Uniquely Decodable Code Pairs (UDCP)

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Uniquely Decodable Code Pairs (UDCP)
Sharper Upper Bounds for Unbalanced
Uniquely Decodable Code Pairs
Jesper Nederlof
ISIT 2016
Joint work with Per Austrin, Petteri Kaski and Mikko Koivisto
KTH, Stockholm
HIIT+Aalto University,
Helsinki
HIIT, Helsinki
Outline
• Introduction + brief overview previous work
• Our approach
– Isoperimetric inequality
– Warm-up bound
– Sketch main bound
• Our motivation for this problem:
– A consequence for additive combinatorics
• Further remarks and research
𝐴 + 𝐵 = {a + b: a, b ∈ 𝐴 × 𝐵},
a + b is addition over ℤ𝑛 .
Uniquely Decodable Code Pairs (UDCP)
• Pair 𝐴, 𝐵 ⊆ {0,1}𝑛 s.t. 𝐴 + 𝐵 = 𝐴 |𝐵|.
• 𝐴 = {10,01}, 𝐵 = {00,01,11} is UDCP:
𝐴 + 𝐵 = {10,11,21,01,02,12}
• Is 𝐴 = 001,010,101 , 𝐵 = {011,110,111}
UDCP?
• No: 001+111=101+011
If 𝐴 + 𝐵 = 𝐴 |𝐵|, then 𝐴 − 𝐵 = 𝐴 |𝐵|:
– If 𝑥1 − 𝑦1 = 𝑥2 − 𝑦2 then 𝑥1 + 𝑦2 = 𝑥2 + 𝑦1
𝐴 + 𝐵 = {a + b: a, b ∈ 𝐴 × 𝐵},
a + b is addition over ℤ𝑛 .
Uniquely Decodable Code Pairs (UDCP)
• Pair 𝐴, 𝐵 ⊆ {0,1}𝑛 s.t. 𝐴 + 𝐵 = 𝐴 |𝐵|.
• 𝐴 = {10,01}, 𝐵 = {00,01,11} is UDCP:
𝐴 + 𝐵 = {10,11,21,01,02,12}
𝐴
1011100
1101101
0000000
1010011
0101010
1111101
1100111
Unbalanced: 𝐴 huge
1 0 1 0 0 1 1
1
0
??
2
1
𝐵
0011001
1010101
0011011
0110110
0 1 1 0 1 1 0
1
0
0
Some (incomplete) History
• 𝐴 = 2𝛼𝑛 , 𝐵 = 2𝛽𝑛 .
• Main Question: How large can 𝛼 + 𝛽 be?
• If (A,B) is UDCP, so is
– ({𝑎1 𝑎2 : 𝑎1 , 𝑎2 ∈ 𝐴}, {𝑏1 𝑏2 : 𝑏1 , 𝑏2 ∈ 𝐵})
• So whenever 𝑛 even, previous example gives
• 𝐴 = 2𝑛/2 , 𝐵 = 3𝑛/2
• 𝛼 = .5, 𝛽 ≈ 0.792, 𝛼 + 𝛽 ≈ 1.292
Kasami & Lin’76
𝛼 + 𝛽 ≈ 1.292
Some (incomplete) History
• 𝐴 = 2𝛼𝑛 , 𝐵 = 2𝛽𝑛 .
sym. difference
• Several works: 𝛼 + 𝛽 ≤ 1.5
• Define 𝑊𝑑 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵: |𝑎Δ𝑏| = 𝑑}
• |𝑊𝑑 | ≤
𝑛
𝑑
2min 𝑑,𝑛−𝑑 (vT’78)
– 𝑎 + 𝑏 or 𝑎 − 𝑏 determines 𝑎, 𝑏
– (𝑎Δ𝑏, 𝑎 ∩ b) or (𝑎Δ𝑏, 𝑎 ∖ 𝑏) determines 𝑎 + 𝑏, 𝑎 − 𝑏
𝛼 + 𝛽 ≤ 1.5
Ahlswede’71
Liao’72
Lindström’72
v Tilborg’78
Kasami & Lin’76
𝛼 + 𝛽 ≈ 1.292
Some (incomplete) History
• 𝐴 = 2𝛼𝑛 , 𝐵 = 2𝛽𝑛 .
