Classical Tensor Codes and Quantum Tanner Codes Achieve Capacity on Erasure Channels

arXiv:2611.14087v1 Announce Type: new Abstract: 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 $\rho\in(0,1/2)$ and every rate gap $\varepsilon>0$, we obtain a family of binary quantum Tanner codes of rate at least $1-2\rho-\varepsilon$. For fixed $\rho$ 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.