List Decoding of Concatenated Codes: Improved Performance Estimates
List Decoding of Concatenated Codes: Improved Performance Estimates
Alexander Barg
ISIT 2004 1/73
Andrew McGregor
List Decoding of Concatenated Codes: Improved Performance Estimates
Alexander Barg
ISIT 2004 1/73
Andrew McGregor
Important Words
ISIT 2004 2/73
Important Words • List Decoding
ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes
ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes Inner: Arbitrary q-ary [n,k,d] code A={x1,x2,…}
ISIT 2004 2/73
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
ISIT 2004 2/73
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” B jhfsoirheqck
ISIT 2004 2/73
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” B jhfsoirheqck A 0 1 0 1 0 ISIT 2004 2/73
1 1 1 0 1
1 0 1 1 0
0 1 1 0 1
1 0 0 1 0
1 1 1 0 1 N
0 0 0 1 0
1 1 1 0 1
0 1 1 0 0
0 1 1 1 1
1 0 1 1 0
1 1 1 n 1 1
Some Previous Work • GMD Decoding (Forney, 1966)
• G-S Decoding (Guruswami, Sudan 2000)
ISIT 2004 3/73
Some Previous Work • GMD Decoding (Forney, 1966)
• G-S Decoding (Guruswami, Sudan 2000)
ISIT 2004 3/73
Some Previous Work • GMD Decoding (Forney, 1966)
• G-S Decoding (Guruswami, Sudan 2000)
ISIT 2004 3/73
Our Work
ISIT 2004 4/73
Our Work • Combined GMD/G-S Decoding
ISIT 2004 4/73
Our Work • Combined GMD/G-S Decoding • Improve Estimates for Random Inner Codes
ISIT 2004 4/73
Decoding Radius of GMD Decoding List Decoding Radius
Outer Code Rate κ
ISIT 2004 5/73
List Decoding Radius
Decoding Radius of GMD Decoding G-S Decoding
Outer Code Rate κ
ISIT 2004 6/73
List Decoding Radius
Decoding Radius of GMD Decoding G-S Decoding Our Result
Outer Code Rate κ
ISIT 2004 7/73
Column Reliability
ISIT 2004 8/73
Column Reliability • Receive word x as in column i
ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if hi=d(x, A) is small
ISIT 2004 8/73
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.
ISIT 2004 8/73
Column Reliability • • • •
ISIT 2004 8/73
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.
Column Reliability • • • • •
ISIT 2004 8/73
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
Column Reliability • • • • •
ISIT 2004 8/73
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
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
ISIT 2004 8/73
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
ISIT 2004 8/73
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
ISIT 2004 8/73
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
ISIT 2004 8/73
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
ISIT 2004 8/73
Most Reliable Columns…
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
• [Dumer ’81] GMD decodes
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
• [Dumer ’81] GMD decodes
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
• [Dumer ’81] GMD decodes
• But what if H is small…
ISIT 2004 9/73
Weight-Settings in G-S • G-S: Set weights such that many c have
• Previously • New
ISIT 2004 10/73
Weight-Settings in G-S • G-S: Set weights such that many c have
• Previously • New
ISIT 2004 10/73
Weight-Settings in G-S • G-S: Set weights such that many c have
• Previously • New
ISIT 2004 10/73
Weight-Settings in G-S • G-S: Set weights such that many c have
Goes down!
Goes up!
• Previously • New
ISIT 2004 10/73
Combing GMD and G-S
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
• Combined G-S/GMD decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
• Combined G-S/GMD decodes:
ISIT 2004 11/73
Random Inner Codes • Analysis of G-S uses: • Using knowledge of the coset distribution:
• With new weight setting, G-S corrects:
ISIT 2004 12/73
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
ISIT 2004 13/73
How big can
be?
x
ISIT 2004 14/73
• Questions…
ISIT 2004
Thank you :-) 15/73
?
