Reed-Solomon codes are very convenient for burst error correction which occurs frequently in applications, but as the number of errors increase, the circuit structure of implementing Reed-Solomon codes becomes very complex. An alternative solution to this problem is the modular and regular structure of cellular automata which can be constructed with VLSI economically. Therefore, in recent years, cellular automata have became an important tool for error correcting codes. For the first time, cellular automata based byte error correcting codes analogous to extended Reed-Solomon codes over binary fields was studied by Chowdhury et al.  and Bhaumik et al.  improved the coding-decoding scheme. In this study cellular automata based double-byte error correcting codes are generalized from binary fields to primitive finite fields Z(p).