We will first prove that any(3,δ)-LCC overRcontains a large subset of small dimension. Later we will
iterate this to get a global dimension bound.
Lemma 10.1.Suppose n>(1/δ)ω(1)and let0<ε<1/50. Let V= (v1, . . . ,vn)∈(Rd)nbe a(3,δ)-LCC.
Then, there exists a subset U⊂V of size at least
|U| ≥(δ3/300)n and dimension at most
Proof. We will prove the lemma by first applyingLemma 8.2to show thatV has a large sub-LCCV0in which the triples cluster. Then, we will applyLemma 9.1to show thatV0has a large low-rank sublist. The details follow.
Setβ1=ε/4 and applyLemma 8.2withβ =β1. To apply the lemma we require thatV does not contain a subsetU of size(δ2/288)nand rank at most max{8δ6d,n1/2−β1/4}= max{8δ6d,n1/2−ε/16}. If this is the case, then our proof is done and there is no need to continue.
Having appliedLemma 8.2, we obtain a(3,δ0)-LCCV0⊂Vwithn0=|V0| ≥(δ/10)n,d0=dim(V0)≤
d,δ0≥δ2/4 and setsS1, . . . ,Smwhich cluster all the triples in the matchingsMv0,v0∈V0used to decode
V0so that
|Si| ≤O(n0/δ06d0)
and
Ω(δ019d03/n01+2β1)≤m≤O(n01+2β1/δ010d0).
We now applyLemma 9.1 withβ =2β1<ε/2 and the same ε to conclude that there exist a subset V00⊂V0of size
n00=|V00| ≥(δ0/2)n0≥(δ2/8)(δ/10)n≥(δ3/80)n and dimension
dim(V00)≤n001/2−ε ≤max{8δ6d,n1/2−ε/16}, as was required.
We now prove an amplification lemma which usesLemma 10.1iteratively. For this lemma we will use the following convenient notation. IfS⊂V is a subset ofV, we denote byspanV(S)⊂V the subset of elements ofV that are spanned by elements ofS. (We think of all these as lists/multisets.)
Lemma 10.2(Amplification lemma). Suppose n>(1/δ)ω(1)and let0<ε<1/50. Let V= (v1, . . . ,vn)∈
(Rd)nbe a linear(3,δ)-LCC. Suppose S⊂V is such thatspanV(S) =S and S6=V . Then there is a set
S⊆S0⊆V withspanV(S0) =S0such that
1. either S0=V or|S0| ≥ |S|+ (δ4/400)n; 2. dim(S0)≤dim(S) +max{δ6d,n1/2−ε/16}.
We defer the proof of the lemma to the end of this section and proceed with the proof ofTheorem 2.3. Proof ofTheorem 2.3. LetV = (v1, . . . ,vn)∈(Rd)nbe a linear(3,δ)-LCC. We will prove the theorem withε =1/1000 We now applyLemma 10.2withε1=1/51 iteratively. Start withS1= /0 and apply
Lemma 10.2repeatedly to obtain setsS2,S3, . . . ,such that for alli,
|Si| ≥ |Si−1|+ (δ4/400)n
and
Since the size ofSicannot grow beyondn, the process will terminate after at mostm=b400/δ4csteps,
yieldingSm=V. We then get that
dim(Sm) =dim(V)≤(400/δ4)max{δ6d,n1/2−ε1/16}=max{(400δ2)d,(400/δ4)n1/2−ε1/16}.
Without loss of generality, for the proof of the theorem we can assume thatδ2<1/500. Thus it must be that
d=dim(V)≤(400/δ4)n1/2−ε1/16≤n1/2−ε. This completes the proof ofTheorem 2.3.
10.1 Proof ofLemma 10.2
Observe that forv∈V\S, all 3 points of any triple inMv cannot be inSsincespanV(S) =S. Thus we
may assume that|S| ≤(1−δ)n, since otherwise each vector inV\Swould be spanned by the points ofS and we would be done.
Case 1: There exists av∈V\Ssuch thatδn/4 of the triples inMvhave two of their points contained
inS. In this case letS0=spanV({v} ∪S). Then|S0| ≥ |S|+ (δ/4)n, and dim(S0)≤dim(S) +1.
If Case 1 does not hold then eachv∈V\S,Mvhas 3δn/4 of its triples intersectingSin either one
or zero points. Let us call a pointv type-zeroif if has at least 3δn/8 of its triples contained inV\Sand
type-oneotherwise. Notice that, ifvis type-one, then it must have at least 3δn/8 of its triples intersecting
Sin exactly one point. We now separate into two additional cases.
Case 2: There are at mostδn/4type-one points. LetV0⊂V\Sbe the set of all type-zero points. Observe that, since|S| ≤(1−δ)n, we have|V0| ≥3δn/4. Also observe that the vectors inV0 form a (3,δ/8)-LCC since each point inV0has at least 3δn/8−δn/4=δn/8≥(δ/8)|V0|triples in its matching contained inV0. UsingLemma 10.1onV0we conclude that there is a subsetU⊂V0of size
|U| ≥(δ3/300)|V0| ≥(δ4/400)n and dimension
dim(U)≤max{8(δ/8)6d0,|V0|1/2−ε/16} ≤max{δ6d,n1/2−ε/16}. SettingS0=S∪U we are done.
Case 3: There are at leastδn/4 type-one points. In this case, there are δn/4 points v inV\S, each having at least 3δn/8 of the triples inMv intersectingSin exactly one point. LetAbe a linear
transformation whose kernel equalsspan(S). After applyingA toV\S we obtain a (2,3δ/4) LDC decoding theδn/4 type-one points. Thus theδn/4 points (after we apply the mappingAto them) must span at mostpoly(1/δ)logn≤max{δ6d,n1/2−ε/16}dimensions byTheorem 4.5. Thus, adding them to Swill increase the dimension of its span by at most this number. This completes the proof also in this case.