Existence of URS codes over F_{2^8} with seven-symbol columns (ℓ=7)

Determine whether Unraveling Reed–Solomon codes over the binary extension field F_{2^8} exist with unraveling order ℓ=7 (i.e., seven-symbol columns). Concretely, construct a degree-7 polynomial G(x) over F_{2^8} and a set of labels α_1,…,α_n such that each polynomial G(x)−α_i splits into seven distinct roots in F_{2^8} for n≥1, producing a valid URS code; or prove that no such construction is possible.

Background

To support bit budgets that are not multiples of the desired symbol size, the paper discusses shortening and fixing bits/symbols. It highlights a practical desire for URS codes with seven-symbol columns over F_{2^8}, which would align certain memory configurations, but indicates such constructions are unknown to the authors.

Known existence results cover ℓ=2 for all prime powers q, cases where ℓ divides q−1 (via G(x)=x^ℓ), and ℓ equal to powers of the characteristic (via additive linear maps G_W). The specific ℓ=7 over F_{2^8} does not fall under these families, motivating a targeted existence question.