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 ⊂ {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
_∞
-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 = ⊖(1), it decays like n
^-⊖(d)
. On the other d = ⊖(n) extreme, it decays like and e
^-⊖(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 ≥ 1 is an integer, then the average L
_∞
-distance is at most min{(e ln (n/2t))
^t
(2t/n)
^(t/2)
, √2e
^-(t/10)
}. For the average L
₁
-distance, we conclude the bound min{(2t + 1)(e ln (n/2t))
^t
(2t/n)
^(t/2)-1
, √2(n + 1)e
^-(t/10)
}, which gives nontrivial results for t ≥ 3. We give 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.