![Page 1: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/1.jpg)
Efficiently Decoding Reed Muller CodesFrom Random Errors
Ben Lee Volk
Joint work with
Ramprasad SaptharishiAmir Shpilka
Tel Aviv University
![Page 2: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/2.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
1 0 1 1 0 0 01 0 1 1 0 0 0 1 1
![Page 3: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/3.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
1 0 1 1 0 0 01 0 1 1 0 0 0 1 1
![Page 4: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/4.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
0 1 0 0 1 1 11 0 1 1 0 0 0 1 1
![Page 5: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/5.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
0 1 0 0 1 1 11 0 1 1 0 0 0 1 1
![Page 6: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/6.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
0 1 0 0 1 1 11 0 1 1 0 0 0 1 1
AdversarialErrors: Impossible.
![Page 7: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/7.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
0 1 0 0 1 1 11 0 1 1 0 0 0 1 1
AdversarialErrors: Impossible.RandomErrors?(each coordinate flipped independently with probability 1/2− o(1))
![Page 8: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/8.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
0 1 0 0 1 1 11 0 1 1 0 0 0 1 1
AdversarialErrors: Impossible.RandomErrors?(each coordinate flipped independently with probability 1/2− o(1))
Theorem: There is an efficient algorithm to recover f ,even for r = o(
pm).
![Page 9: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/9.jpg)
A game!Given the truth-table of a polynomial f ∈ F[x1, . . . , xm] of degree≤ r, with 1/2− o(1) of the entries flipped, recover f efficiently.
0 1 0 0 1 1 11 0 1 1 0 0 0 1 1
AdversarialErrors: Impossible.RandomErrors?(each coordinate flipped independently with probability 1/2− o(1))
Theorem: There is an efficient algorithm to recover f ,even for r = o(
pm).
This talk is about decoding Reed-Muller codes fromrandom errors.
![Page 10: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/10.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 11: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/11.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 12: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/12.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 13: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/13.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with
• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 14: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/14.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 15: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/15.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )
• Dimension:�m
0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 16: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/16.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�
• Rate: dimension/block length =� m≤r
�/2m
![Page 17: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/17.jpg)
Reed-Muller Codes: RM(m, r)• Messages: (coefficient vectors of) degree ≤ r polynomials
f ∈ F2[x1, . . . , xm]
• Encoding: evaluations over Fm2
f (x1, x2, x3, x4) = x1 x2 + x3 x47→
0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0
A linear code, with• Block Length: 2m := n
• Distance: 2m−r (lightest codeword: x1 x2 · · · x r )• Dimension:
�m0
�+�m
1
�+ · · ·+ �mr � := � m≤r
�• Rate: dimension/block length =
� m≤r
�/2m
![Page 18: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/18.jpg)
More Properties of Reed-Muller CodesGenerator Matrix: evaluation matrix of deg≤ r monomials:
v ∈ Fm2
M
:= E(m, r)M(v)
(Every codeword is spanned by the rows.)
Linear codes can be also defined by their parity check matrix: amatrix H whose kernel is the code.Duality: RM(m, r)⊥ = RM(m, m− r − 1).=⇒ parity check matrix of RM(m, r) is E(m, m− r − 1).
![Page 19: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/19.jpg)
More Properties of Reed-Muller CodesGenerator Matrix: evaluation matrix of deg≤ r monomials:
v ∈ Fm2
M
:= E(m, r)M(v)
(Every codeword is spanned by the rows.)
Linear codes can be also defined by their parity check matrix: amatrix H whose kernel is the code.Duality: RM(m, r)⊥ = RM(m, m− r − 1).=⇒ parity check matrix of RM(m, r) is E(m, m− r − 1).
![Page 20: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/20.jpg)
More Properties of Reed-Muller CodesGenerator Matrix: evaluation matrix of deg≤ r monomials:
v ∈ Fm2
M
:= E(m, r)M(v)
(Every codeword is spanned by the rows.)
Linear codes can be also defined by their parity check matrix: amatrix H whose kernel is the code.
