Linear-Time Erasure List-Decoding of Expander Codes Noga Ron-Zewi - - PowerPoint PPT Presentation

Linear-Time Erasure List-Decoding of Expander Codes

Noga Ron-Zewi (University of Haifa) Mary Wootters (Stanford) Gilles Zémor (Université de Bordeaux)

ISIT 2020

Linear-Time Erasure List-Decoding of Expander Codes

• 1. Erasure List-Decoding
• Decoding ECCs from so many erasures that there’s not a unique solution.
• 2. Expander Codes
• Codes based on expander graphs, known for linear-time unique decoding

algorithms.

• 3. Linear-Time
• Main contribution: a linear-time algorithm for erasure list-decoding expander codes.
• One motivation: linear-time list-decoding algorithms from errors.

1 2 3

Erasure List-Decoding

1

Uniquely decoding linear codes from erasures

Alice Bob Message 𝑦 ∈ 𝐺! Generator Matrix 𝐻 ∈ 𝐺"×! Codeword 𝑑 ∈ 𝐷

• A linear code 𝐷 ⊆ 𝐺! is a linear subspace.
• The (relative) distance 𝜀 of 𝐷 is the minimum

relative Hamming distance between 𝑑" ≠ 𝑑# ∈ 𝐷.

= Adversarial Erasure Channel ? ? ? = ? ? ?

• Solve for 𝑦!

Fewer than 𝜀𝑂 erasures ⇒ There is a unique x that could lead to this corrupted codeword.

List decoding linear codes from erasures

Alice Bob Message 𝑦 ∈ 𝐺! Generator Matrix 𝐻 ∈ 𝐺$×! Codeword 𝑑 ∈ 𝐷

• The (relative) generalized r’th distance of 𝐷 is

𝜀$ = 1 𝑂 ⋅ min 𝑗 ∈ [𝑂]: ∃𝑤 ∈ 𝑊 ∶ 𝑤% ≠ 0

= Adversarial Erasure Channel ? ? ? = ? ? ?

• Solve for an r-dimensional

subspace of solutions. ? ?

• Fewer than 𝜀%𝑂 erasures

⇒ There is a r-dimensional subspace of x’s that could lead to this corrupted codeword.

[Guruswami 2003] r-dimensional subspaces 𝑊

Goals in Erasure-List Decoding

• List size isn’t too big (equivalently, 𝜀$ is small).
• Bob can find the list quickly (ideally in time 𝑃(𝑂))
• Before this work, no known algorithms to do this for any code.

Expander Codes

2

Expander Codes: definition-by-picture

d-regular bipartite expander graph 𝐻

d n n

+ linear code 𝐷& ⊆ 𝐺' ⇒ expander code 𝐷 = 𝐷(𝐻, 𝐷&) ⊆ 𝐺('

• Codewords in 𝐷 are given by labelings of the

edges of 𝐻. (So the length is 𝑂 = 𝑜𝑒).

• Constraints are given by the vertices: adjacent

edges must form a codeword of 𝐷*

1 1 1 1 The labels on these edges form a codeword in 𝐷& Also these

Expander Codes

• Introduced by [Sipser, Spielman 1996].
• See also [Zémor 2001] for the formulation we use here.
• If 𝐷& has dimension 𝑙 > 𝑒/2, then 𝐷 has dimension 𝑜 2𝑙 − 𝑒 .
• If 𝐷& has distance 𝜀, then 𝐷 has distance at least 𝜀 𝜀 −

) ' .

• 𝐷 can be uniquely decoded in linear time.
• More precisely, 𝐷 can be decoded from up to a 1 − 𝜗 𝜀 𝜀 − +

, fraction

• f erasures in time 𝑃 𝑂 ⋅ ,!
• for some constant c.

𝜇 is the expansion of 𝐻. Think 𝜇 ≈ 𝑒.

A linear-time algorithm

for list-decoding expander codes from erasures

3

Theorem

• Fix parameters 𝑠, 𝜗 > 0.
• Let 𝐷& have distance 𝜀, r’th generalized distance 𝜀$.
• Let 𝐻 be a 𝑒-regular bipartite expander with expansion 𝜇.
• Suppose that

) ' ≤ *!+! #"#$.

