In this work we study a variant of the Niederreiter cryptosystem. Instead of hiding the parity-check matrix of the secret code with constant matrices we propose to use polynomial matrices. The public key is then the parity-check matrix of a convolutional code. The plaintext is a sequence of vectors of low weight and the ciphertext is the syndrome of this sequence. We propose the use of a Generalized Reed-Solomon code as the secret code and construct the polynomial matrices to obtain a secure scheme. We analyze its security and show that this approach can be of interest to obtain smaller public keys when compared to the NIST proposal Classic McEliece.