Existence conditions for Unraveling Reed–Solomon codes across fields and parameters

Characterize the parameter regimes (finite field F_q, block length n, and unraveling order ℓ) for which Unraveling Reed–Solomon codes exist. Specifically, determine necessary and sufficient conditions on a degree-ℓ polynomial G(x) over F_q and labels α_1,…,α_n ∈ F_q such that each fiber G(x) = α_i splits into ℓ distinct roots in F_q, yielding a generalized Reed–Solomon code whose columns unravel into ℓ interleaved Reed–Solomon row codes of lengths n and dimensions k or k+1 as defined in the URS construction.

Background

Unraveling Reed–Solomon (URS) codes are constructed by choosing a degree-ℓ polynomial G(x) and labels αi such that each polynomial G(x)−α_i splits into ℓ distinct roots β{ij}. Mixing per-column via the corresponding Vandermonde matrices maps a single generalized Reed–Solomon code to an interleaving of ℓ row codes, enabling fast beyond-bound column-oriented decoding while retaining full-block robustness.

The paper provides families guaranteeing existence in certain cases: when ℓ divides q−1 (using G(x)=x^ℓ) and when ℓ is a power of the field characteristic (using additive linear maps G_W with |W|=ℓ). It also shows specific non-collapsing examples (e.g., F_{2^8}, ℓ=6 with limited length). However, a complete characterization across all (F_q, n, ℓ) is not provided.