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Introduction and Motivation List decoding

Uncertainty of Reconstructing Multiple Messages from Uniform-Tandem-Duplication Noise

Yonatan Yehezkeally Moshe Schwartz

Ben-Gurion University of the Negev

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Uniform tandem-duplication noise

Tandem-duplication A substring (template) is duplicated, copy inserted next to template. E.g., x = 1012121 → x′ = 1012012121,

• Dfn. The noise is uniform if the length of duplication window is fixed.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Uniform tandem-duplication noise

Tandem-duplication A substring (template) is duplicated, copy inserted next to template. E.g., x = 1012121 → x′ = 1012012121,

• Dfn. The noise is uniform if the length of duplication window is fixed.

Applications In-vivo DNA storage: Around 3% of the human genome consists of tandem repeats

Mundy, Helbig, Journal of Molecular Evolution, 2004.

Synchronization noise in magnetic media (sticky-insertions)

Mahdavifar and Vardy, ISIT’17, 2017.

In these cases, uniform noise is easier to analyze.

Closely related to the permutation / multiset channel (applications to packet networks and in-vitro DNA storage).

Kovaˇ cevi´ c and Tan, T-IT, 2018. Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Coding redundancy (state-of-the-art)

Unlimited number of errors t = ∞ Rate loss, equivalent to that of an appropriate RLL system.

(0, k − 1)q-RLL, for alphabet size q and duplication window length k.

Jain et.al., T-IT, 2017. Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Coding redundancy (state-of-the-art)

Unlimited number of errors t = ∞ Rate loss, equivalent to that of an appropriate RLL system.

(0, k − 1)q-RLL, for alphabet size q and duplication window length k.

Jain et.al., T-IT, 2017.

Finite number of errors t < ∞ ECC optimal redundancy (lower and upper bounds): t logq(n) + O(1).

Lenz et.al., arXiv, 2018. Kovaˇ cevi´ c and Tan, IEEE Comm. Letters, 2018.

Efficient en/decoding: t logq(n) + o(log(n)) (asymptotically optimal)

Mahdavifar and Vardy, ISIT’17, 2017.

Multiple distinct reads of noisy data: (t − 1) logq(n) + O(1).

(Reconstruction with sublinear uncertainty.)

Yehezkeally and Schwartz, T-IT, 2020. Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Coding redundancy (state-of-the-art)

Unlimited number of errors t = ∞ Rate loss, equivalent to that of an appropriate RLL system.

(0, k − 1)q-RLL, for alphabet size q and duplication window length k.

Jain et.al., T-IT, 2017.

Finite number of errors t < ∞ ECC optimal redundancy (lower and upper bounds): t logq(n) + O(1).

Lenz et.al., arXiv, 2018. Kovaˇ cevi´ c and Tan, IEEE Comm. Letters, 2018.

Efficient en/decoding: t logq(n) + o(log(n)) (asymptotically optimal)

Mahdavifar and Vardy, ISIT’17, 2017.

Multiple distinct reads of noisy data: (t − 1) logq(n) + O(1).

(Reconstruction with sublinear uncertainty.)

Yehezkeally and Schwartz, T-IT, 2020.

Question: At what cost may redundancy be further reduced?

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Setting

Setting: Space of finite strings Σ∗ over an alphabet Σ of size q > 1. Noise: Strings affected by uniform tandem duplication noise (duplication window length k) E.g., for x ∈ Σn: x = ⇒ y ∈ Σn+k Error spheres: Dt(x), with error cones D∗(x) ∞

t=0 Dt(x)

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Setting

Setting: Space of finite strings Σ∗ over an alphabet Σ of size q > 1. Noise: Strings affected by uniform tandem duplication noise (duplication window length k) E.g., for x ∈ Σn: x = ⇒ y ∈ Σn+k Error spheres: Dt(x), with error cones D∗(x) ∞

t=0 Dt(x)

Thm.: Error cones (for y, z ∈ Σ∗)

Jain et.al., T-IT, 2017.

