Linear complexity is an important design index for assessing the cryptographic strength of a sequence.Pseudorandom sequences with large linear complexity and large k-error linear complexity is a hot topic in cryptography and communications. Niederreiter found many such periodic sequences over F_q firstly. In this paper,the authors construct some periodic sequences over F_q with very large 1-error linear complexity by the GDFT of a periodic sequence. The result is much better than the known ones.