Asymptotically good homological error correcting codes

Abstract

Let $\Delta$ be an abstract simplicial complex. We study classical homological error correcting codes associated to $\Delta$, which generalize the cycle codes of simple graphs. It is well-known that cycle codes of graphs do not yield asymptotically good families of codes. We show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least $2$. We also prove general bounds and formulas for (co-)cycle and (co-)boundary codes for arbitrary simplicial complexes over arbitrary fields.