We study the expected number of random linear observations required to recover a prescribed information symbol. For ensembles with uniformly random supports, independently sampled nonzero coefficients, and uniformly bounded column weights, we prove that the asymptotic normalized expected recovery time is at least $π/4$ under optimal decoding. The result holds over finite or infinite fields and extends the lower bound previously established for binary fully symmetric codes under peeling decoding. For singleton - pair observations, we determine the exact limiting expectation and show that optimal and peeling decoding have the same limit, independently of the field and coefficient distribution. The minimum within this family is approximately $0.793933445$, which explicit constructions attain asymptotically over every fixed finite field.

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