List Decoding of Concatenated Codes: Improved Performance Estimates - - PowerPoint PPT Presentation

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List Decoding of Concatenated Codes: Improved Performance Estimates - - PowerPoint PPT Presentation

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List Decoding of Concatenated Codes: Improved Performance Estimates Alexander Barg Andrew McGregor ISIT 2004 1/73 List Decoding of Concatenated Codes: Improved Performance Estimates Alexander Barg Andrew McGregor ISIT 2004 1/73 Important

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ISIT 2004

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List Decoding of Concatenated Codes: Improved Performance Estimates

Andrew McGregor Alexander Barg

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ISIT 2004

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List Decoding of Concatenated Codes: Improved Performance Estimates

Andrew McGregor Alexander Barg

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ISIT 2004

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Important Words

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Important Words

  • List Decoding
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Important Words

  • List Decoding
  • Concatenated Codes
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Important Words

  • List Decoding
  • Concatenated Codes

Inner: Arbitrary q-ary [n,k,d] code A={x1,x2,…}

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Important Words

  • List Decoding
  • Concatenated Codes

Inner: Arbitrary q-ary [n,k,d] code A={x1,x2,…} Outer: Reed-Solomon qk-ary [N, K=κN, D] code B

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Important Words

  • List Decoding
  • Concatenated Codes

Inner: Arbitrary q-ary [n,k,d] code A={x1,x2,…} Outer: Reed-Solomon qk-ary [N, K=κN, D] code B

“Message” jhfsoirheqck

B

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Important Words

  • List Decoding
  • Concatenated Codes

Inner: Arbitrary q-ary [n,k,d] code A={x1,x2,…} Outer: Reed-Solomon qk-ary [N, K=κN, D] code B

0 1 1 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1

A

n N “Message” jhfsoirheqck

B

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Some Previous Work

  • GMD Decoding (Forney, 1966)
  • G-S Decoding (Guruswami, Sudan 2000)
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Some Previous Work

  • GMD Decoding (Forney, 1966)
  • G-S Decoding (Guruswami, Sudan 2000)
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Some Previous Work

  • GMD Decoding (Forney, 1966)
  • G-S Decoding (Guruswami, Sudan 2000)
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Our Work

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Our Work

  • Combined GMD/G-S Decoding
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Our Work

  • Combined GMD/G-S Decoding
  • Improve Estimates for Random Inner Codes
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List Decoding Radius Outer Code Rate κ

Decoding Radius of GMD Decoding

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Decoding Radius of GMD Decoding G-S Decoding

List Decoding Radius Outer Code Rate κ

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Decoding Radius of GMD Decoding G-S Decoding Our Result

List Decoding Radius Outer Code Rate κ

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Column Reliability

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Column Reliability

  • Receive word x as in column i
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
  • G-S decoding will output codeword c if
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
  • G-S decoding will output codeword c if
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
  • G-S decoding will output codeword c if
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
  • G-S decoding will output codeword c if
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Column Reliability

  • Receive word x as in column i
  • Consider column reliable if hi=d(x, A) is small
  • GMD: Inner Decoder passes 1 inner codeword and hi.
  • G-S: Inner Decoder passes weights wi(xj) on each xj ∈ A.
  • The weight of an outer codeword c=a1a2…aN is
  • G-S decoding will output codeword c if
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Most Reliable Columns…

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Most Reliable Columns…

  • Consider the N-D most reliable columns
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Most Reliable Columns…

  • Consider the N-D most reliable columns
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Most Reliable Columns…

  • Consider the N-D most reliable columns
  • [Dumer ’81] GMD decodes
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Most Reliable Columns…

  • Consider the N-D most reliable columns
  • [Dumer ’81] GMD decodes
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Most Reliable Columns…

  • Consider the N-D most reliable columns
  • [Dumer ’81] GMD decodes
  • But what if H is small…
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Weight-Settings in G-S

  • G-S: Set weights such that many c have
  • Previously
  • New
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Weight-Settings in G-S

  • G-S: Set weights such that many c have
  • Previously
  • New
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Weight-Settings in G-S

  • G-S: Set weights such that many c have
  • Previously
  • New
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Weight-Settings in G-S

  • G-S: Set weights such that many c have
  • Previously
  • New

Goes down! Goes up!

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Combing GMD and G-S

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Combing GMD and G-S

  • G-S decodes:
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Combing GMD and G-S

  • G-S decodes:
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Combing GMD and G-S

  • G-S decodes:
  • GMD decodes:
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Combing GMD and G-S

  • G-S decodes:
  • GMD decodes:
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Combing GMD and G-S

  • G-S decodes:
  • GMD decodes:
  • Combined G-S/GMD decodes:
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Combing GMD and G-S

  • G-S decodes:
  • GMD decodes:
  • Combined G-S/GMD decodes:
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Random Inner Codes

  • Analysis of G-S uses:
  • Using knowledge of the coset distribution:
  • With new weight setting, G-S corrects:
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Using Coset Distribution

  • Coset Distribution Result [Zyablov & Pinsker ’81]: For

almost all [n,rn] linear codes the number of codewords in a sphere of radius n(δ−ε), is at most

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x How big can be?

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?

  • Questions…

Thank you :-)

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UGLY EXPRESSIONS!

(since you asked…)