Recently, sequences of error-correcting codes with doubly-transitive permutation groups were shown to achieve capacity on erasure channels under symbol-wise maximum a posteriori (MAP) decoding. From this, it follows that Reed-Muller and primitive narrow-sense BCH codes achieve capacity in the same setting. We extend this result to a large family of cyclic codes by considering codes whose permutation groups satisfy a condition weaker than double transitivity.
This extension is based on two small technical contributions. First, we show that the transition width of a monotone boolean function is O(1/log(k)), where k is the size of the smallest orbit induced by its symmetry group. The proof is based on Talagrand's lower bound on influences for monotone boolean functions. Second, we consider the extrinsic information transfer (EXIT) function of an linear cyclic code over GF(q) whose blocklength N is coprime with q-1. We show that this EXIT function is a monotone boolean function and that the smallest orbit of its symmetry group has size at least min{ s>0 | gcd(q^s - 1,N)>1 }.  Combining the results, we find that sequences of cyclic codes achieve capacity on the q-ary erasure channel if their blocklengths have the right numerology (e.g., increasing prime lengths are sufficient).