  The bilateral minimum distance of a binary linear code is the maximum $d$
such that all nonzero codewords have weights between $d$ and $n-d$. Let
$Q\subset \{0,1\}^n$ be a binary linear code whose dual has bilateral minimum
distance at least $d$, where $d$ is odd. Roughly speaking, we show that the
average $L_\infty$-distance -- and consequently the $L_1$-distance -- between
the weight distribution of a random cosets of $Q$ and the binomial distribution
decays quickly as the bilateral minimum distance $d$ of the dual of $Q$
increases. For $d = \Theta(1)$, it decays like $n^{-\Theta(d)}$. On the other
$d=\Theta(n)$ extreme, it decays like and $e^{-\Theta(d)}$. It follows that,
almost all cosets of $Q$ have weight distributions very close to the to the
binomial distribution. In particular, we establish the following bounds. If the
dual of $Q$ has bilateral minimum distance at least $d=2t+1$, where $t\geq 1$
is an integer, then the average $L_\infty$-distance is at most
$\min\{\left(e\ln{\frac{n}{2t}}\right)^{t}\left(\frac{2t}{n}\right)^{\frac{t}{2}
}, \sqrt{2} e^{-\frac{t}{10}}\}$. For the average $L_1$-distance, we conclude
the bound $\min\{(2t+1)\left(e\ln{\frac{n}{2t}}\right)^{t}
\left(\frac{2t}{n}\right)^{\frac{t}{2}-1},\sqrt{2}(n+1)e^{-\frac{t}{10}}\}$,
which gives nontrivial results for $t\geq 3$. We given applications to the
weight distribution of cosets of extended Hadamard codes and extended dual BCH
codes. Our argument is based on Fourier analysis, linear programming, and
polynomial approximation techniques.
