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 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={x 1 ,x 2 ,…} ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes Inner: Arbitrary q -ary [ n,k,d ] code A={x 1 ,x 2 ,…} Outer: Reed-Solomon q k -ary [ N, K= κ N, D ] code B ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes Inner: Arbitrary q -ary [ n,k,d ] code A={x 1 ,x 2 ,…} Outer: Reed-Solomon q k -ary [ N, K= κ N, D ] code B B “Message” jhfsoirheqck ISIT 2004 2/73
Important Words • List Decoding • Concatenated Codes Inner: Arbitrary q -ary [ n,k,d ] code A={x 1 ,x 2 ,…} Outer: Reed-Solomon q k -ary [ N, K= κ N, D ] code B B “Message” jhfsoirheqck A 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 n 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 ISIT 2004 N 2/73
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
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
Decoding Radius of GMD Decoding G-S Decoding List Decoding Radius Outer Code Rate κ ISIT 2004 6/73
Decoding Radius of GMD Decoding G-S Decoding List Decoding Our Result Radius 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 h i =d(x, A) is small ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N is ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N is ISIT 2004 8/73
Column Reliability • Receive word x as in column i • Consider column reliable if h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N 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 h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N 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 h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N 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 h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N 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 h i =d(x, A) is small • GMD: Inner Decoder passes 1 inner codeword and h i . • G-S: Inner Decoder passes weights w i ( x j ) on each x j ∈ A . • The weight of an outer codeword c=a 1 a 2 …a N 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
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