We show that on the classical erasure channel, the tensor product of two capacity-achieving code families is itself capacity-achieving under mild conditions. We also prove that duals of tensor products of independent uniform random binary linear codes achieve erasure capacity. Using these local codes, for every erasure probability $ρ\in(0,1/2)$ and every rate gap $\varepsilon>0$, we obtain a family of binary quantum Tanner codes of rate at least $1-2ρ-\varepsilon$. For fixed $ρ$ and $\varepsilon$, these codes have bounded check weights, minimum distance growing linearly with the blocklength, and a linear-time erasure decoder whose decoding error decays exponentially with the blocklength.

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