• Then the expander code 𝐷 = 𝐷 𝐷&, 𝐻 can be list-decoded

from up to 1 − 𝜗 𝜀𝜀$𝑂 erasures in time O 𝑂 ⋅

#"' *+ ,

and returns a list of dimension at most

#!"#% *$+$ .

Theorem

• Fix parameters 𝑠, 𝜗 > 0.
• Let 𝐷& have distance 𝜀, r’th generalized distance 𝜀$.
• Let 𝐻 be a 𝑒-regular bipartite expander with expansion 𝜇.
• Suppose that

) ' ≤ *!+! #"#$.

• Then the expander code 𝐷 = 𝐷 𝐷&, 𝐻 can be list-decoded

from up to 1 − 𝜗 𝜀𝜀$𝑂 erasures in time O 𝑂 ⋅

#"' *+ ,

and returns a list of dimension at most

#!"#% *$+$ . Ideally this would be 𝜀! 𝐷 𝑂. For 𝑠 = 2, 𝜀" 𝐷 ≈ 𝜀𝜀" under appropriate conditions. For larger 𝑠, getting a handle on 𝜀!(𝐷) seems tricky. aka, linear in 𝑂 Ideally, this would be r. If 𝜀! 𝐷 = 𝜀𝜀!, then our algorithm returns a list of dimension r.

Algorithm idea (inspired by [Hemenway-W. 2015])

1 1 ? ? ? ?

• 1. List-decode 𝐷* at every vertex where

there are fewer than 𝜀2𝑒 erasures.

r-dimensional subspace: 𝐵 𝑐 + 𝑒 𝑠 ? ?

Observation

r-dimensional subspace: 𝐵 𝑐 + 𝑒 𝑠

Suppose these two rows are the same. Then a symbol on one of these edges determines a symbol on the other.

Algorithm idea (inspired by [Hemenway-W. 2015])

• 1. List-decode 𝐷* at every vertex where there are

fewer than 𝜀2𝑒 erasures.

• 2. Use the observation to define equivalence classes
• n the edges.
• Determining the symbol on any edge in the equivalence

class determines them all.

• Because of expander-ness, a few large equivalence

classes cover most of the graph.

• 3. Slow thing to do: exhaust over all assignments for

all classes.

• This is technically linear-time in 𝑂, but it runs in time

doubly-exponential in 𝑠. L

Algorithm idea (inspired by [Hemenway-W. 2015])

• 1. List-decode 𝐷* at every vertex where there are

fewer than 𝜀2𝑒 erasures.

• 2. Use the observation to define equivalence classes
• n the edges.
• Determining the symbol on any edge in the equivalence

class determines them all.

• Because of expander-ness, a few large equivalence

classes cover most of the graph.

• 3. Set up and solve a linear system to find labels for

each equivalence class that satisfy all the constraints.

• This is fast because (a) there are not too many

equivalence classes and (b) the constraints are sparse.

• 4. Unique decode to fix the few remaining erasures.

Theorem

• Fix parameters 𝑠, 𝜗 > 0.
• Let 𝐷& have distance 𝜀, r’th generalized distance 𝜀$.
• Let 𝐻 be a 𝑒-regular bipartite expander with expansion 𝜇.
• Suppose that

) ' ≤ *!+! #"#$.

• Then the expander code 𝐷 = 𝐷 𝐷&, 𝐻 can be list-decoded

from up to 1 − 𝜗 𝜀𝜀$𝑂 erasures in time O 𝑂 ⋅

#"' *+ ,

and returns a list of dimension at most

#!"#% *$+$ . Ideally this would be 𝜀! 𝐷 𝑂. For 𝑠 = 2, 𝜀" 𝐷 ≈ 𝜀𝜀" under appropriate conditions. For larger 𝑠, getting a handle on 𝜀!(𝐷) seems tricky. aka, linear in 𝑂 Ideally, this would be r. If 𝜀! 𝐷 = 𝜀𝜀!, then our algorithm returns a list of dimension r.

Open questions

• What is 𝜀$(𝐷) for expander codes? Can we decode in linear-time up

to that?

• Linear-time list-decoding algorithms from errors?
• Please ask me at marykw@stanford.edu!

Other questions?