Improved efficiency for covering codes matching the sphere- - PowerPoint PPT Presentation

Improved efficiency for covering codes matching the sphere- covering bound Aditya Potukuchi and Yihan Zhang ISIT 2020

Introduction: Covering codes

Introduction: Covering codes • A subset 𝒟 ⊆ {0,1} n is said to be a covering code with relative covering z ∈ {0,1} n radius if for every , we have that min c ∈𝒟 d ( c , z ) ≤ δ n δ

Introduction: Covering codes • A subset 𝒟 ⊆ {0,1} n is said to be a covering code with relative covering z ∈ {0,1} n radius if for every , we have that min c ∈𝒟 d ( c , z ) ≤ δ n δ Hamming distance

Introduction: Covering codes • A subset 𝒟 ⊆ {0,1} n is said to be a covering code with relative covering z ∈ {0,1} n radius if for every , we have that min c ∈𝒟 d ( c , z ) ≤ δ n δ Hamming distance • ``every point is close to some point in the code''

Introduction: Covering codes • A subset 𝒟 ⊆ {0,1} n is said to be a covering code with relative covering z ∈ {0,1} n radius if for every , we have that min c ∈𝒟 d ( c , z ) ≤ δ n δ Hamming distance • ``every point is close to some point in the code'' • Play a role in rate distortion theory and source coding

Introduction: Covering codes • A subset 𝒟 ⊆ {0,1} n is said to be a covering code with relative covering z ∈ {0,1} n radius if for every , we have that min c ∈𝒟 d ( c , z ) ≤ δ n δ Hamming distance • ``every point is close to some point in the code'' • Play a role in rate distortion theory and source coding • Dual notion of the usual codes, interesting combinatorial objects

Upper and lower bounds on size

Upper and lower bounds on size • Sphere-covering bound: Every covering code of block length and (relative n 2 n covering) radius must have size at least δ ( ≤ δ n ) n

Upper and lower bounds on size • Sphere-covering bound: Every covering code of block length and (relative n 2 n covering) radius must have size at least δ ( ≤ δ n ) n

Upper and lower bounds on size • Sphere-covering bound: Every covering code of block length and (relative n 2 n covering) radius must have size at least δ ( ≤ δ n ) n

Upper and lower bounds on size • Sphere-covering bound: Every covering code of block length and (relative n 2 n covering) radius must have size at least δ ( ≤ δ n ) n 2 n 𝒟 ⊂ {0,1} n Existence: A random subset of size is almost surely 100 n ⋅ • ( ≤ δ n ) n a covering code of radius δ

Upper and lower bounds on rate

Upper and lower bounds on rate • Sphere-covering bound: Any covering code of radius must have rate at δ least 1 − H ( δ ) + o (1)

Upper and lower bounds on rate • Sphere-covering bound: Any covering code of radius must have rate at δ least 1 − H ( δ ) + o (1) • Existence: There exist covering codes of radius and rate 1 − H ( δ ) + o (1) δ

Upper and lower bounds on rate • Sphere-covering bound: Any covering code of radius must have rate at δ least 1 − H ( δ ) + o (1) • Existence: There exist covering codes of radius and rate 1 − H ( δ ) + o (1) δ • So, the optimal rate-radius tradeo ff is well understood

Upper and lower bounds on rate • Sphere-covering bound: Any covering code of radius must have rate at δ least 1 − H ( δ ) + o (1) • Existence: There exist covering codes of radius and rate 1 − H ( δ ) + o (1) δ • So, the optimal rate-radius tradeo ff is well understood • Interested in constructing such codes

Concatenation preserves covering radius

Concatenation preserves covering radius 𝒟 1 , 𝒟 2 ⊆ {0,1} n • For , the concatenated code is defined as 𝒟 1 ⊕ 𝒟 2 := {( c 1 , c 2 ) | c 1 ∈ 𝒟 1 , c 2 ∈ 𝒟 2 }

Concatenation preserves covering radius 𝒟 1 , 𝒟 2 ⊆ {0,1} n • For , the concatenated code is defined as 𝒟 1 ⊕ 𝒟 2 := {( c 1 , c 2 ) | c 1 ∈ 𝒟 1 , c 2 ∈ 𝒟 2 } 𝒟 1 , 𝒟 2 ⊆ {0,1} n • Fact: If are covering codes of radius respectively, then δ 1 , δ 2 δ 1 + δ 2 has radius 𝒟 1 ⊕ 𝒟 2 2

Concatenation preserves covering radius 𝒟 1 , 𝒟 2 ⊆ {0,1} n • For , the concatenated code is defined as 𝒟 1 ⊕ 𝒟 2 := {( c 1 , c 2 ) | c 1 ∈ 𝒟 1 , c 2 ∈ 𝒟 2 } 𝒟 1 , 𝒟 2 ⊆ {0,1} n • Fact: If are covering codes of radius respectively, then δ 1 , δ 2 δ 1 + δ 2 has radius 𝒟 1 ⊕ 𝒟 2 2 • If , concatenation preserves radius (and also rate) δ 1 = δ 2

