This paper presents a new cipher called the hexi cipher which makes use of a new class of codes called quasi cyclic partial hexi codes. This hexi cipher is a modification of the Advanced Encryption Standard (AES). It alters the mixed column operation of the AES and makes use of quasi cyclic partial hexi codes for error correction instead of the Maximum Distance Separable (MDS) code that has no error correcting capacity within the AES. The hexi cipher has the capacity to detect 99% of the errors and correct nearly 25% of the errors that occur, thus, it has a better error correcting capacity than existing error correcting ciphers.

This paper introduces a new class of hexi codes namely, hexi polynomial codes, hexi Rank Distance codes, hexi Maximum Rank Distance codes, hexi Goppa codes and hexi wild Goppa codes. These codes are useful to create variants of the McEliece public key cryptosystem known as the hexi McEliece public key cryptosystem and its variants; these cryptosystems are secure against attacks carried out on the existing variants of the McEliece public key cryptosystem. This newly introduced cryptosystem has better error correcting capacity and lesser time complexity making it more feasible to use. The security and possible attacks on these variants of the hexi McEliece public key cryptosystem are analysed.