On the nonperiodic cyclic equivalence classes of Reed-Solomon codes

Picking up exactly one member from each of the nonperiodic cyclic equivalence classes of an (n, k+1) Reed-Solomon code E over GF(q) gives a code, E", which has bounded Hamming correlation values and the self-synchronizing property. The exact size of E" is shown to be (1/n) Σ_d|n μ(d)q^1+kd/, where μ(d) is the Mobius function, (x) is the integer part of x, and the summation is over all the divisors d of n=q-1. A construction for a subset V of E is given to prove that |E"|⩾|V|=(q^k+1-q ^k+1-N)/(q-1) where N is the number of integers from 1 to k which are relatively prime to q-1. A necessary and sufficient condition for |E"|=| V| is proved and some special cases are presented with examples. For all possible values of q>2, a number B(q) is determined such that |E"|=|V| for 1 ⩽k⩽B(q ) and |E"|>|V| for k>B( q)