Pseudocodewords of quantum, quasi-cyclic, and spatially-coupled LDPC codes: a fundamental cone perspective

TL;DR

This paper investigates the structure of pseudocodewords in various LDPC codes, including quantum, quasi-cyclic, and spatially-coupled types, using a fundamental cone perspective to understand decoding performance.

Contribution

It introduces a fundamental cone analysis of pseudocodewords for these codes, linking LP decoding to their structural properties and decoding behavior.

Findings

Pseudocodewords depend on the code's parity-check matrix and decoding algorithm.

Quantum stabilizer codes exhibit specific pseudocodeword structures under LP decoding.

Spatially-coupled LDPC codes have pseudocodewords influencing their decoding performance.

Abstract

While low-density parity-check (LDPC) codes are near capacity-achieving when paired with iterative decoders, these decoders may not output a codeword due to the existence of pseudocodewords. Thus, pseudocodewords have been studied to give insight into the performance of modern decoders including iterative and linear programming decoders. These pseudocodewords are found to be dependent on the parity-check matrix of the code and the particular decoding algorithm used. In this paper, we consider LP decoding, which has been linked to graph cover decoding, providing functions which capture these pseudocodewords. In particular, we analyze the underlying structure of pseudocodewords from quantum stabilizer codes that arise from LP decoding, quasi-cyclic LDPC codes, and spatially-coupled LDPC codes.