A Family Of Special Cyclic Codes

Since linear code has many good properties, so many coding experts devoted themselves to the linear properties, especially the linearity of the binary code. Recently, many coding experts have studied much of the Z₄-cyclic code(which is not necessarily linear) and its Gray map, and obtained a set of equivalent conditions in which a family of cyclic codes are linear, and the condition that this family codes are self-dual cyclic, and in this condition Nechaev-Gray image of the cyclic codes are self-dual cyclic.Enlightened by those results, by using the properties of the p-adic code ( is a prime), the differential form and the cardinality property of the codes, we study the Nechaev-Gray image of a family of Z_p²-cyclic codes, and obtain a set of equivalent conditions in which this family of cyclic codes are linear, and we also obtain that these cyclic codes are equivalent to some family of the linear cyclic codes which have the same Nechaev-Gray image. so we learn that the two families of codes are equivalent. So this family of codes are linear cyclic.Moreover, we also study a family of Z_2^k+1cyclic codes. In the process of proving their Nechaev-Gray image, we use the method of inclusion. The proof of the right inclusion is similar to the proof over Z p2; but in the proof of the left inclusion, we only choose one solution as its left inclusion item. Moreover, this paper also prove the conditions that the Nechaev-Gray image of the cyclic codes are self-dual quasi-cyclic codes.In this paper, we expand binary code to p-adic code, i.e., we extended the research rang of this family.