Duality: RM(m, r)⊥ = RM(m, m− r − 1).=⇒ parity check matrix of RM(m, r) is E(m, m− r − 1).
![Page 21: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/21.jpg)
More Properties of Reed-Muller CodesGenerator Matrix: evaluation matrix of deg≤ r monomials:
v ∈ Fm2
M
:= E(m, r)M(v)
(Every codeword is spanned by the rows.)
Linear codes can be also defined by their parity check matrix: amatrix H whose kernel is the code.Duality: RM(m, r)⊥ = RM(m, m− r − 1).
=⇒ parity check matrix of RM(m, r) is E(m, m− r − 1).
![Page 22: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/22.jpg)
More Properties of Reed-Muller CodesGenerator Matrix: evaluation matrix of deg≤ r monomials:
v ∈ Fm2
M
:= E(m, r)M(v)
(Every codeword is spanned by the rows.)
Linear codes can be also defined by their parity check matrix: amatrix H whose kernel is the code.Duality: RM(m, r)⊥ = RM(m, m− r − 1).=⇒ parity check matrix of RM(m, r) is E(m, m− r − 1).
![Page 23: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/23.jpg)
Decoding Reed-Muller CodesWorstCaseErrors: Up to d/2 (d is minimal distance).
d d/2
(algorithm by Reed54)
ListDecoding: max radius with constant # of words
Gopalan-Klivans-Zuckerman08,Bhowmick-Lovett15:List decoding radius = d.
![Page 24: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/24.jpg)
Decoding Reed-Muller CodesWorstCaseErrors: Up to d/2 (d is minimal distance).
d d/2
(algorithm by Reed54)ListDecoding: max radius with constant # of words
Gopalan-Klivans-Zuckerman08,Bhowmick-Lovett15:List decoding radius = d.
![Page 25: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/25.jpg)
Decoding Reed-Muller CodesWorstCaseErrors: Up to d/2 (d is minimal distance).
d d/2
(algorithm by Reed54)ListDecoding: max radius with constant # of words
d Gopalan-Klivans-Zuckerman08,Bhowmick-Lovett15:List decoding radius = d.
![Page 26: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/26.jpg)
Decoding Reed-Muller Codes: Average Case• Reed-Muller Codes are not very good with respect to
worst-case errors (can’t have constant rate and mindistance at the same time)
• What about the Shannon model of random corruptions?• Standard model in coding theory with recent breakthroughs
in the last few years (e.g. Arıkan’s polar codes)• An ongoing research endeavor: how do Reed-Muller
perform in Shannon’s random error model?
![Page 27: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/27.jpg)
Decoding Reed-Muller Codes: Average Case• Reed-Muller Codes are not very good with respect to
worst-case errors (can’t have constant rate and mindistance at the same time)
• What about the Shannon model of random corruptions?
• Standard model in coding theory with recent breakthroughsin the last few years (e.g. Arıkan’s polar codes)
• An ongoing research endeavor: how do Reed-Mullerperform in Shannon’s random error model?
![Page 28: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/28.jpg)
Decoding Reed-Muller Codes: Average Case• Reed-Muller Codes are not very good with respect to
worst-case errors (can’t have constant rate and mindistance at the same time)
• What about the Shannon model of random corruptions?• Standard model in coding theory with recent breakthroughs
in the last few years (e.g. Arıkan’s polar codes)
• An ongoing research endeavor: how do Reed-Mullerperform in Shannon’s random error model?
![Page 29: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/29.jpg)
Decoding Reed-Muller Codes: Average Case• Reed-Muller Codes are not very good with respect to
worst-case errors (can’t have constant rate and mindistance at the same time)
• What about the Shannon model of random corruptions?• Standard model in coding theory with recent breakthroughs
in the last few years (e.g. Arıkan’s polar codes)• An ongoing research endeavor: how do Reed-Muller
perform in Shannon’s random error model?
![Page 30: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/30.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0
![Page 31: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/31.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 00 0 1
![Page 32: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/32.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0? ? ?
