In this paper we present a variant of the McEliece cryptosystem that possesses several interesting properties, including a reduction of the public key for a given security level. In contrast to the classical McEliece cryptosystems, where block codes are used, we propose the use of a convolutional encoder to be part of the public key. The permutation matrix is substituted by a polynomial matrix whose coefficient matrices have columns with weight zero or at least weight two. This allows the use of Generalized Reed-Solomon (GRS) codes which translates into shorter keys for a given security level. Hence, the private key is constituted by a generator matrix of a GRS code and two polynomial matrices containing large parts generated completely at random. In this setting the message is a sequence of messages instead of a single block message and the errors are added throughout the sequence. We discuss possible structural and ISD attacks to this scheme. We conclude presenting the key sizes obtained for different parameters and estimating the computational cost of encryption and decryption process.