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arXiv:2402.00562 · IEEE International Symposium on Information Theory (ISIT)

Endomorphisms of Linear Block Codes

Jonathan Mandelbaum, Sisi Miao, Holger Jäkel, Laurent Schmalen

Published Jul 7, 2024• 4 citations
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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 endomor-phisms, 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.

DOI
10.1109/isit57864.2024.10619704

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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 endomor-phisms, 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.
Low-density parity-check codes together with belief propagation (BP) decoding are known to be well-performing for large block lengths. However, for short block lengths there is still a considerable gap between the performance of BP decoding and maximum likelihood decoding. Different ensemble decoding schemes such as, e.g., automorphism ensemble decoding (AED), can reduce this gap in short block length regime. We propose generalized AED (GAED) that uses automorphisms according to the definition in linear algebra. Here, an automorphism of a vector space is defined as a linear, bijective self-mapping, whereas in coding theory self-mappings that are scaled permutations are commonly used. We show that the more general definition leads to an explicit joint construction of codes and automorphisms, and significantly enlarges the search space for automorphisms of existing linear codes. Furthermore, we prove the concept that generalized automorphisms can indeed be used to improve decoding. Additionally, we propose a code construction of linear codes enabling the construction of codes with suitably designed automorphisms. Finally, we analyze the decoding performances of GAED for some of our constructed codes.
Low-density parity-check (LDPC) codes together with belief propagation (BP) decoding yield exceptional error correction capabilities in the large block length regime. Yet, there remains a gap between BP decoding and maximum likelihood decoding for short block length LDPC codes. In this context, ensemble decoding schemes yield both reduced latency and good error rates. In this paper, we propose subcode ensemble decoding (SCED), which employs an ensemble of decodings on different subcodes of the code. To ensure that all codewords are decodable, we use the concept of linear coverings and explore approaches for sampling suitable ensembles for short block length LDPC codes. Monte-Carlo simulations conducted for three LDPC codes demonstrate that SCED improves decoding performance compared to stand-alone decoding and automorphism ensemble decoding. In particular, in contrast to existing schemes, e.g., multiple bases belief propagation and automorphism ensemble decoding, SCED does not require the NP-complete search for low-weight dual codewords or knowledge of the automorphism group of the code, which is often unknown.
In this paper, we propose an analysis of the automorphism group of polar codes, with the aim of designing codes tailored forautomorphism ensemble(AE) decoding. Using a novel description of polar codes as monomial codes through negative monomials, we prove the equivalence between the notion ofdecreasing monomial codesand the universal partial order (UPO) framework for polar codes; this property is widely believed to hold true but a formal proof was missing. We further provide a rigorous mathematical connection between code word permutations and affine transformations, an important link to understand the considered automorphisms. Based on this mathematical formalisms, we analyze the algebraic properties of theaffine automorphisms groupof polar codes, providing a novel description of its structure. We classify automorphisms such that all automorphisms in the same class lead to the same result under permutation decoding, which gives rise to the concept ofredundantautomorphisms. Mathematically this is achieved by introducing equivalence classes of affine automorphisms under AE-based decoding. For practical application, we provide an algorithm to compute representatives for the equivalence classes, such that one automorphism from each equivalence class can be selected for use in AE decoding. A numerical analysis of the error correction performance of AE decoding of polar codes, based on equivalence classes, concludes the paper.
Automorphism ensemble (AE) decoding for polar codes was proposed by decoding permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to that of successive cancellation list (SCL) decoding. However, some automorphisms are SC-invariant, thus are redundant in AE decoding. In this paper, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, we provide an algorithm to determine the complete SC-invariant affine automorphisms under a specific polar code construction.
A parameterized universal decoding framework based on polar transformations was recently proposed to enable polar-style decoding of general binary linear block codes (BLBCs). However, existing parallelization methods for polar codes cannot be directly applied to this framework. The key obstacle is that these methods rely on static frozen-set structures, which are incompatible with the dynamic frozen constraints induced by polar transformations. To address this challenge, we revisit code automorphisms from a new perspective: instead of applying them to the codeword space, we let them act on the transformation parameter itself. We show that the BLBC automorphism group induces equivalence polar transformation classes. Crucially, all transformations within an equivalence class share the same polar subcode structure, eliminating the need for separate decoder designs. This insight enables a novel parallel decoding framework termed polar ensemble decoding (PED). By decoding multiple equivalent transformations simultaneously, PED exploits transformation diversity while maintaining decoder compatibility. Simulation results on extended BCH and extended Golay codes demonstrate that PED achieves near maximum-likelihood performance while significantly reducing the decoding latency compared to successive cancellation list (SCL) decoding.
Abstract — We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász’s Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed prop-erties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.