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

Wittawat Kositwattanarerk, Gretchen L. Matthews, Emily McMillon, Tunchanok Yutitumsatit

TL;DR

The paper develops a fundamental-cone framework to study LP decoding pseudocodewords across quantum stabilizer (CSS), quasi-cyclic, and spatially-coupled LDPC codes. It shows that pseudocodewords often factor via product constructions, enabling tractable analysis through generating functions and block-wise decompositions. For CSS codes, pseudocodewords decompose as products of component codes; quasi-cyclic codes preserve LP performance under shifts and admit redundancy-based improvements; spatially-coupled codes yield an interpretable L-fold product structure. The results provide a structural lens on graph-cover/LP decoding and guide parity-check design to control pseudocodeword behavior, with concrete illustrations like Steane codes.

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.

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

TL;DR

The paper develops a fundamental-cone framework to study LP decoding pseudocodewords across quantum stabilizer (CSS), quasi-cyclic, and spatially-coupled LDPC codes. It shows that pseudocodewords often factor via product constructions, enabling tractable analysis through generating functions and block-wise decompositions. For CSS codes, pseudocodewords decompose as products of component codes; quasi-cyclic codes preserve LP performance under shifts and admit redundancy-based improvements; spatially-coupled codes yield an interpretable L-fold product structure. The results provide a structural lens on graph-cover/LP decoding and guide parity-check design to control pseudocodeword behavior, with concrete illustrations like Steane codes.

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.

Paper Structure

This paper contains 8 sections, 14 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of the fundamental cone for general codes
  4. Properties of the fundamental cone for specific classes of codes
  5. Pseudocodewords of quantum stabilizer codes
  6. Pseudocodewords of quasi-cyclic codes
  7. Pseudocodewords of spatially-coupled codes
  8. Conclusion

Key Result

Theorem 7

KLVW Let $H \in \mathbb{F}_2^{r \times n}$ and $\mathbf{p} \in \mathbb{N}^n$. Then $\mathbf{p} \in \mathcal{P}(H)$ if and only if $\mathbf{p} \in \mathcal{K}(H)$ and $H \mathbf{p}^T = \mathbf{0} \mod 2$.

Figures (2)

  • Figure 1: The Tanner graph associated with the parity check matrix in Example \ref{['ex:Tanner_graph']}.
  • Figure 2: A graph cover of $T(H)$ from Example \ref{['ex:Tanner_graph']} where there is an extra copy of each vertex.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 5
  • Definition 6
  • Theorem 7
  • Lemma 8
  • proof
  • Theorem 9
  • ...and 25 more