On the number of information symbols in Bose-Chaudhuri codes

Let α be a primitive root of G. F. (q^m). Let I(m, v) be the number of information symbols of the code with parity check matrix (α^\mij), i = 1, …, v, j = 0, …, q^m − 2. Let v = q^λ, m − λ = r. Then for sufficiently large m we have {fx0153-1} where 〈c〉 denotes the nearest integer to c and ρ is the positive root of the equation x^r = (q − 1) (x^r−1 + … + 1). For small values of m we have I(m, v) = ρ^m + ε where | ε \t| ≦ (r − 1)τ^m, ″>τ 〈< 1. Estimates for τ are given in the Appendix. Two other formulas, exact for all values of m ≧ r − 1, are also given. The first contains r terms, the second [m/r + 1] terms, where [c] denotes the largest integer not exceeding c.