We propose an appropriate approach of defining the linear complexities (LC) of sequences over unknown symbol set. We are able to characterize those p-ary sequences whose R-tuple versions now eve. GF($p^{R}$ ) have the same characteristic polynomial as the original with respect to any basis. This leads to a construction of $p^{R}$ -ary sequences whose characteristic polynomial is essentially over GF(p). In addition, we can characterize those $p^{R}$ -ary sequences whose characteristic polynomials are uniquely determined when symbols are represented as R-tuples over GF(p) with respect to any basis.