Concatenation preserves covering radius 𝒟 1 , 𝒟 2 ⊆ {0,1} n • For , the concatenated code is defined as 𝒟 1 ⊕ 𝒟 2 := {( c 1 , c 2 ) | c 1 ∈ 𝒟 1 , c 2 ∈ 𝒟 2 } 𝒟 1 , 𝒟 2 ⊆ {0,1} n • Fact: If are covering codes of radius respectively, then δ 1 , δ 2 δ 1 + δ 2 has radius 𝒟 1 ⊕ 𝒟 2 2 • If , concatenation preserves radius (and also rate) δ 1 = δ 2 • Enough to construct codes of small block length and bootstrap

Linear covering codes

Linear covering codes • Theorem [Blinovsky '90]: A random linear code of rate is 1 − H ( δ ) + O (1/ n ) a covering code with radius with high probability ( ). 1 − o (1) δ

Linear covering codes • Theorem [Blinovsky '90]: A random linear code of rate is 1 − H ( δ ) + O (1/ n ) a covering code with radius with high probability ( ). 1 − o (1) δ • Corollary (Folklore): For every , the concatenation of all linear codes of ϵ > 0 block length and rate gives a code of rate n 1 − H ( δ ) + Θ (1/ n ) and radius 1 − H ( δ ) + ϵ δ

Linear covering codes • Theorem [Blinovsky '90]: A random linear code of rate is 1 − H ( δ ) + O (1/ n ) a covering code with radius with high probability ( ). 1 − o (1) δ • Corollary (Folklore): For every , the concatenation of all linear codes of ϵ > 0 block length and rate gives a code of rate n 1 − H ( δ ) + Θ (1/ n ) and radius 1 − H ( δ ) + ϵ δ • Gives a construction of covering codes that is ``explicit''

An issue with the construction

An issue with the construction • Concatenation of all linear codes of block length and rate n gives a code of rate and radius 1 − H ( δ ) + Θ (1/ n ) 1 − H ( δ ) + ϵ δ

An issue with the construction • Concatenation of all linear codes of block length and rate n gives a code of rate and radius 1 − H ( δ ) + Θ (1/ n ) 1 − H ( δ ) + ϵ δ • Concatenation of codes of the same rate preserves rate

An issue with the construction • Concatenation of all linear codes of block length and rate n gives a code of rate and radius 1 − H ( δ ) + Θ (1/ n ) 1 − H ( δ ) + ϵ δ • Concatenation of codes of the same rate preserves rate • This gives guarantees for constructions as long as n = Ω (1/ ϵ )

An issue with the construction • Concatenation of all linear codes of block length and rate n gives a code of rate and radius 1 − H ( δ ) + Θ (1/ n ) 1 − H ( δ ) + ϵ δ • Concatenation of codes of the same rate preserves rate • This gives guarantees for constructions as long as n = Ω (1/ ϵ ) • Suppose we wanted to construct codes of block length , we need to N N ≥ n ⋅ 2 Ω δ ( n 2 ) = exp(1/ ϵ 2 ) 2 Ω δ ( n 2 ) concatenate codes so

Main motivation

Main motivation • What the actually have: To obtain codes of block length and radius and N δ N ≥ exp(1/ ϵ 2 ) rate , the previous construction requires that 1 − H ( δ ) + ϵ

Main motivation • What the actually have: To obtain codes of block length and radius and N δ N ≥ exp(1/ ϵ 2 ) rate , the previous construction requires that 1 − H ( δ ) + ϵ • We want a better dependence on and (we call this ``e ffi ciency'') N ϵ

Main motivation • What the actually have: To obtain codes of block length and radius and N δ N ≥ exp(1/ ϵ 2 ) rate , the previous construction requires that 1 − H ( δ ) + ϵ • We want a better dependence on and (we call this ``e ffi ciency'') N ϵ • Open question: Obtain an explicit construction where N = poly (1/ ϵ )

Main motivation • What the actually have: To obtain codes of block length and radius and N δ N ≥ exp(1/ ϵ 2 ) rate , the previous construction requires that 1 − H ( δ ) + ϵ • We want a better dependence on and (we call this ``e ffi ciency'') N ϵ • Open question: Obtain an explicit construction where N = poly (1/ ϵ ) • We know that is possible N = 1/ ϵ

Revisiting the previous construction

Revisiting the previous construction • Why was the e ffi ciency so bad?

Revisiting the previous construction • Why was the e ffi ciency so bad? 2 Ω δ ( n 2 ) • Need to concatenate codes of block length . We know that most of n these have optimal rate-radius tradeo ff

Revisiting the previous construction • Why was the e ffi ciency so bad? 2 Ω δ ( n 2 ) • Need to concatenate codes of block length . We know that most of n these have optimal rate-radius tradeo ff • Can improve tradeo ff if we could concatenate fewer codes, each of block length , where we know most of these have optimal rate-radius tradeo ff n

Our main result

Our main result • Main Theorem [informal]: For every , there is a construction of a code of ϵ > 0 block length with rate and radius as long as N 1 − H ( δ ) + ϵ δ N ≥ exp(1/ ϵ log(1/ ϵ ))