sym. difference
• Several works: 𝛼 + 𝛽 ≤ 1.5
• Define 𝑊𝑑 = { 𝑎, 𝑏 ∈ 𝐴 × 𝐵: |𝑎Δ𝑏| = 𝑑}
• |𝑊𝑑 | ≤
𝑛
𝑑
2min 𝑑,𝑛−𝑑 (vT’78)
– 𝑎 + 𝑏 or 𝑎 − 𝑏 determines 𝑎, 𝑏
– (𝑎Δ𝑏, 𝑎 ∩ b) or (𝑎Δ𝑏, 𝑎 ∖ 𝑏) determines 𝑎 + 𝑏, 𝑎 − 𝑏
𝛼 + 𝛽 ≤ 1.5
Ahlswede’71
Liao’72
Lindström’72
v Tilborg’78
𝛼 ≥1−𝑜 1
𝛽 ≥ 0.25
Kasami et al.’82
v Tilborg’83
Kasami & Lin’76
vd Braak ‘84
𝛼 + 𝛽 ≈ 1.292
v Tilborg & vd Braak’85
𝛼 + 𝛽 ≈ 1.30366
Mattas & Östergård’05
𝛼 + 𝛽 ≈ 1.31781
Some (incomplete) History
𝛼 + 𝛽 ≤ 1.5
Ahlswede’71
Liao’72
Lindström’72
v Tilborg’78
𝛼 = 𝑅1 ≈ 1 gives 𝛽 = 𝑅2 < 0.4922.
𝛼 ≥1−𝑜 1
𝛽 ≥ 0.25
Kasami et al.’82
v Tilborg’83
Kasami & Lin’76
vd Braak ‘84
𝛼 + 𝛽 ≈ 1.292
v Tilborg & vd Braak’85
𝛼 + 𝛽 ≈ 1.30366
Urbanke & Li’98
Mattas & Östergård’05
𝛼 + 𝛽 ≈ 1.31781
Some (incomplete) History
𝛼 + 𝛽 ≤ 1.5
Ahlswede’71
Liao’72
Lindström’72
v Tilborg’78
𝛼 = 𝑅1 ≈ 1 gives 𝛽 = 𝑅2 < 0.4798.
𝛼 ≥1−𝑜 1
𝛽 ≥ 0.25
Kasami et al.’82
v Tilborg’83
Kasami & Lin’76
vd Braak ‘84
𝛼 + 𝛽 ≈ 1.292
v Tilborg & vd Braak’85
𝛼 + 𝛽 ≈ 1.30366
Urbanke & Li’98
Mattas & Östergård’05
𝛼 + 𝛽 ≈ 1.31781
Ordentlich &
Shayevitz’15
Some (incomplete) History
𝛼 + 𝛽 ≤ 1.5
𝛼 + 𝛽 ≤ 1.5
Ahlswede’71
Liao’72
Lindström’72
v Tilborg’78
0
𝛼 ≥1−𝑜 1
𝛽 ≥ 0.25
Kasami et al.’82
v Tilborg’83
Kasami & Lin’76
vd Braak ‘84
𝛼 + 𝛽 ≈ 1.292
v Tilborg & vd Braak’85
𝛼 + 𝛽 ≈ 1.30366
From Schleger & Grant
`coordinated multiuser communications’
Urbanke & Li’98
Mattas & Östergård’05
𝛼 + 𝛽 ≈ 1.31781
Ordentlich &
Shayevitz’15
Some (incomplete) History
.42 .44 .46 .48 .5
Our main result:
𝜷 ≤0.4229+ 𝟏 − 𝜶
𝛼 + 𝛽 ≤ 1.5
Ahlswede’71
Liao’72
Lindström’72
v Tilborg’78
𝛼 + 𝛽 ≤ 1.5
0
𝛼 ≥1−𝑜 1
𝛽 ≥ 0.25
Kasami et al.’82
v Tilborg’83
Kasami & Lin’76
vd Braak ‘84
𝛼 + 𝛽 ≈ 1.292
v Tilborg & vd Braak’85
𝛼 + 𝛽 ≈ 1.30366
From Schleger & Grant
`coordinated multiuser communications’
Urbanke & Li’98
Mattas & Östergård’05
𝛼 + 𝛽 ≈ 1.31781
We
Ordentlich &
Shayevitz’15
Our Approach
Isoperimetric Inequality
• For 𝑥 ∈ {0,1}𝑛 ,0 ≤ 𝜌 ≤ 1, y ∼𝜌 𝑥 is a 𝜌-noisy copy of y:
1+𝜌
𝑥𝑒 ,
with probability
,
2
𝑦𝑒 =
1−𝜌
1 − 𝑥𝑒 , with probability
.
