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 \(m\ge 2\), let \(\mathcal {C}(q,m)\) 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 \(\big (\mathcal {C}(q,m)\big )^2\), whose dual has \(\lfloor \frac{m}{2}\rfloor +1\) zeros. Using the character sums method and the known result of counting \(m\times m\) symmetric matrices over \(\mathbb {F}_q\) with given rank, we explicitly determine the complete weight enumerator of \(\left( \mathcal {C}(q,m)\right) ^2\) and show that \(\left( \mathcal {C}(q,m)\right) ^2\) is a \((2\lfloor \frac{m}{2}\rfloor +1)\)-weight cyclic code with parameters \([q^{m}-1,\frac{m(m+1)}{2},(q-1)(q^{m-1}-q^{m-2})]\). Moreover, we get the weight distribution of the square of the simplex code by puncturing the last \(\frac{(q-2)(q^m-1)}{q-1}\) coordinates of \(\left( \mathcal {C}(q,m)\right) ^2\).