Recursive decoding of projective Reed-Muller codes

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.

💡 Research Summary

The paper presents a new recursive decoding framework for projective Reed–Muller (PRM) codes that leverages existing decoders for affine Reed–Muller (RM) codes. After reviewing the basic parameters of PRM and RM codes, the authors introduce a specific ordering of points in the projective space P^m based on a primitive element ξ of the finite field F_q. This ordering is essential for the recursive structure used throughout the algorithm.

A central theoretical contribution is Theorem 2.8, which shows that any PRM code can be expressed as a “(u, u + v ξ^d)” construction:
PRM_d(m) = { (u + v·ξ^d, v) | u ∈ RM_{d‑1}(m), v ∈ PRM_d(m‑1) }.
This representation mirrors the classic (u, u+v) construction used for concatenated codes but includes a scaling by powers of ξ. Using this recursion, the authors define a lower bound η_d(m) on the minimum distance of PRM_d(m):

η_d(m) = Σ_{i=0}^{m‑ν‑1} wt(RM_d(m‑i)) + 1,

where d‑1 = ν(q‑1) + μ (0 ≤ μ < q‑1). Lemma 2.9 proves that η_d(m) = wt(PRM_d(m)) − μ·q^{m‑ν‑1} − 1/(q‑1), and that η_d(m) coincides with the true minimum distance when μ = 0 or ν = m‑1. Consequently, η_d(m) is a guaranteed, often tight, lower bound that is larger than the bound used in the only previously known PRM decoder