2
B
• 𝑦 ∼0 𝑥: 𝑦 uniform
• 𝑦 ∼1 𝑥: 𝑥 = 𝑦
• 𝑦 ∼𝜌 𝑥: 𝔼 𝑥Δ𝑦 =
1−𝜌
2
A
• Theorem (Mossel et al.):
−𝑈
2
1−𝛼 + 1−𝛽 +2𝜌 1−𝛼 1−𝛽
1−𝜌2
≤ Pr [𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵]
𝑥∼𝜌 𝑦
Warm-up bound
• Large fraction of a, b ∈ 𝐴 × 𝐵 are close in HD by iso. ineq
• But only few can be close pairs by vT’s bound
(*similar tension between vT’s bound and iso. ineq used by UL)
−𝑛
2
1−𝛼 + 1−𝛽 +2𝜌 1−𝛼 1−𝛽
1−𝜌2
Pr [𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵] =
𝑥∼𝜌 𝑦
2−𝑛
𝑑
≤
≤
1+𝜌
2
𝑛−𝑑
1−𝜌
2
𝑛−𝑑 1−𝜌 𝑑
1+𝜌
𝑛
2−𝑛 𝑑
𝑑
2
2
2−2𝑛 3 − 𝜌 𝑛 .
𝑑
|𝑊𝑑 |
2𝑑
≤
• Setting 𝜌 = 0.3838 gives 𝛽 ≤ 0.4777 + 2 𝜖
• Intuitively, we lower and upper bound |𝑊𝑑 | for some 𝑑
Main Bound
• Let 𝐴 = 2𝛼𝑛 , 𝐵 = 2𝛽𝑛 . Think 𝛼 = 1 − 𝜖 for small 𝜖.
• Isoperimetric inequality is tight if A and B Hamming balls
far away from each other.
• But then A,B cannot be a good UDCP
B
• We argue in two steps:
1. 𝐵 has to be spread out
using an encoding argument
2. Use this for a refined version
of the warm-up bound
A
Main Bound
𝐵 projected to 𝐿
• Step 1: with an encoding argument, find a partition 𝐿, 𝑅
of {1, … , 𝑛} such that 𝐿 , 𝑅 ≈ 𝑛/2, 𝐵𝐿 ≥ 2 𝛽−𝜖 𝑛
• Step 2: Study the number of pairs 𝑥, 𝑦 ∈ 𝐴 × 𝐵 with
|𝑥𝐿 Δ𝑦𝐿 | small and 𝑥𝑅 Δ𝑦𝑅 ≈ |𝑅|/2
– Lower bound with iso. ineq. using 𝜌 = 0.654 on L, 𝜌 = 0 on R
– Upper bound by encoding argument
𝑩𝑳
𝑩𝑹
𝐀𝑳
𝐀𝑹
Our Motivation
• Let 𝑤 ∈ ℕ𝑛 . We study two parameters
– 𝛽 𝑤 = max 𝑥 ∈ {0,1}𝑛 : 𝑤 ⋅ 𝑥 = 𝑖
𝑖
– 𝜎(𝑤) = |{𝑤 ⋅ 𝑥: 𝑥 ∈ {0,1}𝑛 }|
• If 𝜎 𝑤 ≥ 2 1−𝜖 𝑛 for small 𝜖, upper bound 𝛽(𝑤)
• Useful for finding faster algo’s for Subset Sum problem
• Similar to `Inverse Littlewood-Offord’ questions in
additive combinatorics
w
𝜷(𝒘) 𝝈(𝒘)