![Page 33: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/33.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0? ? ?
Binary Symmetric Channel — BSC(p)Each bit independently flipped with probability p
![Page 34: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/34.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0? ? ?
Binary Symmetric Channel — BSC(p)Each bit independently flipped with probability p
0 0 1 1 00 0 1
![Page 35: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/35.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0? ? ?
Binary Symmetric Channel — BSC(p)Each bit independently flipped with probability p
0 0 1 1 01 1 0
![Page 36: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/36.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0? ? ?
Binary Symmetric Channel — BSC(p)Each bit independently flipped with probability p
0 0 1 1 01 1 0
(almost) equiv: fixed number t = pn of random errors
![Page 37: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/37.jpg)
Models for random corruptions (channels)Binary Erasure Channel — BEC(p)
Each bit independently replaced by ’?’ with probability p
0 0 1 1 0? ? ?
Binary Symmetric Channel — BSC(p)Each bit independently flipped with probability p
0 0 1 1 01 1 0
(almost) equiv: fixed number t = pn of random errors
Shannon48: max rate that enables decoding (w.h.p.) is 1− p (forBEC) and 1−H(p) (for BSC). Codes achieving bound calledcapacity achieving.
![Page 38: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/38.jpg)
![Page 39: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/39.jpg)
Decoding Erasures• How many random erasures can RM(m, m− r − 1) tolerate?
• Fact: for any linear code, a subset S ⊆ [n] of erasures isuniquely decodable ⇐⇒ columns indexed by S inparity-check matrix are linearly independentdecoding algo: solve system of linear equations
• Reminder: PCM of RM(m, m− r − 1) is E(m, r)
• =⇒ can’t correct more than� m≤r
�erasures
(also follows from Shannon’s theorem)• Question: Can we correct (1− o(1))
� m≤r
�erasures?
• if yes, RM(m, m− r − 1) achieves capacity for the BEC
![Page 40: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/40.jpg)
Decoding Erasures• How many random erasures can RM(m, m− r − 1) tolerate?• Fact: for any linear code, a subset S ⊆ [n] of erasures is
uniquely decodable ⇐⇒ columns indexed by S inparity-check matrix are linearly independentdecoding algo: solve system of linear equations
• Reminder: PCM of RM(m, m− r − 1) is E(m, r)
• =⇒ can’t correct more than� m≤r
�erasures
(also follows from Shannon’s theorem)• Question: Can we correct (1− o(1))
� m≤r
�erasures?
• if yes, RM(m, m− r − 1) achieves capacity for the BEC
![Page 41: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/41.jpg)
Decoding Erasures• How many random erasures can RM(m, m− r − 1) tolerate?• Fact: for any linear code, a subset S ⊆ [n] of erasures is
uniquely decodable ⇐⇒ columns indexed by S inparity-check matrix are linearly independentdecoding algo: solve system of linear equations
• Reminder: PCM of RM(m, m− r − 1) is E(m, r)
• =⇒ can’t correct more than� m≤r
�erasures
(also follows from Shannon’s theorem)• Question: Can we correct (1− o(1))
� m≤r
�erasures?
• if yes, RM(m, m− r − 1) achieves capacity for the BEC
![Page 42: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/42.jpg)
Decoding Erasures• How many random erasures can RM(m, m− r − 1) tolerate?• Fact: for any linear code, a subset S ⊆ [n] of erasures is
uniquely decodable ⇐⇒ columns indexed by S inparity-check matrix are linearly independentdecoding algo: solve system of linear equations
• Reminder: PCM of RM(m, m− r − 1) is E(m, r)
• =⇒ can’t correct more than� m≤r
�erasures
(also follows from Shannon’s theorem)• Question: Can we correct (1− o(1))
� m≤r
�erasures?
