Algebraic construction of cyclic codes over Z8 with a good Euclidean minimum distance

Let S(8) denote the set of the eight admissible signals of an 8PSK communication system. The alphabet S(8) is endowed with the structure of Z₈, the set of integers taken modulo 8, and codes are defined to be Z₈-submodules of Z₈ⁿ. Three cyclic codes over Z₈ are then constructed. Their length is equal to 6, 8, and 7, and they, respectively, contain 64, 64, and 512 codewords. The square of their Euclidean minimum distance is equal to 8, 16-4√2 and 10-2√2, respectively. The size of the codes of length 6 and 7 can be doubled while the Euclidean minimum distance remains the same