00000
32
1
1 2 4 8 16
32 64 128
256 512
1024
1
2048
12345
3
16
3 20 58 90
267 493 869
961
1000 1153
9
1246 1598
1766 1922
7005
Histogram
Connection to UDCPs
• Let 𝐴 ⊆ {0,1}𝑛 be s.t. for all 𝑥1 , 𝑥2 ∈ 𝐴:
𝑤 ⋅ 𝑥1 = 𝑤 ⋅ 𝑥2 implies 𝑥1 = 𝑥2
• Let 𝐵 ⊆ {0,1}𝑛 be s.t. for all 𝑦1 , 𝑦2 ∈ 𝐵: 𝑤 ⋅
𝑦1 = 𝑤 ⋅ 𝑦2
• Can take 𝐴 = |𝜎(𝑤)|, 𝐵 = 𝛽 𝑤
• (𝐴, 𝐵) is UDCP: suppose 𝑥1 , 𝑥2 ∈ 𝐴, 𝑦1 , 𝑦2 ∈ 𝐵
𝑥1 + 𝑦1 = 𝑥2 + 𝑦2
𝑤 ⋅ (𝑥1 + 𝑤 ⋅ 𝑦1 ) = 𝑤 ⋅ (𝑥2 + 𝑤 ⋅ 𝑦2 )
Connection to UDCPs
• Let 𝐴 ⊆ {0,1}𝑛 be s.t. for all 𝑥1 , 𝑥2 ∈ 𝐴:
𝑤 ⋅ 𝑥1 = 𝑤 ⋅ 𝑥2 implies 𝑥1 = 𝑥2
• Let 𝐵 ⊆ {0,1}𝑛 be s.t. for all 𝑦1 , 𝑦2 ∈ 𝐵: 𝑤 ⋅
𝑦1 = 𝑤 ⋅ 𝑦2
• Can take 𝐴 = |𝜎(𝑤)|, 𝐵 = 𝛽 𝑤
• (𝐴, 𝐵) is UDCP: suppose 𝑥1 , 𝑥2 ∈ 𝐴, 𝑦1 , 𝑦2 ∈ 𝐵
𝑥1 + 𝑦1 = 𝑥2 + 𝑦2
𝑤 ⋅ (𝑥1 + 𝑤 ⋅ 𝑦1 ) = 𝑤 ⋅ (𝑥2 + 𝑤 ⋅ 𝑦2 )
Connection to UDCPs
• Let 𝐴 ⊆ {0,1}𝑛 be s.t. for all 𝑥1 , 𝑥2 ∈ 𝐴:
𝑤 ⋅ 𝑥1 = 𝑤 ⋅ 𝑥2 implies 𝑥1 = 𝑥2
• Let 𝐵 ⊆ {0,1}𝑛 be s.t. for all 𝑦1 , 𝑦2 ∈ 𝐵: 𝑤 ⋅
𝑦1 = 𝑤 ⋅ 𝑦2
• Can take 𝐴 = |𝜎(𝑤)|, 𝐵 = 𝛽 𝑤
• (𝐴, 𝐵) is UDCP: suppose 𝑥1 , 𝑥2 ∈ 𝐴, 𝑦1 , 𝑦2 ∈ 𝐵
𝑥1 + 𝑦1 = 𝑥2 + 𝑦2
𝑤 ⋅ (𝑥1 + 𝑤 ⋅ 𝑦1 ) = 𝑤 ⋅ (𝑥2 + 𝑤 ⋅ 𝑦2 )
𝑤 ⋅ (𝑥1 +→ 𝑥1 = 𝑥2 → 𝑦1 = 𝑦2
Connection to UDCPs
• Let 𝐴 ⊆ {0,1}𝑛 be s.t. for all 𝑥1 , 𝑥2 ∈ 𝐴:
𝑤 ⋅ 𝑥1 = 𝑤 ⋅ 𝑥2 implies 𝑥1 = 𝑥2
• Let 𝐵 ⊆ {0,1}𝑛 be s.t. for all 𝑦1 , 𝑦2 ∈ 𝐵:
𝑦1 = 𝑤 ⋅ 𝑦2
• Corollaries:
– 𝛽 𝑤 𝜎 𝑤 ≤ 21.5𝑛
– If 𝜎 𝑤 ≥ 2
1−𝜖 𝑛 ,
𝛽 𝑤 ≤ 2(0.4223+
𝜖)𝑛
𝑤⋅
Further Remarks
• OS also used a projection property
– We use UDCP property to prove projection property
• Can 𝛼 + 𝛽 = 1.5?
– Can provided methods be used to exclude this?
• If 𝛼 → 1, sharpen 0.25 ≤ 𝛽 ≤ 0.4223
• Let 𝛽 𝑤 ≥ 2𝛽𝑛 , 𝛽 > 0. Is there 𝜖 ≔ 𝜖 𝛽 s.t.
𝜎 𝑤 ≤ 2 1−𝜖 𝑛 ?
• Thanks for listening!

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