D∗(y) ∩ D∗(z) = ∅ ⇐ ⇒ ∃x ∈ Σ∗ : y, z ∈ D∗(x) Thus, space is partitioned into disjoint descendant cones of irreducible strings.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Setting

Setting: Space of finite strings Σ∗ over an alphabet Σ of size q > 1. Noise: Strings affected by uniform tandem duplication noise (duplication window length k) E.g., for x ∈ Σn: x = ⇒ y ∈ Σn+k Error spheres: Dt(x), with error cones D∗(x) ∞

t=0 Dt(x)

Thm.: Error cones (for y, z ∈ Σ∗)

Jain et.al., T-IT, 2017.

D∗(y) ∩ D∗(z) = ∅ ⇐ ⇒ ∃x ∈ Σ∗ : y, z ∈ D∗(x) Thus, space is partitioned into disjoint descendant cones of irreducible strings. Metric: Dfn.: For x ∈ Σ∗, y, z ∈ Dr(x) d(y, z) min

• t ∈ N : Dt(y) ∩ Dt(z) = ∅
• Yehezkeally and Schwartz, ISIT’2020

Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Associative memory Principles

Item retreived by association with a set of other items Dfn.: Uncertainty N(m) is the cardinality of largest set whose members are associated with an

m-subset of the memory code-book.

Yaakobi and Bruck, T-IT, 2019. Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Associative memory Principles

Item retreived by association with a set of other items Dfn.: Uncertainty N(m) is the cardinality of largest set whose members are associated with an

m-subset of the memory code-book.

Yaakobi and Bruck, T-IT, 2019.

Generalizes the reconstruction schema: multiple distinct noisy version of data are available to decoder (e.g., cell replication in in-vivo DNA cannel) ≤ t errors occur in transmission N largest intrsection of two t-balls

• =

⇒ N + 1 noisy outputs suffice to

decode transmitted code-word.

Levenshtein, T-IT, 2001. Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Associative memory Principles

Item retreived by association with a set of other items Dfn.: Uncertainty N(m) is the cardinality of largest set whose members are associated with an

m-subset of the memory code-book.

Yaakobi and Bruck, T-IT, 2019.

Generalizes the reconstruction schema: multiple distinct noisy version of data are available to decoder (e.g., cell replication in in-vivo DNA cannel) ≤ t errors occur in transmission N largest intrsection of two t-balls

• =

⇒ N + 1 noisy outputs suffice to

decode transmitted code-word.

Levenshtein, T-IT, 2001.

Reduction to m = 2, enables reconstruction of unique (m − 1 = 1) input. m > 2 = ⇒ N(m) + 1 outputs yield set of l < m code-words, i.e., enables list decoding

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Results

Aim Find the trade-off, as the message length n grows, between N Uncertainty (or required number of reads–minus one) t Number of uniform tandem duplication errors m Maximal list size (plus one) d Designed minimum distance ((d − 1) logq(n) + O(1) redundancy)

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Tandem-duplication noise Setting and results

Results

Aim Find the trade-off, as the message length n grows, between N Uncertainty (or required number of reads–minus one) t Number of uniform tandem duplication errors m Maximal list size (plus one) d Designed minimum distance ((d − 1) logq(n) + O(1) redundancy) Results(1)(2) logn N + ⌈logn m⌉ + d = t + ǫ + o(1)

(1) Coding is done in a typical subspace of Σn, asymptotically achieving full space size (2) ǫ ∈ {0, 1} is (generally) an implicit non-increasing function of m Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Associative memory Definitions

Dfn.: Given m, n, t ∈ N and x1, . . . , xm ∈ Σn: St(x1, . . . , xm)

m

• i=1

Dt(xi)

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Associative memory Definitions

Dfn.: Given m, n, t ∈ N and x1, . . . , xm ∈ Σn: St(x1, . . . , xm)

m

• i=1

Dt(xi) The Uncertainty of C ⊆ Σn Nt(m, C) max

x1,...,xm∈C xi=xj

|St(x1, . . . , xm)|

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Associative memory Definitions

Dfn.: Given m, n, t ∈ N and x1, . . . , xm ∈ Σn: St(x1, . . . , xm)

m

• i=1

Dt(xi) The Uncertainty of C ⊆ Σn Nt(m, C) max

x1,...,xm∈C xi=xj

|St(x1, . . . , xm)|

Uncertainty assocd.(1) with minimum distance d

NTyp

t

(m, n, d) max

x1,...,xm∈Typn d(xi,xj)≥d

|St(x1, . . . , xm)|

(1) Typn ⊆ Σn, |Typn|/|Σn| → 1 Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Embedding in integer orthants (x ∈ Σ∗ is irreducible)

Nw+1 (w > 0)

Endow with the product order ≤ Define the 1-norm u1 ∑w+1

i=1 u(i)

Define the metric d1(u, v) 1

2 u − v1

∃w = w(x) s.t.:

Jain et.al., T-IT, 2017. Yehezkeally and Schwartz, T-IT, 2020.

