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Endomorphisms of Linear Block Codes

Jonathan Mandelbaum, Sisi Miao, Holger Jäkel, Laurent Schmalen

TL;DR

The paper extends the study of code automorphisms to endomorphisms of linear block codes, clarifying the algebraic structure of endomorphism transformation matrices through a construction $\bm{T}=\bm{A}\bm{Z}\bm{A}^{-1}$ and linking endomorphisms to a larger code $\mathcal{C}_{\mathrm{E}}(n^2,2kn-k^2)$. It shows a one-to-one mapping between endomorphism matrices and $\mathcal{C}_{\mathrm{E}}$, enabling practical search for endomorphisms, and demonstrates how to build endomorphisms from automorphisms. The authors then propose Endomorphism Ensemble Decoding (EED), a decoding framework that uses multiple endomorphism paths, a reconstruction matrix $\bm{R}$, and ML-in-the-list selection to improve performance over traditional AED/GAED in short-block-length scenarios. Numerical results on Golay and short polar codes illustrate FER gains, highlighting EED's potential to enhance decoding where automorphism-based methods alone fall short. This work thus provides both structural insights into endomorphisms and a versatile decoding paradigm leveraging them.

Abstract

The automorphism groups of various linear codes are extensively studied yielding insights into the respective code structure. This knowledge is used in, e.g., theoretical analysis and in improving decoding performance, motivating the analyses of endomorphisms of linear codes. In this work, we discuss the structure of the set of transformation matrices of code endomorphisms, defined as a generalization of code automorphisms, and provide an explicit construction of a bijective mapping between the image of an endomorphism and its canonical quotient space. Furthermore, we introduce a one-to-one mapping between the set of transformation matrices of endomorphisms and a larger linear block code enabling the use of well-known algorithms for the search for suitable endomorphisms. Additionally, we propose an approach to obtain unknown code endomorphisms based on automorphisms of the code. Furthermore, we consider ensemble decoding as a possible use case for endomorphisms by introducing endomorphism ensemble decoding. Interestingly, EED can improve decoding performance when other ensemble decoding schemes are not applicable.

Endomorphisms of Linear Block Codes

TL;DR

The paper extends the study of code automorphisms to endomorphisms of linear block codes, clarifying the algebraic structure of endomorphism transformation matrices through a construction and linking endomorphisms to a larger code . It shows a one-to-one mapping between endomorphism matrices and , enabling practical search for endomorphisms, and demonstrates how to build endomorphisms from automorphisms. The authors then propose Endomorphism Ensemble Decoding (EED), a decoding framework that uses multiple endomorphism paths, a reconstruction matrix , and ML-in-the-list selection to improve performance over traditional AED/GAED in short-block-length scenarios. Numerical results on Golay and short polar codes illustrate FER gains, highlighting EED's potential to enhance decoding where automorphism-based methods alone fall short. This work thus provides both structural insights into endomorphisms and a versatile decoding paradigm leveraging them.

Abstract

The automorphism groups of various linear codes are extensively studied yielding insights into the respective code structure. This knowledge is used in, e.g., theoretical analysis and in improving decoding performance, motivating the analyses of endomorphisms of linear codes. In this work, we discuss the structure of the set of transformation matrices of code endomorphisms, defined as a generalization of code automorphisms, and provide an explicit construction of a bijective mapping between the image of an endomorphism and its canonical quotient space. Furthermore, we introduce a one-to-one mapping between the set of transformation matrices of endomorphisms and a larger linear block code enabling the use of well-known algorithms for the search for suitable endomorphisms. Additionally, we propose an approach to obtain unknown code endomorphisms based on automorphisms of the code. Furthermore, we consider ensemble decoding as a possible use case for endomorphisms by introducing endomorphism ensemble decoding. Interestingly, EED can improve decoding performance when other ensemble decoding schemes are not applicable.
Paper Structure (9 sections, 4 theorems, 41 equations, 2 figures)

This paper contains 9 sections, 4 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Endomorphisms of Codes
  4. Endomorphisms of Codes
  5. Transformation Matrices form a Linear Code
  6. Endomorphism based on Automorphisms
  7. Endomorphism Ensemble Decoding
  8. Numerical Results
  9. Conclusion

Key Result

Theorem 1

Let $\mathcal{C}$ be a linear code with PCM $\bm{H}\in \mathbb{F}_q^{(n-k)\times n}$ of rank $n-k$ and CCM $\bm{A}\in \mathcal{A}$. Then, the mapping $\tau:\mathcal{C}\rightarrow\mathcal{C}$ with transformation matrix $\bm{T}$ is an element of $\mathop{\mathrm{\mathrm{End}(\mathcal{C})}}\limits$ if

Figures (2)

  • Figure 1: Block diagram of EED using $K$ different endomorphisms with transformation matrices ${\bm{T}_i \in \mathop{\mathrm{\mathcal{T}_E(\mathcal{C})}}\limits}$.
  • Figure 2: Performance of different decoders for the extended Golay code $\mathcal{C}_{\mathrm{G}}(24,12)$. All BP decodings are based on an overcomplete PCM following the construction proposed in MBBP2. The auxiliary paths of EED use endomorphisms with weight over permutation ${\Delta(\bm{T})=8}$ and rank deficiency $s=8$.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof