序列的块状两行投影。对于更高阶情形，我们利用最后一列到第一列（LF）置换分析了自然边界合并候选序列$L_{σ,k}$：该方案在$k=4$且所有$σ\geq 3$时失败，而$k=5$时对$σ\in {3,4}$验证的实例中$χ/r$比值超过$1.96$。