The complete weight enumerator of the square of one-weight irreducible cyclic codes

In this paper, for an odd prime power q and an integer m >= 2, let C(q,m) be a one-weight irreducible cyclic code with parameters [q(m)-1,m,(q-1)q(m-1)], we consider the complete weight enumerator and the weight distribution of the square (C(q,m))(2), whose dual has & LeftFloor;(m)/(2)& RightFloor;+1 zeros. Using the character sums method and the known result of counting mxm symmetric matrices over F-q with given rank, we explicitly determine the complete weight enumerator of (C(q,m))(2) and show that (C(q,m))(2) is a (2 & LeftFloor;(m)/(2)& RightFloor;+1)-weight cyclic code with parameters [q(m)-1,(m(m+1))/(2),(q-1)(q(m)-1-q(m-2))]. Moreover, we get the weight distribution of the square of the simplex code by puncturing the last (q-2)(q(m)-1)/(q-1) coordinates of (C(q,m))(2).