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 + e where | e \t| ≦ ( r − 1) τ m , ″> τ 〈 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 .