• if yes, RM(m, m− r − 1) achieves capacity for the BEC
![Page 43: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/43.jpg)
Decoding Erasures• How many random erasures can RM(m, m− r − 1) tolerate?• Fact: for any linear code, a subset S ⊆ [n] of erasures is
uniquely decodable ⇐⇒ columns indexed by S inparity-check matrix are linearly independentdecoding algo: solve system of linear equations
• Reminder: PCM of RM(m, m− r − 1) is E(m, r)
• =⇒ can’t correct more than� m≤r
�erasures
(also follows from Shannon’s theorem)• Question: Can we correct (1− o(1))
� m≤r
�erasures?
• if yes, RM(m, m− r − 1) achieves capacity for the BEC
![Page 44: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/44.jpg)
Decoding Erasures• How many random erasures can RM(m, m− r − 1) tolerate?• Fact: for any linear code, a subset S ⊆ [n] of erasures is
uniquely decodable ⇐⇒ columns indexed by S inparity-check matrix are linearly independentdecoding algo: solve system of linear equations
• Reminder: PCM of RM(m, m− r − 1) is E(m, r)
• =⇒ can’t correct more than� m≤r
�erasures
(also follows from Shannon’s theorem)• Question: Can we correct (1− o(1))
� m≤r
�erasures?
• if yes, RM(m, m− r − 1) achieves capacity for the BEC
![Page 45: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/45.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 46: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/46.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 47: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/47.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 48: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/48.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 49: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/49.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 50: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/50.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 51: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/51.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
![Page 52: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/52.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
If t = (1− o(1))� m≤r
�, RM(m, m− r −1) achieves capacity for BEC.
![Page 53: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/53.jpg)
Reed-Muller Codes and the BECFor v ∈ Fm
2 , let vr = the column indexed by v in E(m, r)(all evals of monoms of deg≤ r on v )
What’s the max t such that forrandomly picked u1, . . . ,ut ∈ Fm
2 ,ur
1, . . . ,urt are linearly independent?
E(m, r)
If t = (1− o(1))� m≤r
�, RM(m, m− r −1) achieves capacity for BEC.
sanity check: r = 1 is good
![Page 54: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/54.jpg)
Reed-Muller Codes and the BECMainQuestion: Does RM(m, r) achieve capacity for BEC?
Abbe-Shpilka-Wigderson15: YES if r = o(m) orr = m− o(p
m/ log m).Kumar-Pfister15, Kudekar-Mondelli-Şaşoğlu-Urbanke15:YES if rate is constant (i.e. r = m/2±O(
pm)).
m/20 m
o(m) o(p
m/ log m)O(p
m)
OpenProblem: Prove for every degree r.
![Page 55: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/55.jpg)
Reed-Muller Codes and the BECMainQuestion: Does RM(m, r) achieve capacity for BEC?Abbe-Shpilka-Wigderson15: YES if r = o(m) orr = m− o(p
m/ log m).
Kumar-Pfister15, Kudekar-Mondelli-Şaşoğlu-Urbanke15:YES if rate is constant (i.e. r = m/2±O(
pm)).
m/20 m
o(m) o(p
m/ log m)O(p
m)
OpenProblem: Prove for every degree r.
![Page 56: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/56.jpg)
Reed-Muller Codes and the BECMainQuestion: Does RM(m, r) achieve capacity for BEC?Abbe-Shpilka-Wigderson15: YES if r = o(m) orr = m− o(p
m/ log m).Kumar-Pfister15, Kudekar-Mondelli-Şaşoğlu-Urbanke15:YES if rate is constant (i.e. r = m/2±O(
pm)).
m/20 m
o(m) o(p
m/ log m)O(p
m)
OpenProblem: Prove for every degree r.
![Page 57: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/57.jpg)
Reed-Muller Codes and the BECMainQuestion: Does RM(m, r) achieve capacity for BEC?Abbe-Shpilka-Wigderson15: YES if r = o(m) orr = m− o(p
m/ log m).Kumar-Pfister15, Kudekar-Mondelli-Şaşoğlu-Urbanke15:YES if rate is constant (i.e. r = m/2±O(
pm)).
m/20 m
o(m) o(p
m/ log m)O(p
m)
OpenProblem: Prove for every degree r.