D∗(x) isomorphic to Nw+1 as a poset (ordered by descendancy) Dr(x) isometric to its image ∆w

r

• u ∈ Nw+1 : u1 = r
• Yehezkeally and Schwartz, ISIT’2020

Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Embedding in integer orthants (x ∈ Σ∗ is irreducible)

Nw+1 (w > 0)

Endow with the product order ≤ Define the 1-norm u1 ∑w+1

i=1 u(i)

Define the metric d1(u, v) 1

2 u − v1

∃w = w(x) s.t.:

Jain et.al., T-IT, 2017. Yehezkeally and Schwartz, T-IT, 2020.

D∗(x) isomorphic to Nw+1 as a poset (ordered by descendancy) Dr(x) isometric to its image ∆w

r

• u ∈ Nw+1 : u1 = r
• Dfn.: For y ∈ Dr(x)

r(y) r w(y) w(x)

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Typical set

Space Σn contain some pathological extreme cases. Analyze instead: Typn

• x ∈ Σn :
• w(x)− q−1

q (n−k)

• <n3/4,
• r(x)−

q−1 q(qk −1) (n−k)

• <2n3/4

Thm.: |Typn|/|Σn| ≥ 1 − 4e−√n/2

• Prf. (sketch): If wtH(x − x′) = 1, then |w(x) − w(x′)|, |r(x) − r(x′)| ≤ 2.

McDiarmid’s inequality justifies the required concentration result around E(w(x)),E(r(x)).

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Uncertainty of typical codes

Define the orthant-analogues: For m, w, r ∈ N and u1, . . . , um ∈ ∆w

r

¯ St(u1, . . . , um)

m

• i=1
• v ∈ Nw+1 : v ≥ ui, v − ui1 = t
• For m, w, r, d ∈ N

¯ Nt(m, w, r, d) max

u1,...,um∈∆w

r

d1(ui,uj)≥d

• ¯

St(u1, . . . , um)

• Yehezkeally and Schwartz, ISIT’2020

Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Uncertainty of typical codes

Define the orthant-analogues: For m, w, r ∈ N and u1, . . . , um ∈ ∆w

r

¯ St(u1, . . . , um)

m

• i=1
• v ∈ Nw+1 : v ≥ ui, v − ui1 = t
• For m, w, r, d ∈ N

¯ Nt(m, w, r, d) max

u1,...,um∈∆w

r

d1(ui,uj)≥d

• ¯

St(u1, . . . , um)

• Lem.: For sufficiently large n:

NTyp

t

(m, n, d) = max

• ¯

Nt(m, w, r, d) :

• w− q−1

q (n−k)

• <n3/4,
• r−

q−1 q(qk −1) (n−k)

• <2n3/4

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Intersection of spheres

u1, . . . , um ∈ ∆w

r , u m i=1 ui

Lem.:

• ¯

St(u1, . . . , um)

• =
• u1 > r + t,

(w+r+t−u1

w

)

• therwise.
• Prf. (sketch): follows from the representation ¯

St(u1, . . . , um) =

• v ∈ Nw+1 : v ≥ u, v − u11 = t
• Yehezkeally and Schwartz, ISIT’2020

Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Intersection of spheres

u1, . . . , um ∈ ∆w

r , u m i=1 ui

Lem.:

• ¯

St(u1, . . . , um)

• =
• u1 > r + t,

(w+r+t−u1

w

)

• therwise.
• Prf. (sketch): follows from the representation ¯

St(u1, . . . , um) =

• v ∈ Nw+1 : v ≥ u, v − u11 = t
• Conc.: ¯

Nt(m, w, r, d) = (w+t−σ

w

), where σ = σ(m, w, r, d) min u1,...,um∈∆w

r

d1(ui,uj)≥d

• m

i=1ui

• 1 − r

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Sets of lower bounds

How does one find σ(m, w, r, d)?

