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Soft Decision Decoding of Recursive Plotkin Constructions Based on Hidden Code Words

Martin Bossert

TL;DR

The paper introduces a novel soft-decision decoding framework for recursive Plotkin constructions that uncovers hidden code words by combining blocks in multiple ways. By running several independent decoding variants—each starting from a different hidden word—and selecting the best by correlation with the received vector, the approach achieves near-ML performance while keeping decoding complexity low thanks to short recursive blocks. The method is validated on RM- and BCH-based double Plotkin constructions, and extended to half-rate, low-rate, and high-rate codes, with results showing substantial gains from variant diversity and list decoding. This work offers a scalable, parallelizable decoding strategy for recursive code constructions and demonstrates practical gains for a broad class of codes beyond RM codes.

Abstract

The Plotkin construction combines two codes to a code of doubled length. It can be applied recursively. The class of Reed-Muller (RM) codes is a particular example. Also, a special class of generalized concatenated codes (GCC) can be described as recursive Plotkin construction. Exploiting a property of the code words constructed by the recursive Plotkin construction, we present novel soft-decision decoders. These are based on the decoding of hidden code words which are inherent contained in the constructed code words and can be uncovered by adding particular parts of the overall code word. The main idea is to use more than one decoding variant where each variant starts with the decoding of a different hidden code word. The final decoding decision selects the best of the decisions of the used variants. The more variants are used the closer the performance gets to the maximum-likelihood (ML) decoding performance. This is verified by an ML-bound for the cases where the ML performance is not known. The decoding algorithms use only additions, comparisons, and sign operations. Further, due to the recursive structure, only relatively short codes have to be decoded, thus, the decoding complexity is very low. In addition, we introduce two novel classes of half-rate codes based on recursive Plotkin constructions with RM codes.

Soft Decision Decoding of Recursive Plotkin Constructions Based on Hidden Code Words

TL;DR

The paper introduces a novel soft-decision decoding framework for recursive Plotkin constructions that uncovers hidden code words by combining blocks in multiple ways. By running several independent decoding variants—each starting from a different hidden word—and selecting the best by correlation with the received vector, the approach achieves near-ML performance while keeping decoding complexity low thanks to short recursive blocks. The method is validated on RM- and BCH-based double Plotkin constructions, and extended to half-rate, low-rate, and high-rate codes, with results showing substantial gains from variant diversity and list decoding. This work offers a scalable, parallelizable decoding strategy for recursive code constructions and demonstrates practical gains for a broad class of codes beyond RM codes.

Abstract

The Plotkin construction combines two codes to a code of doubled length. It can be applied recursively. The class of Reed-Muller (RM) codes is a particular example. Also, a special class of generalized concatenated codes (GCC) can be described as recursive Plotkin construction. Exploiting a property of the code words constructed by the recursive Plotkin construction, we present novel soft-decision decoders. These are based on the decoding of hidden code words which are inherent contained in the constructed code words and can be uncovered by adding particular parts of the overall code word. The main idea is to use more than one decoding variant where each variant starts with the decoding of a different hidden code word. The final decoding decision selects the best of the decisions of the used variants. The more variants are used the closer the performance gets to the maximum-likelihood (ML) decoding performance. This is verified by an ML-bound for the cases where the ML performance is not known. The decoding algorithms use only additions, comparisons, and sign operations. Further, due to the recursive structure, only relatively short codes have to be decoded, thus, the decoding complexity is very low. In addition, we introduce two novel classes of half-rate codes based on recursive Plotkin constructions with RM codes.
Paper Structure (30 sections, 12 theorems, 47 equations, 15 figures, 6 tables)

This paper contains 30 sections, 12 theorems, 47 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction and Notations
  2. The Plotkin Construction
  3. Recursive Plotkin Construction
  4. Reed--Muller Codes
  5. Generalized Code Concatenation
  6. Soft Decision Decoding
  7. Decoding Functions
  8. Correlation and ML Decoding
  9. ML Decoding Bound
  10. Decoding Strategies for Plotkin Constructions
  11. Error Cancellation
  12. Decoding Variants
  13. Variants Starting Decoding with $\mathcal{C}_3$
  14. Variants Starting Decoding with $\mathcal{C}_1$, $\mathcal{C}_2$
  15. Variants Starting Decoding with $\mathcal{C}_0$
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\mathcal{C}_0(n,k_0,d_0)$ and $\mathcal{C}_1(n,k_1,d_1)$ be two binary linear component codes. Then, the code $\mathcal{C} = \{ \mathbf c = |\mathbf u_0|\mathbf u_0+\mathbf u_1|:\ \mathbf u_0\in \mathcal{C}_0,\ \mathbf u_1 \in \mathcal{C}_1 \}$ has length $2n$, dimension $k= k_0 + k_1$, and min

Figures (15)

  • Figure 1: Recursive Plotkin Construction of RM codes
  • Figure 2: Partitioning of $(2,2,1)$
  • Figure 3: Partitioning of $(4,4,1)$
  • Figure 4: BER for Different (join, add) Combinations
  • Figure 5: WER $\mathcal{R}(2,5)$ starting with $\mathcal{C}_3$
  • ...and 10 more figures

Theorems & Definitions (33)

  • Theorem 1: Plotkin Construction
  • proof
  • Example 1: Recursive Plotkin Construction
  • Lemma 1: Hidden Code Words
  • proof
  • Example 2: Hidden Code Words
  • Definition 1: Binary RM-Code
  • Remark 1
  • Example 3: GCC and Plotkin Construction
  • Lemma 2
  • ...and 23 more