![Page 58: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/58.jpg)
Reed-Muller Codes and the BECMainQuestion: Does RM(m, r) achieve capacity for BEC?Abbe-Shpilka-Wigderson15: YES if r = o(m) orr = m− o(p
m/ log m).Kumar-Pfister15, Kudekar-Mondelli-Şaşoğlu-Urbanke15:YES if rate is constant (i.e. r = m/2±O(
pm)).
m/20 m
o(m) o(p
m/ log m)O(p
m)
OpenProblem: Prove for every degree r.
![Page 59: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/59.jpg)
Decoding Erasures to Decoding ErrorsHow is this relevant for error correction?
Theorem: (ASW) Any erasure pattern correctable from erasuresin RM(m, m− r − 1) is correctable from errors inRM(m, m− 2r − 2).Thistalk: There is an efficient algorithm that corrects fromerrors, in RM(m, m− 2r − 2), any pattern which is correctablefrom erasures in RM(m, m− r − 1).(+ extensions to larger alphabets and other codes)
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 60: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/60.jpg)
Decoding Erasures to Decoding ErrorsHow is this relevant for error correction?Theorem: (ASW) Any erasure pattern correctable from erasuresin RM(m, m− r − 1) is correctable from errors inRM(m, m− 2r − 2).
Thistalk: There is an efficient algorithm that corrects fromerrors, in RM(m, m− 2r − 2), any pattern which is correctablefrom erasures in RM(m, m− r − 1).(+ extensions to larger alphabets and other codes)
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 61: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/61.jpg)
Decoding Erasures to Decoding ErrorsHow is this relevant for error correction?Theorem: (ASW) Any erasure pattern correctable from erasuresin RM(m, m− r − 1) is correctable from errors inRM(m, m− 2r − 2).Thistalk: There is an efficient algorithm that corrects fromerrors, in RM(m, m− 2r − 2), any pattern which is correctablefrom erasures in RM(m, m− r − 1).
(+ extensions to larger alphabets and other codes)
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 62: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/62.jpg)
Decoding Erasures to Decoding ErrorsHow is this relevant for error correction?Theorem: (ASW) Any erasure pattern correctable from erasuresin RM(m, m− r − 1) is correctable from errors inRM(m, m− 2r − 2).Thistalk: There is an efficient algorithm that corrects fromerrors, in RM(m, m− 2r − 2), any pattern which is correctablefrom erasures in RM(m, m− r − 1).(+ extensions to larger alphabets and other codes)
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 63: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/63.jpg)
Decoding Erasures to Decoding ErrorsHow is this relevant for error correction?Theorem: (ASW) Any erasure pattern correctable from erasuresin RM(m, m− r − 1) is correctable from errors inRM(m, m− 2r − 2).Thistalk: There is an efficient algorithm that corrects fromerrors, in RM(m, m− 2r − 2), any pattern which is correctablefrom erasures in RM(m, m− r − 1).(+ extensions to larger alphabets and other codes)
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 64: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/64.jpg)
Decoding Errors in RM CodesCorollary#1: (low-rate) efficient decoding algo for (1
2 − o(1))nrandom errors in RM(m, o(
pm)) (min distance is 2m−pm) .
Corollary#2: (high-rate) efficient decoding algo for(1− o(1))� m≤r
�random errors in RM(m, m− r) if r = o(
pm/ log m)
(min distance is 2r ) .Corollary#3: If RM(m,
�1+ρ
2
�m) achieves capacity, efficient
decoding algo for 2h�
1−ρ2
�m random errors in RM(m,ρm).
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ρ
# of
erro
rs c
orre
cted
1−ρh� 1−ρ
2
�
![Page 65: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/65.jpg)
Decoding Errors in RM CodesCorollary#1: (low-rate) efficient decoding algo for (1
2 − o(1))nrandom errors in RM(m, o(
pm)) (min distance is 2m−pm) .
Corollary#2: (high-rate) efficient decoding algo for(1− o(1))� m≤r
�random errors in RM(m, m− r) if r = o(
pm/ log m)
(min distance is 2r ) .
