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Cyclic Group Projection for Enumerating Quasi-Cyclic Codes Trapping Sets

Vasiliy Usatyuk, Yury Kuznetsov, Sergey Egorov

TL;DR

This work addresses efficient enumeration and weighing of trapping sets in quasi-cyclic codes with composite circulant sizes. It introduces a cyclic-group projection framework that relates trapping sets across circulant sizes via projections $P_{z\to z_*}$ and lifting, leveraging $l$-coverings and the automorphism group $\mathcal{G}_{n,z}$. A tabular importance-sampling approach is developed to estimate pseudo-codeword weights, exploiting QC structure and orbit decompositions to reduce variance and computation. The combined method enables scalable TS analysis and error-floor estimation for QC codes used in standards, such as 5G QC-LDPC and DVB codes, enhancing decoding reliability in practical channels.

Abstract

This paper introduces a novel approach to enumerate and assess Trapping sets in quasi-cyclic codes, those with circulant sizes that are non-prime numbers. Leveraging the quasi-cyclic properties, the method employs a tabular technique to streamline the importance sampling step for estimating the pseudo-codeword weight of Trapping sets. The presented methodology draws on the mathematical framework established in the provided theorem, which elucidates the behavior of projection and lifting transformations on pseudo-codewords

Cyclic Group Projection for Enumerating Quasi-Cyclic Codes Trapping Sets

TL;DR

This work addresses efficient enumeration and weighing of trapping sets in quasi-cyclic codes with composite circulant sizes. It introduces a cyclic-group projection framework that relates trapping sets across circulant sizes via projections and lifting, leveraging -coverings and the automorphism group . A tabular importance-sampling approach is developed to estimate pseudo-codeword weights, exploiting QC structure and orbit decompositions to reduce variance and computation. The combined method enables scalable TS analysis and error-floor estimation for QC codes used in standards, such as 5G QC-LDPC and DVB codes, enhancing decoding reliability in practical channels.

Abstract

This paper introduces a novel approach to enumerate and assess Trapping sets in quasi-cyclic codes, those with circulant sizes that are non-prime numbers. Leveraging the quasi-cyclic properties, the method employs a tabular technique to streamline the importance sampling step for estimating the pseudo-codeword weight of Trapping sets. The presented methodology draws on the mathematical framework established in the provided theorem, which elucidates the behavior of projection and lifting transformations on pseudo-codewords
Paper Structure (6 sections, 3 theorems, 47 equations, 4 tables, 1 algorithm)

This paper contains 6 sections, 3 theorems, 47 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic definition
  3. Cyclic Group Decomposition
  4. QC Codes TS Enumerating
  5. Quasi-Cyclic Codes Trapping Set Weighing
  6. CONCLUSION

Key Result

Theorem 1

For any pseudocode word $(w, v)$-pseudocode word $x\in \mathbb{F}_2^{nz}$ relative to the check matrix $H$ vector $x_\ast = P_{z\rightarrow z_\ast}x\in \mathbb{F}_2^{nz_\ast}$ will $(w', v')$-TS pseudo-code word relative to the parity check matrix $H_\ast = \mathbb{P}_{z\rightarrow z_\ast}H$, and $w As a consequence, the linear transformation maps the linear code $C(H)$ to some subset of code $C(

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Theorem 1
  • ...and 10 more