Coded Compressed Sensing With List Recoverable Codes for the Unsourced Random Access

Kirill V. Andreev, Pavel S. Rybin, Alexey A. Frolov

Abstract: We consider a coded compressed sensing approach for the unsourced random access and replace the outer tree code proposed by Amalladinne et al. (2020) with the list recoverable code capable of correcting <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> errors. A finite-length random coding bound for such codes is derived. The numerical experiments in the single-antenna quasi-static Rayleigh fading channel show that transition to list recoverable codes correcting <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> errors improves the performance of the coded compressed sensing scheme by 7–10 dB compared to the tree code-based scheme. We propose two practical constructions of outer codes. The first is a modification of the tree code called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> -tree code. It utilizes the same code structure, and a key difference is a decoder capable of correcting up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> errors. The second is based on the Reed–Solomon codes and Guruswami–Sudan list decoding algorithm. The first scheme provides energy efficiency very close to the random coding bound when the decoding complexity (number of decoding paths) is unbounded. But when we restrict the number of decoding paths with a practical value, the second scheme outperforms the first one. Both schemes improve the performance of a tree code-based scheme for a small and moderate number of active users.