Near-Optimal Erasure List- Decodable · PDF file Near-Optimal Erasure List-Decodable Codes...

Click here to load reader

  • date post

    21-Apr-2020
  • Category

    Documents

  • view

    4
  • download

    0

Embed Size (px)

Transcript of Near-Optimal Erasure List- Decodable · PDF file Near-Optimal Erasure List-Decodable Codes...

  • Near-Optimal Erasure List- Decodable Codes

    Dean Doron (Tel-Aviv University) Workshop on Coding and Information Theory, CMSA

    Joint work with Avi Ben-Aroya Amnon Ta-Shma

  • • A code C : {0,1}n → {0,1}n̄ is (1-ε,L) erasure list- decodable, if

    • Given a codeword

    Erasure List-Decodable Codes

    0 1 1 0 0 0 1 0 1 1

  • • A code C : {0,1}n → {0,1}n̄ is (1-ε,L) erasure list- decodable, if

    • Given a codeword where all but ε fraction of its coordinates were (adversarially) erased,

    Erasure List-Decodable Codes

    0 1 1 0 0 0 1 0 1 1

  • • A code C : {0,1}n → {0,1}n̄ is (1-ε,L) erasure list- decodable, if

    • Given a codeword where all but ε fraction of its coordinates were (adversarially) erased,

    Erasure List-Decodable Codes

    ? 1 ? ? ? 0 1 ? ? 1

  • • A code C : {0,1}n → {0,1}n̄ is (1-ε,L) erasure list- decodable, if

    • Given a codeword where all but ε fraction of its coordinates were (adversarially) erased,

    Erasure List-Decodable Codes

    ? 1 ? ? ? 0 1 ? ? 1

    1 1 1 0 1 0 1 0 0 1

    0 1 1 0 0 0 1 0 1 1

    0 1 0 0 0 0 1 1 1 1

    ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

    0 1 1 1 1 0 1 1 0 1

    n̄there exists a list of size at most L that contains the original codeword.

  • • A code C : {0,1}n → {0,1}n̄ is (1-ε,L) erasure list- decodable, if

    • Given a codeword where all but ε fraction of its coordinates were (adversarially) erased,

    Erasure List-Decodable Codes

    ? 1 ? ? ? 0 1 ? ? 1

    1 1 1 0 1 0 1 0 0 1

    0 1 1 0 0 0 1 0 1 1

    0 1 0 0 0 0 1 1 1 1

    ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

    0 1 1 1 1 0 1 1 0 1

    n̄there exists a list of size at most L that contains the original codeword.

  • • We work over the binary field.

    • The two key parameters:

    • The code’s rate R = n̄/n.

    • The list-size L.

    Bounds for the binary case

  • Bounds on the rate

  • • The code’s rate — R.

    Bounds on the rate

  • • The code’s rate — R.

    • Can be as large as ε [Guruswami 03] (recall that ε is the fraction of coordinates we keep).

    • Compare with the errors model, where ~ ε2 is necessary for a code of distance ½ - ε.

    Bounds on the rate

  • Bounds on the list-size

  • • The list-size — L.

    Bounds on the list-size

  • • The list-size — L.

    • Can be as tiny as log(1/ε) [Guruswami 03].

    • Compare with the errors model, where poly(1/ε) is necessary.

    Bounds on the list-size

  • • The list-size — L.

    • Can be as tiny as log(1/ε) [Guruswami 03].

    • Compare with the errors model, where poly(1/ε) is necessary.

    • For linear codes, the list-size is at least 1/ε [Guruswami 03].

    Bounds on the list-size

  • Previous Results

  • • Any binary code of distance ½ - ε can handle 1-2ε erasures with polynomial list size.

    Previous Results

  • • Any binary code of distance ½ - ε can handle 1-2ε erasures with polynomial list size.

    • Drawback: We have upper bounds. Cannot break the rate-ε2 bound that way [MRRW,…].

    Previous Results

  • • Any binary code of distance ½ - ε can handle 1-2ε erasures with polynomial list size.

    • Drawback: We have upper bounds. Cannot break the rate-ε2 bound that way [MRRW,…].

    • Few explicit constructions (especially for binary codes).

    Previous Results

  • Previous Results

    Rate R List-Size L

    Non-explicit ε log(1/ε)

    Optimal codes with distance ½-ε ε

    2 poly(1/ε)

    Guruswami 01 (constant ε) ε

    2/log(1/ε) 1/ε

    Guruswami-Indyk 03 (constant ε) ε

    2/log(1/ε) log(1/ε)

  • • In the small ε regime, we break the ε2 barrier, and manage to do this with nearly optimal parameters.

    Our Result

  • • In the small ε regime, we break the ε2 barrier, and manage to do this with nearly optimal parameters.

