Weight distribution of cosets of small codes with good dual properties

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 $L1$-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 $L1$-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.