We study the $(K,N,S,M,K_c)$ distributed linearly separable computation (DLSC) setting, where a master employs $N$ workers with a per-worker computation load of $M$ to recover $K_c$ deterministic linear combinations of $K$ messages in the presence of up to $S$ stragglers. We propose a DLSC framework based on nested cyclic codes that achieves $M=\frac{K}{N}(S+K_c)$. For sufficiently large $K$, we establish the matching converse $M\geq\frac{K}{N}(S+K_c)$, proving the asymptotic computation load optimality of the proposed framework. By instantiating cyclic locally recoverable codes (LRCs) as base codes in the framework, the master can recover the desired $K_c$ linear combinations by contacting fewer than $N-S$ workers under many straggler scenarios. This essentially softens the straggler threshold without increasing the computation load or communication cost. Finally, we extend the construction to reduce the per-worker communication load by a factor of $m$, while achieving an asymptotically optimal computation load of $M=\frac{K}{N}(S+mK_c)$.

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