UGLY EXPRESSIONS! (since you asked…)
ISIT 2004 16/73
Alexander Barg
ISIT 2004 1/73
Andrew McGregor
List Decoding of Concatenated Codes: Improved Performance Estimates
Alexander Barg
ISIT 2004 1/73
Andrew McGregor
Important Words
ISIT 2004 2/73
Important Words • List Decoding
ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes
ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes Inner: Arbitrary q-ary [n,k,d] code A={x1,x2,…}
ISIT 2004 2/73
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
ISIT 2004 2/73
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” B jhfsoirheqck
ISIT 2004 2/73
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” B jhfsoirheqck A 0 1 0 1 0 ISIT 2004 2/73
1 1 1 0 1
1 0 1 1 0
0 1 1 0 1
1 0 0 1 0
1 1 1 0 1 N
0 0 0 1 0
1 1 1 0 1
0 1 1 0 0
0 1 1 1 1
1 0 1 1 0
1 1 1 n 1 1
Some Previous Work • GMD Decoding (Forney, 1966)
• G-S Decoding (Guruswami, Sudan 2000)
ISIT 2004 3/73
Some Previous Work • GMD Decoding (Forney, 1966)
• G-S Decoding (Guruswami, Sudan 2000)
ISIT 2004 3/73
Some Previous Work • GMD Decoding (Forney, 1966)
• G-S Decoding (Guruswami, Sudan 2000)
ISIT 2004 3/73
Our Work
ISIT 2004 4/73
Our Work • Combined GMD/G-S Decoding
ISIT 2004 4/73
Our Work • Combined GMD/G-S Decoding • Improve Estimates for Random Inner Codes
ISIT 2004 4/73
Decoding Radius of GMD Decoding List Decoding Radius
Outer Code Rate κ
ISIT 2004 5/73
List Decoding Radius
Decoding Radius of GMD Decoding G-S Decoding
Outer Code Rate κ
ISIT 2004 6/73
List Decoding Radius
Decoding Radius of GMD Decoding G-S Decoding Our Result
Outer Code Rate κ
ISIT 2004 7/73
Column Reliability
ISIT 2004 8/73
Column Reliability • Receive word x as in column i
ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if hi=d(x, A) is small
ISIT 2004 8/73
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.
ISIT 2004 8/73
Column Reliability • • • •
ISIT 2004 8/73
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.
Column Reliability • • • • •
ISIT 2004 8/73
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
Column Reliability • • • • •
ISIT 2004 8/73
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
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
ISIT 2004 8/73
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
ISIT 2004 8/73
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
ISIT 2004 8/73
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
ISIT 2004 8/73
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
ISIT 2004 8/73
Most Reliable Columns…
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
• [Dumer ’81] GMD decodes
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
• [Dumer ’81] GMD decodes
ISIT 2004 9/73
Most Reliable Columns… • Consider the N-D most reliable columns
• [Dumer ’81] GMD decodes
• But what if H is small…
ISIT 2004 9/73
Weight-Settings in G-S • G-S: Set weights such that many c have
• Previously • New
ISIT 2004 10/73
Weight-Settings in G-S • G-S: Set weights such that many c have
• Previously • New
ISIT 2004 10/73
Weight-Settings in G-S • G-S: Set weights such that many c have
• Previously • New
ISIT 2004 10/73
Weight-Settings in G-S • G-S: Set weights such that many c have
Goes down!
Goes up!
• Previously • New
ISIT 2004 10/73
Combing GMD and G-S
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
• Combined G-S/GMD decodes:
ISIT 2004 11/73
Combing GMD and G-S • G-S decodes:
• GMD decodes:
• Combined G-S/GMD decodes:
ISIT 2004 11/73
Random Inner Codes • Analysis of G-S uses: • Using knowledge of the coset distribution:
• With new weight setting, G-S corrects:
ISIT 2004 12/73
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
ISIT 2004 13/73
How big can
be?
x
ISIT 2004 14/73
• Questions…
ISIT 2004
Thank you :-) 15/73
?
UGLY EXPRESSIONS! (since you asked…)
ISIT 2004 16/73
