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Z8-Cyclic Codes and Quadratic Residue Codes

A set of n-tuples over Z8 is called a code over Z8 or a Z8 code if it is a Z8 module. A particularly interesting family of Z8-cyclic codes are quadratic residue codes. We define such codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over a field. In particular we show that the quadratic residuce codes over Z8 have large automorphism groups which will be useful in decoding these codes by using the powerful permutation decoding methods described by F. J. MacWilliams and N. J. A. Sloane (1978, “Theory of Error-Correcting Codes,” North-Holland, Amsterdam). We also define a distance preserving map from ZN8 (Lee distance) to Z4N2 (Hamming distance).