Enumeration results for the codewords having no inner periods in Reed- Solomon codes (Q1191673)

A finite sequence \(x\) in \(GF(q)\) is said to have generalized period \(r\) if \(S^ r x\), its cyclic shift over \(r\) positions, satisfies \(S^ r x=x+(u,u,\dots,u)\) for some \(u\in GF(q)\). If \(u=0\) in the above definition, \(x\) has in fact (ordinary) period \(r\). A finite sequence is said to have no inner period if its minimum generalized period equals its length. Reed-Solomon codewords with no inner period are useful for constructing sequences with perfect generalized Hamming correlation properties. In the present article, the author counts the number of such Reed-Solomon codewords, using the Fourier transform over \(GF(q)\) and Polya's enumeration formula. Up to now, only a lower bound on this number was known. Also the random sequences in \(GF(q)\) with generalized period \(n\) are counted. Although not obvious from the text, the reviewer thinks that this is meant to be the number of sequences in \(GF(q)\) of length \(n\) with no inner period. If this is so, the method of counting (application of the Möbius inversion formula) is basically correct, but the details are not worked out properly which results in an incorrect conclusion. In the reviewer's opinion, the number of length \(n\) sequences in \(GF(q)\) with no inner period equals \[ \sum_{d\mid n, \text{gcd}(q,d)\neq 1}\mu(d)q^{{n\over d}+1}+\sum_{d\mid n, \text{gcd}(q,d)=1}\mu(d)q^{{n\over d}}, \] where \(\mu(.)\) denotes the Möbius function.