Graphical Analysis of Lifted Product Code Constructions
For quantum error correction researchers, this work provides foundational graph-theoretic understanding of lifted product codes, though it is incremental as it builds on known code constructions.
The paper proves that the Tanner graphs of the X and Z parity-check matrices of lifted product codes are isomorphic, establishes conditions for their connectivity, and bounds their minimal absorbing sets, providing new insight into decoding performance.
Lifted product codes are an important family of quantum low-density parity-check (QLDPC) codes, as they were the first QLDPC code family shown to be asymptotically good. Understanding the structure of their parity-check matrices $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$, as well as the associated Tanner graphs, is essential for analyzing their decoding behavior and error-floor performance. In this work, we show that the Tanner graphs of $H_{\mathsf{X}}$ and $H_{\mathsf{Z}}$ are indeed isomorphic, and investigate their graph-theoretical structure. We establish conditions ensuring the connectivity of these graphs and provide bounds on their minimal absorbing sets, providing new insight into the combinatorial structures influencing decoding performance.