Largest BCH error-budget growth compatible with asymptotically unit storage rate

Determine the largest asymptotic growth range of the BCH error-correction parameter s=s(m) for which the full-parity storage codes constructed from primitive binary BCH coset graphs have storage rate tending to one as the extension degree m tends to infinity.

Background

The paper proves asymptotically unit storage rate for primitive binary BCH coset graphs when the error budget is fixed, and also when the growing parameter satisfies s(log(s+1))^2=o(m). This condition arises from the polynomial-rank estimate and the bound on the Boolean coordinate degree of the BCH power map.

The conclusion does not establish that this sufficient growth condition is optimal. Determining the maximal allowable growth of s(m) remains unresolved.