Corollary#3: If RM(m,�
1+ρ2
�m) achieves capacity, efficient
decoding algo for 2h�
1−ρ2
�m random errors in RM(m,ρm).
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ρ
# of
erro
rs c
orre
cted
1−ρh� 1−ρ
2
�
![Page 66: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/66.jpg)
Decoding Errors in RM CodesCorollary#1: (low-rate) efficient decoding algo for (1
2 − o(1))nrandom errors in RM(m, o(
pm)) (min distance is 2m−pm) .
Corollary#2: (high-rate) efficient decoding algo for(1− o(1))� m≤r
�random errors in RM(m, m− r) if r = o(
pm/ log m)
(min distance is 2r ) .Corollary#3: If RM(m,
�1+ρ
2
�m) achieves capacity, efficient
decoding algo for 2h�
1−ρ2
�m random errors in RM(m,ρm).
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ρ
# of
erro
rs c
orre
cted
1−ρh� 1−ρ
2
�
![Page 67: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/67.jpg)
Decoding Errors in RM CodesCorollary#1: (low-rate) efficient decoding algo for (1
2 − o(1))nrandom errors in RM(m, o(
pm)) (min distance is 2m−pm) .
Corollary#2: (high-rate) efficient decoding algo for(1− o(1))� m≤r
�random errors in RM(m, m− r) if r = o(
pm/ log m)
(min distance is 2r ) .Corollary#3: If RM(m,
�1+ρ
2
�m) achieves capacity, efficient
decoding algo for 2h�
1−ρ2
�m random errors in RM(m,ρm).
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ρ
# of
erro
rs c
orre
cted
1−ρh� 1−ρ
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![Page 68: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/68.jpg)
Proof IdeaGoal: decode in RM(m, m− 2r − 2) every pattern which iscorrectable from erasures in RM(m, m− r − 1).
0 0 1 1 01 1 0
recall: {u1, . . . ,ut} correctable from erasures iff {ur1, . . . ,ur
t } arelinearly independent.
E(m, r)
![Page 69: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/69.jpg)
Proof IdeaGoal: decode in RM(m, m− 2r − 2) every pattern which iscorrectable from erasures in RM(m, m− r − 1).
0 0 1 1 01 1 0
recall: {u1, . . . ,ut} correctable from erasures iff {ur1, . . . ,ur
t } arelinearly independent.
E(m, r)
![Page 70: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/70.jpg)
Dual PolynomialsFact: If {ur
1, . . . ,urt } lin. indep., ∃ polys { f1, . . . , ft} of deg≤ r
such that
fi(u j) =
¨1 i = j
0 i ̸= j.
Proof: —— ur1 ——...
—— urt ——
· |fi|
= ei
Solve this system for fi.
Our approach would be to find those polynomials.
![Page 71: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/71.jpg)
Dual PolynomialsFact: If {ur
1, . . . ,urt } lin. indep., ∃ polys { f1, . . . , ft} of deg≤ r
such that
fi(u j) =
¨1 i = j
0 i ̸= j.
Proof: —— ur1 ——...
—— urt ——
· |fi|
= ei
Solve this system for fi.
Our approach would be to find those polynomials.
![Page 72: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/72.jpg)
Dual PolynomialsFact: If {ur
1, . . . ,urt } lin. indep., ∃ polys { f1, . . . , ft} of deg≤ r
such that
fi(u j) =
¨1 i = j
0 i ̸= j.
Proof: —— ur1 ——...
—— urt ——
· |fi|
= ei
Solve this system for fi.
Our approach would be to find those polynomials.
![Page 73: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/73.jpg)
Dual PolynomialsFact: If {ur
1, . . . ,urt } lin. indep., ∃ polys { f1, . . . , ft} of deg≤ r
such that
fi(u j) =
¨1 i = j
0 i ̸= j.
Proof: —— ur1 ——...
—— urt ——
· |fi|
= ei
Solve this system for fi.
Our approach would be to find those polynomials.