    Our Result

    For every small enough ε and a constant γ there exists an explicit binary erasure list-decodable code with rate

    R = ε1+ γ

    and list-size

    L = poly(log(1/ε))

  • Previous Results

    Rate R List-Size L

    Non-explicit ε log(1/ε)

    Optimal codes with distance ½-ε ε

    2 poly(1/ε)

    Guruswami 01 (constant ε) ε

    2/log(1/ε) 1/ε

    Guruswami-Indyk 03 (constant ε) ε

    2/log(1/ε) log(1/ε)

    Our Result
 (small ε) ε

    1+γ poly(log(1/ε))

  • Different Perspectives

    Erasure List-

    Decodable Codes

    Ramsey Graphs

    Strong 1- bit

    Dispersers

    [Guruswami 04]

    [Gradwohl, Kindler, 
 Reingold, Ta-Shma 05]

  • Different Perspectives

    Erasure List-

    Decodable Codes

    Ramsey Graphs

    Strong 1- bit

    Dispersers

    [Guruswami 04]

    [Gradwohl, Kindler, 
 Reingold, Ta-Shma 05]

  • Different Perspectives

    Erasure List-

    Decodable Codes

    Ramsey Graphs

    Strong 1-bit

    Dispersers

    [Guruswami 04]

    [Gradwohl, Kindler, 
 Reingold, Ta-Shma 05]

  • Different Perspectives

    Erasure List-

    Decodable Codes

    Ramsey Graphs

    Strong 1-bit

    Dispersers

    [Guruswami 04]

    [Gradwohl, Kindler, 
 Reingold, Ta-Shma 05]

  • Different Perspectives

    Erasure List-

    Decodable Codes

    Ramsey Graphs

    Strong 1-bit

    Dispersers

    [Guruswami 04]

    [Gradwohl, Kindler, 
 Reingold, Ta-Shma 05]

  • 1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    Disp(x,y)

    {0,1}

    D=2dx

  • • For every (n,k)-source X, or a set X⊆{0,1}n of size at least K=2k,

    1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    X Disp(x,y)

    {0,1}

    D=2dx

  • • For every (n,k)-source X, or a set X⊆{0,1}n of size at least K=2k,

    1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    X

    For every x, Pr[X = x] ≤ 2-k.

    Disp(x,y)

    {0,1}

    D=2dx

  • • For every (n,k)-source X, or a set X⊆{0,1}n of size at least K=2k,

    1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    X Disp(x,y)

    {0,1}

    D=2dx

  • • For every (n,k)-source X, or a set X⊆{0,1}n of size at least K=2k,

    • For all but ε-fraction of the seeds y∈{0,1}d,

    Disp(X,y) = {0,1}

    1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    X Disp(x,y)

    {0,1}

    D=2dx

  • • For every (n,k)-source X, or a set X⊆{0,1}n of size at least K=2k,

    • For all but ε-fraction of the seeds y∈{0,1}d,

    Disp(X,y) = {0,1}

    1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    X Disp(x1,ybad)=1

    Disp(xK,ybad)=1

    {0,1}

  • • For every (n,k)-source X, or a set X⊆{0,1}n of size at least K=2k,

    • For all but ε-fraction of the seeds y∈{0,1}d,

    Disp(X,y) = {0,1}

    1-bit Strong Dispersers

    Disp: {0,1}n × {0,1}d →{0,1}

    {0,1}n

    0

    1

    X

    Disp(x1,y)=0

    Disp(xK,y)=1

    {0,1}

  • Bounds for1-bit Strong Dispersers

  • • Dispersers are an important tool in derandomization.

    Bounds for1-bit Strong Dispersers

  • • Dispersers are an important tool in derandomization.

    • The two key parameters:

    Bounds for1-bit Strong Dispersers

  • • Dispersers are an important tool in derandomization.

    • The two key parameters:

    • The disperser’s seed length — d.

    • The disperser’s entropy requirement — k (or, how small can X be).

    Bounds for1-bit Strong Dispersers

  • Bounds on the seed length

  • • The disperser’s seed length — d.

    Bounds on the seed length

  • • The disperser’s seed length — d.

    • Can be as small as log(n)+log(1/ε) [Radhakrishnan,Ta- Shma 00, Meka,Reingold,Zhou 14].

    Bounds on the seed length

  • Bounds on the entropy

  • • The disperser’s entropy requirement — k (or, how small can X be).

    • Can be as tiny as loglog(1/ε). That is, an optimal disperser works for extremely small sets — of size O(log(1/ε)) [Radhakrishnan,Ta-Shma 00, Meka,Reingold,Zhou 14].

    Bounds on the entropy

  • Comparison with extractors

  • • Dispersers’ parameters outperform seeded extractors, in which we don