Expand–accumulate (EA) codes are sparse linear codes underlying constructions of correlated pseudorandomness and field-agnostic succinct arguments. In “Field-Agnostic SNARKs from Expand–Accumulate Codes” (CRYPTO 2024), Block et al. conjectured that a single fixed-row-weight EA component already achieves constant relative distance with inverse-polynomial failure probability.

We prove this conjecture in a stronger, field-uniform form. For every rate R∈(0,1)R\in(0,1), there exists δR>0\delta_R>0 such that, for every target exponent C>0C>0, one can choose γ=γ(R,C)>0\gamma=\gamma(R,C)>0 for which the fixed-row ensemble with t=⌈γlog⁡N⌉t=\lceil\gamma\log N\rceil satisfies

[ \mathbb{P}!\left[ \min_{x\in\mathbb{F}_q^{\lfloor RN\rfloor}\setminus{0}} \operatorname{wt}(xEA) \le \delta_R N \right] \le N^{-C} ]

for all sufficiently large NN. The same constants work for every prime power qq; in particular, the field may vary arbitrarily with the block length. Thus, a single fixed-row-weight EA component is asymptotically good over all finite fields, and its polynomial reliability exponent can be made arbitrarily large by increasing the row-weight constant.

The proof separates sparse and high-weight messages. Sparse messages are handled through expansion and a compact analysis of accumulator cancellations, while high-weight messages are controlled by a surplus of linear constraints over large fields and a stochastic accumulator analysis over bounded fields. A terminal-boundary obstruction shows that, for t=Θ(log⁡N)t=\Theta(\log N), inverse-polynomial failure is qualitatively optimal.