![Page 74: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/74.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:∀ monomial M , deg M ≤ r:
t∑i=1
f (ui) = f (v) = 1 ( f non-trivial)t∑
i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 75: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/75.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:
∀ monomial M , deg M ≤ r:t∑
i=1f (ui) = f (v) = 1 ( f non-trivial)
t∑i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 76: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/76.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:∀ monomial M , deg M ≤ r:
t∑i=1
f (ui) = f (v) = 1 ( f non-trivial)t∑
i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 77: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/77.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:∀ monomial M , deg M ≤ r:
t∑i=1
f (ui) = f (v) = 1 ( f non-trivial)
t∑i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 78: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/78.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:∀ monomial M , deg M ≤ r:
t∑i=1
f (ui) = f (v) = 1 ( f non-trivial)t∑
i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 79: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/79.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:∀ monomial M , deg M ≤ r:
t∑i=1
f (ui) = f (v) = 1 ( f non-trivial)t∑
i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)
If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 80: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/80.jpg)
Algorithm: Solve a system of Linear EquationsU = {u1, . . . ,ut} ∈ Fm
2 (unknown) set of error locations.
MainClaim: For every v ∈ {0,1}m, v ∈ U ⇐⇒ the followinglinear system, in unknown coeffs of degree r poly f , is solvable:∀ monomial M , deg M ≤ r:
t∑i=1
f (ui) = f (v) = 1 ( f non-trivial)t∑
i=1( f ·M)(ui) = M(v) (v spanned by U )
t∑i=1( f ·M · (xℓ + vℓ + 1))(ui) = M(v)
(v spanned by vectors in U that agree with its ℓ-th coordinate)If U lin. indep. and v= ui ∈ U , fi is a solution. Conversely, ifsolvable and U lin. indep., can show v ∈ U .
![Page 81: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/81.jpg)
Setting up system of EquationsEvery coefficient in the system is of the form
∑ti=1 g(ui) for poly g
of degree ≤ 2r + 1.
How to compute the coefficients?
Input to the algo: y= c+ e with c ∈ RM(m, m− 2r − 2), and echaracteristic vector of U = {u1, . . . ,ut}.Syndrome of y: E(m, 2r + 1) · y= E(m, 2r + 1) · e.(recall: E(m, 2r + 1) is PCM of RM(m, m− 2r − 2) )
Corollary: The syndrome of y is a� m≤2r+1
�long vector α, where
αM =∑t
i=1 M(ui), for every monom M , deg M ≤ 2r + 1.
![Page 82: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/82.jpg)
Setting up system of EquationsEvery coefficient in the system is of the form
∑ti=1 g(ui) for poly g
of degree ≤ 2r + 1.How to compute the coefficients?
Input to the algo: y= c+ e with c ∈ RM(m, m− 2r − 2), and echaracteristic vector of U = {u1, . . . ,ut}.Syndrome of y: E(m, 2r + 1) · y= E(m, 2r + 1) · e.(recall: E(m, 2r + 1) is PCM of RM(m, m− 2r − 2) )
Corollary: The syndrome of y is a� m≤2r+1
�long vector α, where
αM =∑t
i=1 M(ui), for every monom M , deg M ≤ 2r + 1.
![Page 83: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/83.jpg)
Setting up system of EquationsEvery coefficient in the system is of the form
∑ti=1 g(ui) for poly g
of degree ≤ 2r + 1.How to compute the coefficients?
Input to the algo: y= c+ e with c ∈ RM(m, m− 2r − 2), and echaracteristic vector of U = {u1, . . . ,ut}.
Syndrome of y: E(m, 2r + 1) · y= E(m, 2r + 1) · e.(recall: E(m, 2r + 1) is PCM of RM(m, m− 2r − 2) )
Corollary: The syndrome of y is a� m≤2r+1
�long vector α, where
αM =∑t
i=1 M(ui), for every monom M , deg M ≤ 2r + 1.
![Page 84: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/84.jpg)
Setting up system of EquationsEvery coefficient in the system is of the form
∑ti=1 g(ui) for poly g
of degree ≤ 2r + 1.How to compute the coefficients?
