We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in n dimensions,
the existence of efficient streaming algorithms which can process \Omega(n^2) updates implies efficient linear sketching algorithms with comparable cost.
This improves upon the previous work of Li, Nguyen and Woodruff [LNW14] and Ai, Hu, Li and Woodruff [AHLW16] which required a triple-exponential number of updates to achieve a similar result for updates over integers. We extend our results to updates modulo p for integers p \ge 2, and to approximation instead of exact computation.