We study approximate integration of a function
f
f
over
[
0
,
1
]
s
[0,1]^s
based on taking the median of
2
r
−
1
2r-1
integral estimates derived from independently randomized
(
t
,
m
,
s
)
(t,m,s)
-nets in base
2
2
. The nets are randomized by Matousek’s random linear scramble with a random digital shift. If
f
f
is analytic over
[
0
,
1
]
s
[0,1]^s
, then the probability that any one randomized net’s estimate has an error larger than
2
−
c
m
2
/
s
2^{-cm^2/s}
times a quantity depending on
f
f
is
O
(
1
/
m
)
O(1/\sqrt {m})
for any
c
>
3
log
(
2
)
/
π
2
≈
0.21
c>3\log (2)/\pi ^2\approx 0.21
. As a result, the median of the distribution of these scrambled nets has an error that is
O
(
n
−
c
log
(
n
)
/
s
)
O(n^{-c\log (n)/s})
for
n
=
2
m
n=2^m
function evaluations. The sample median of
2
r
−
1
2r-1
independent draws attains this rate too, so long as
r
/
m
2
r/m^2
is bounded away from zero as
m
→
∞
m\to \infty
. We include results for finite precision estimates and some nonasymptotic comparisons to taking the mean of
2
r
−
1
2r-1
independent draws.