Input to the algo: y= c+ e with c ∈ RM(m, m− 2r − 2), and echaracteristic vector of U = {u1, . . . ,ut}.Syndrome of y: E(m, 2r + 1) · y= E(m, 2r + 1) · e.(recall: E(m, 2r + 1) is PCM of RM(m, m− 2r − 2) )
Corollary: The syndrome of y is a� m≤2r+1
�long vector α, where
αM =∑t
i=1 M(ui), for every monom M , deg M ≤ 2r + 1.
![Page 85: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/85.jpg)
Setting up system of EquationsEvery coefficient in the system is of the form
∑ti=1 g(ui) for poly g
of degree ≤ 2r + 1.How to compute the coefficients?
Input to the algo: y= c+ e with c ∈ RM(m, m− 2r − 2), and echaracteristic vector of U = {u1, . . . ,ut}.Syndrome of y: E(m, 2r + 1) · y= E(m, 2r + 1) · e.(recall: E(m, 2r + 1) is PCM of RM(m, m− 2r − 2) )
Corollary: The syndrome of y is a� m≤2r+1
�long vector α, where
αM =∑t
i=1 M(ui), for every monom M , deg M ≤ 2r + 1.
![Page 86: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/86.jpg)
Back to BECThe algorithm works whenever ur
1, . . . ,urt lin. indep., which
happens (w.h.p.) whenever RM(m, m− r − 1) achieves capacity.
Restating the main question: for which values of r, ur1, . . . ,ur
t arelinearly independent with high probability for t = (1− o(1))
� m≤r
�?
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 87: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/87.jpg)
Back to BECThe algorithm works whenever ur
1, . . . ,urt lin. indep., which
happens (w.h.p.) whenever RM(m, m− r − 1) achieves capacity.
Restating the main question: for which values of r, ur1, . . . ,ur
t arelinearly independent with high probability for t = (1− o(1))
� m≤r
�?
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 88: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/88.jpg)
Back to BECThe algorithm works whenever ur
1, . . . ,urt lin. indep., which
happens (w.h.p.) whenever RM(m, m− r − 1) achieves capacity.
Restating the main question: for which values of r, ur1, . . . ,ur
t arelinearly independent with high probability for t = (1− o(1))
� m≤r
�?
m/20 m
o(m) o(p
m/ log m)O(p
m)
![Page 89: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/89.jpg)
SummaryDecoding algo for RM far beyond the minimal distance, both forhigh-rate and low-rate regimes.
OpenProblems:• Prove RM achieves capacity for BEC for all degrees
• RM for BSC: much less is known! (ASW proved it achievescapacity for small rates)
m/20 m
![Page 90: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/90.jpg)
SummaryDecoding algo for RM far beyond the minimal distance, both forhigh-rate and low-rate regimes.
OpenProblems:• Prove RM achieves capacity for BEC for all degrees
• RM for BSC: much less is known! (ASW proved it achievescapacity for small rates)
m/20 m
![Page 91: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/91.jpg)
SummaryDecoding algo for RM far beyond the minimal distance, both forhigh-rate and low-rate regimes.
OpenProblems:• Prove RM achieves capacity for BEC for all degrees• RM for BSC: much less is known! (ASW proved it achieves
capacity for small rates)
m/20 m
![Page 92: EfficientlyDecodingReedMullerCodes FromRandomErrorsbenleevolk/pdf/rm_slides.pdf · Errors: Impossible. Random Errors? (each coordinate flipped independently with probability1=2 o(1))](https://reader034.vdocuments.net/reader034/viewer/2022052518/5f0b42687e708231d42fa1d4/html5/thumbnails/92.jpg)
SummaryDecoding algo for RM far beyond the minimal distance, both forhigh-rate and low-rate regimes.
OpenProblems:• Prove RM achieves capacity for BEC for all degrees• RM for BSC: much less is known! (ASW proved it achieves
capacity for small rates)
m/20 m
Thank You