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Recursive decoding of projective Reed-Muller codes

Rodrigo San-José

TL;DR

This work addresses decoding Projective Reed-Muller (PRM) codes by exploiting a recursive decomposition that links PRM and affine RM codes. The authors develop a decoder that corrects up to $\eta_d(m)/2$ errors, using decoders for affine RM codes at each recursion level, and extend the approach to Projective Reed-Solomon codes and general PRM cases. They show the proposed method improves upon the previously known PRM decoding bounds and analyze its complexity, which remains tied to the complexity of the affine RM decoders. The results enable practical, scalable decoding of PRM codes and open avenues for enhancements such as list decoding and subfield subcodes.

Abstract

We give a recursive decoding algorithm for projective Reed-Muller codes making use of a decoder for affine Reed-Muller codes. We determine the number of errors that can be corrected in this way, which is the current highest for decoders of projective Reed-Muller codes. We show when we can decode up to the error correction capability of these codes, and we compute the order of complexity of the algorithm, which is given by that of the chosen decoder for affine Reed-Muller codes.

Recursive decoding of projective Reed-Muller codes

TL;DR

This work addresses decoding Projective Reed-Muller (PRM) codes by exploiting a recursive decomposition that links PRM and affine RM codes. The authors develop a decoder that corrects up to errors, using decoders for affine RM codes at each recursion level, and extend the approach to Projective Reed-Solomon codes and general PRM cases. They show the proposed method improves upon the previously known PRM decoding bounds and analyze its complexity, which remains tied to the complexity of the affine RM decoders. The results enable practical, scalable decoding of PRM codes and open avenues for enhancements such as list decoding and subfield subcodes.

Abstract

We give a recursive decoding algorithm for projective Reed-Muller codes making use of a decoder for affine Reed-Muller codes. We determine the number of errors that can be corrected in this way, which is the current highest for decoders of projective Reed-Muller codes. We show when we can decode up to the error correction capability of these codes, and we compute the order of complexity of the algorithm, which is given by that of the chosen decoder for affine Reed-Muller codes.
Paper Structure (9 sections, 10 theorems, 85 equations, 2 figures, 2 algorithms)

This paper contains 9 sections, 10 theorems, 85 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recursive decoding algorithm
  4. Projective Reed-Solomon codes
  5. Projective Reed-Muller codes over the projective plane
  6. Projective Reed-Muller codes, general case
  7. Analysis of the algorithm and improvements
  8. Complexity
  9. Conclusion

Key Result

Theorem 2.2

The code $\mathop{\mathrm{PRM}}\nolimits_d(m)$, for $1\leq d\leq m(q-1)$, is an $[n,k]$-code with For the minimum distance, we have

Figures (2)

  • Figure 1: $T_0/T$ as a function of $1\leq d \leq m(q-1)$, for $m=2$.
  • Figure 2: $T_0/T$ as a function of $1\leq d \leq m(q-1)$, for $m=5$.

Theorems & Definitions (26)

  • Example 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8
  • Lemma 2.9
  • ...and 16 more