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Sets of lower bounds

How does one find σ(m, w, r, d)? Directly analogous to the following question: Given s ≤ t, what u ∈ ∆w

r+s has a maximum-size d1-error-correcting code,

with minimum distance d, in the set of lower bounds: {v ∈ ∆w

r : v ≤ u}?

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Sets of lower bounds

How does one find σ(m, w, r, d)? Directly analogous to the following question: Given s ≤ t, what u ∈ ∆w

r+s has a maximum-size d1-error-correcting code,

with minimum distance d, in the set of lower bounds: {v ∈ ∆w

r : v ≤ u}?

Fix w, r, t s.t. r + t ≤ w + 1; For all s ≤ t: Lem.: Such u ∈ ∆w

r+s satisfies u ≤ (1, 1, . . . , 1), and the maximum-size of such

code is A(r + s, 2d, s),

where A(ν, 2δ, ω) is the size of the largest length ν binary code with minimum Hamming distance 2δ and constant Hamming weight ω.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Lower bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≥ ⌈logn m⌉ + d − 1

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Lower bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≥ ⌈logn m⌉ + d − 1 Applying the first Johnson bound:

Johnson, (IRE) T-IT, 1962.

A(r + s − 1, 2d, s − 1) ≤ r + s − 1 s − d s − 1 s − d

• < (d − 1)!

(s − 1)! (r + s − 1)s−d,

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Lower bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≥ ⌈logn m⌉ + d − 1 Applying the first Johnson bound:

Johnson, (IRE) T-IT, 1962.

A(r + s − 1, 2d, s − 1) ≤ r + s − 1 s − d s − 1 s − d

• < (d − 1)!

(s − 1)! (r + s − 1)s−d, hence for r satisfying

• r −

q−1 q(qk−1)(n − k)

• < 2n3/4 and sufficiently large n:

logn A(r + s − 1, 2d, s − 1) < s − d,

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Lower bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≥ ⌈logn m⌉ + d − 1 Applying the first Johnson bound:

Johnson, (IRE) T-IT, 1962.

A(r + s − 1, 2d, s − 1) ≤ r + s − 1 s − d s − 1 s − d

• < (d − 1)!

(s − 1)! (r + s − 1)s−d, hence for r satisfying

• r −

q−1 q(qk−1)(n − k)

• < 2n3/4 and sufficiently large n:

logn A(r + s − 1, 2d, s − 1) < s − d, implying that A(r + s − 1, 2d, s − 1) < m, and therefore σ(m, w, r, d)

(∗)

≥ s = ⌈logn m⌉ + d − 1.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Upper bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≤ ⌈logn m⌉ + d

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Upper bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≤ ⌈logn m⌉ + d For any prime power p, p > r + s: A(r + s + 1, 2d, s + 1) ≥ 1 pd−1 r + s + 1 s + 1

• Graham and Sloane, T-IT, 1980.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Upper bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≤ ⌈logn m⌉ + d For any prime power p, p > r + s: A(r + s + 1, 2d, s + 1) ≥ 1 pd−1 r + s + 1 s + 1

• Graham and Sloane, T-IT, 1980.

For sufficiently large n and r satisfying

• r −

q−1 q(qk −1)(n − k)

• < 2n3/4, find a prime number

r + s < p ≤ n (Bertrand’s postulate), implying A(r + s + 1, 2d, s + 1) ≥ 1 nd−1 r + s + 1 s + 1

• >

rs+1 nd−1(s + 1)!.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Upper bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≤ ⌈logn m⌉ + d For any prime power p, p > r + s: A(r + s + 1, 2d, s + 1) ≥ 1 pd−1 r + s + 1 s + 1

• Graham and Sloane, T-IT, 1980.

For sufficiently large n and r satisfying

• r −

q−1 q(qk −1)(n − k)

• < 2n3/4, find a prime number

r + s < p ≤ n (Bertrand’s postulate), implying A(r + s + 1, 2d, s + 1) ≥ 1 nd−1 r + s + 1 s + 1

• >

rs+1 nd−1(s + 1)!. hence logn A(r + s + 1, 2d, s + 1) > s − d + 2 + o(1),

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Upper bound (denote s ⌈logn m⌉ + d − 1)

σ(m, w, r, d) ≤ ⌈logn m⌉ + d For any prime power p, p > r + s: A(r + s + 1, 2d, s + 1) ≥ 1 pd−1 r + s + 1 s + 1

• Graham and Sloane, T-IT, 1980.

