Asymptotically good homological error correcting codes
Keywords:
Error correcting codes, Simplicial complexes, Simplicial homology
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.
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A. R. Calderbank, P. W. Shor, Good quantum error–correcting codes exist, Physical Review A 54(2) (1996) 1098–1105.
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R. G. Gallager, Low–density parity–check codes, IRE Trans. 8(1) (1962) 21–28.
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Steven Roman, Coding and Information Theory, Graduate Texts in Mathematics, vol. 134, Springer– Verlag, New York, 1992.
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G. Zémor, On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction, Coding and cryptology, Lecture Notes in Comput. Sci., vol. 5557, Springer, Berlin (2009) 259–273.
