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On Decoding First- and Second-Order BiD Codes

Devansh Jain, Lakshmi Prasad Natarajan

TL;DR

This work introduces BiD codes as a flexible, length-$3^m$ family of abelian codes with competitive dual-distance properties relative to Reed-Muller codes. It develops a rigorous algebraic framework via the polynomial ring $\\mathscr{R}$ and spectral (DFT) view, identifies the minimum-weight parity checks for the dual of the second-order BiD code, and proves a projection property that enables efficient decoding. The paper then designs fast ML and max-log-MAP decoders for BiD$(m,1,1)$ and BiD$(m,0,1)$, and constructs a belief propagation decoder for BiD$(m,2,2)$ that leverages projections and minimum-weight checks to approach ML performance for moderate block lengths. Simulation results demonstrate near-ML performance for $N=81$ and $N=243$, with BP decoding on BiD$(6,2,2)$ at $N=729$ competitive with CRC-aided Polar codes. Overall, the work highlights BiD codes as a promising algebraic family offering flexible lengths, strong distance properties, and practical decoding strategies that can rival established code families under realistic decoding budgets.

Abstract

BiD codes, which are a new family of algebraic codes of length $3^m$, achieve the erasure channel capacity under bit-MAP decoding and offer asymptotically larger minimum distance than Reed-Muller (RM) codes. In this paper we propose fast maximum-likelihood (ML) and max-log-MAP decoders for first-order BiD codes. For second-order codes, we identify their minimum-weight parity checks and ascertain a code property known as 'projection' in the RM coding literature. We use these results to design a belief propagation decoder that performs within 1 dB of ML decoder for block lengths 81 and 243.

On Decoding First- and Second-Order BiD Codes

TL;DR

This work introduces BiD codes as a flexible, length- family of abelian codes with competitive dual-distance properties relative to Reed-Muller codes. It develops a rigorous algebraic framework via the polynomial ring and spectral (DFT) view, identifies the minimum-weight parity checks for the dual of the second-order BiD code, and proves a projection property that enables efficient decoding. The paper then designs fast ML and max-log-MAP decoders for BiD and BiD, and constructs a belief propagation decoder for BiD that leverages projections and minimum-weight checks to approach ML performance for moderate block lengths. Simulation results demonstrate near-ML performance for and , with BP decoding on BiD at competitive with CRC-aided Polar codes. Overall, the work highlights BiD codes as a promising algebraic family offering flexible lengths, strong distance properties, and practical decoding strategies that can rival established code families under realistic decoding budgets.

Abstract

BiD codes, which are a new family of algebraic codes of length , achieve the erasure channel capacity under bit-MAP decoding and offer asymptotically larger minimum distance than Reed-Muller (RM) codes. In this paper we propose fast maximum-likelihood (ML) and max-log-MAP decoders for first-order BiD codes. For second-order codes, we identify their minimum-weight parity checks and ascertain a code property known as 'projection' in the RM coding literature. We use these results to design a belief propagation decoder that performs within 1 dB of ML decoder for block lengths 81 and 243.
Paper Structure (29 sections, 9 theorems, 34 equations, 4 figures, 3 tables)

This paper contains 29 sections, 9 theorems, 34 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. BiD Codes: Construction & Properties
  3. Algebraic Framework
  4. Codes from $\mathscr{R}$
  5. BiD Codes
  6. Minimum-Weight Parity Checks of ${\rm BiD}(m,2,2)$
  7. Main Idea
  8. Minimum-Weight Parity Checks
  9. On the Dual Distance of Other BiD Codes
  10. Projection Property of BiD Codes
  11. Decoding First- and Second-Order Codes
  12. Decoding ${\rm BiD}(m,1,1)$ and ${\rm BiD}(m,0,1)$
  13. Belief Propagation Decoding of ${\rm BiD}(m,2,2)$
  14. Simulation Results
  15. First-Order Codes
  16. ...and 14 more sections

Key Result

Lemma 2.1

For any $m \geq 1$ and $\mathscr{W} \subseteq \{0,\dots,m\}$ the following permutations are automorphisms of $\mathscr{C}_{\rm A}(m,\mathscr{W})$

Figures (4)

  • Figure 1: Comparison of first-order BiD and RM codes (ML decoding).
  • Figure 2: Second-order BiD code of length $81$.
  • Figure 3: Second-order BiD code of length $243$.
  • Figure 4: BLER comparison of codes with length $729$

Theorems & Definitions (10)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Lemma 4.1
  • Theorem 4.2
  • Lemma 1
  • Theorem 2