6.896 Sublinear Time Algorithms
September 11, 2004
Lecture 17
Lecturer: Eli BenSasson
Scribe: Rafael Pass
1
Lower Bounds for Property Testing: Rest of the Proof
This lecture is the continuation of the previous one, in which we prove the following Theorem, stated
last time:
Theorem 1 ([1])
There exists integers
c, d
, and values
δ >
0
,
0
<
<
1
/
2
, γ >
0
such that for all
suﬃciently large
n
, with high probability, for a random low density parity check code (LDPC) every
(
δ,
0
, , q
)
linear test requires
q
≥
γn
queries.
As mentioned in the previous lecture, this result also extends to
any
test, and not just linear tests.
Towards the goal of proving the above theorem we rely on Yao’s minmax principle, which (in our case)
boils down to showing one “bad” distribution over inputs, such that every deterministic linear test fails
with probability larger than
. We continue to use the notation and definitions from previous lecture.
1.1
The “bad” distribution
Recall that the code
V
defined by the constraint matrix
A
is
V
= ker(
A
) =
{
x
∈
F
n
2

Ax
= 0
}
.
Let
B
i
be the uniform distribution over strings that
falsify
constraint
A
i
, but satify the rest, i.e.,
B
i
= ker(
A
−
i
)
−
ker(
A
), where
A
−
i
is the matrix obtained by removing row
i
from
A
. Define the “bad”
distribution to be the uniform convex combination of
B
i
’s.
In other words, the “bad” distribution is
obtained by uniformly chosing
i
∈
[
m
] (recall
m
is the number of rows in
A
) and then uniformly chosing
an element in
B
i
. Note that it follows from the definition of
B
that for every
w
∈
B
,
w /
∈
V
. In fact,
every
w
∈
B
is “far” from
V
, as is shown in the following lemma (a proof sketch of the lemma is given
in the next section).
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 Spring '04
 RonittRubinfeld
 Algorithms, Error detection and correction, Lemma, Lowdensity paritycheck code, high probability, linear test

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