For sufficiently large n and r satisfying

• r −

q−1 q(qk −1)(n − k)

• < 2n3/4, find a prime number

r + s < p ≤ n (Bertrand’s postulate), implying A(r + s + 1, 2d, s + 1) ≥ 1 nd−1 r + s + 1 s + 1

• >

rs+1 nd−1(s + 1)!. hence logn A(r + s + 1, 2d, s + 1) > s − d + 2 + o(1), implying that m < A(r + s + 1, 2d, s + 1), and therefore σ(m, w, r, d)

(∗)

≤ s + 1 = ⌈logn m⌉ + d.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Results revisited

Conc.: For sufficiently large n and

• w− q−1

q (n−k)

• <n3/4,
• r−

q−1 q(qk −1) (n−k)

• <2n3/4 :

¯ Nt(m, w, r, d) =

1+o(1) (t−⌈logn m⌉−d+ǫ)!

• q−1

q n

t−⌈logn m⌉−d+ǫ , where ǫ =

• 1

m ≤ A(r + ⌈logn m⌉ + d − 1, 2d, ⌈logn m⌉ + d − 1),

• therwise.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Efficient decoding

Fix n, m and d ≤ t. Take C ⊆ Typn with minimum distance d, and assume a decoding scheme D : Σn+k(d−1) → C is provided for recovering up to d − 1 tandem-duplication errors. Denote N NTyp

t

(n, m, d) and assume as input distinct y1, . . . , yN+1 ∈ Dt(x), for some (unknown) x ∈ C.

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Efficient decoding

Fix n, m and d ≤ t. Take C ⊆ Typn with minimum distance d, and assume a decoding scheme D : Σn+k(d−1) → C is provided for recovering up to d − 1 tandem-duplication errors. Denote N NTyp

t

(n, m, d) and assume as input distinct y1, . . . , yN+1 ∈ Dt(x), for some (unknown) x ∈ C.

1

Map them to v1, . . . , vN+1 ∈ ∆w

r+t (appropriate w, r)

2

Calculate v N+1

i=1 vi ∈ ∆w r ′

3

Find (e.g., using combination generators) the set

• u ∈ ∆w

r+d−1 : u ≤ v

• 4

Map these back to z1, . . . , zl ∈ Dd−1(x) (x still unknown)

5

For each zi, decode xi D(zi) ∈ C

6

If zi ∈ Dd−1(xi), return xi in the list; otherwise, discard it

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Efficient decoding

Fix n, m and d ≤ t. Take C ⊆ Typn with minimum distance d, and assume a decoding scheme D : Σn+k(d−1) → C is provided for recovering up to d − 1 tandem-duplication errors. Denote N NTyp

t

(n, m, d) and assume as input distinct y1, . . . , yN+1 ∈ Dt(x), for some (unknown) x ∈ C.

1

Map them to v1, . . . , vN+1 ∈ ∆w

r+t (appropriate w, r)

2

Calculate v N+1

i=1 vi ∈ ∆w r ′

3

Find (e.g., using combination generators) the set

• u ∈ ∆w

r+d−1 : u ≤ v

• 4

Map these back to z1, . . . , zl ∈ Dd−1(x) (x still unknown)

5

For each zi, decode xi D(zi) ∈ C

6

If zi ∈ Dd−1(xi), return xi in the list; otherwise, discard it Thm.: The algorithm above returns correct output, and operates in O(nt + nt−d+1C) steps (C the run-time complexity of D).

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages

Introduction and Motivation List decoding Typical set Finding uncertainty / Efficient decoding

Summary

Suggested a list decoding approach to further reduce coding redundancy under uniform tandem-duplication noise. Found the asymptotic trade-off between uncertainty, number of errors, list size and coding redundancy: logn N + ⌈logn m⌉ + d = t + ǫ + o(1). Developed explicit and efficient list decoding algorithm (given decoding schemes for ECCs).

Thanks for your attention!

Yehezkeally and Schwartz, ISIT’2020 Reconstructing Multiple Messages