We strengthen the notion of double samplers, first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders.
The ABNNR code construction [IEEE Trans. Inform. Theory, 38(2):509–516] achieves large distance by starting with a base code C with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler then the construction has an efficient list-decoding algorithm. Our algorithm works even if the ABNNR construction is not applied to a base code C but to any string. In this case the resulting code is approximate-list-decodable, i.e. the output list contains an approximation to the original input.
Our list-decoding algorithm works as follows: it uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list-decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results.
Double samplers and high dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.
We develop the notion of double samplers, first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders.
We show how double samplers give a generic way of amplifying distance in a way that enables efficient list-decoding. There are many error correcting code constructions that achieve large distance by starting with a base code $C$ with moderate distance, and then amplifying the distance using a sampler, e.g., the ABNNR code construction [IEEE Trans. Inform. Theory, 38(2):509--516, 1992.]. We show that if the sampler is part of a larger double sampler then the construction has an efficient list-decoding algorithm and the list decoding algorithm is oblivious to the base code $C$ (i.e., it runs the unique decoder for $C$ in a black box way).
Our list-decoding algorithm works as follows: it uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results.
Double samplers and high dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.
