  Let $M_n$ be a random matrix of size $n\times n$ and let
$\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral
distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}#
\{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}.$$
  The circular law theorem in random matrix theory asserts that if the entries
of $M_n$ are i.i.d. copies of a random variable with mean zero and variance
$\sigma^2$, then the empirical spectral distribution of the normalized matrix
$\frac{1}{\sigma\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform
distribution $\mu_\cir$ over the unit disk as $n$ tends to infinity.
  In this paper we show that the empirical spectral distribution of the
normalized matrix of $M_n$, a random matrix whose rows are independent random
$(-1,1)$ vectors of given row-sum $s$ with some fixed integer $s$ satisfying
$|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new
polynomial estimate on the least singular value of $M_n$.
