Parametric Linear System Solving with Error Correction. Cleveland - - PowerPoint PPT Presentation

Parametric Linear System Solving with Error Correction. Cleveland Waddell SYMBOLIC-NUMERIC COMPUTING SEMINAR September 26, 2019

2

Overview

— Error Correction: — Parametric Linear System Solving — Relating Error Correction to Parametric Linear System Solving

2

Overview

— Error Correction:

— Reed-Solomon Codes (Gemmel/Sudan view of Welch/Berlekamp decoding).

— Parametric Linear System Solving — Relating Error Correction to Parametric Linear System Solving

2

Overview

— Error Correction:

— Reed-Solomon Codes (Gemmel/Sudan view of Welch/Berlekamp decoding).

— Parametric Linear System Solving — Relating Error Correction to Parametric Linear System Solving

2

Overview

— Error Correction:

— Reed-Solomon Codes (Gemmel/Sudan view of Welch/Berlekamp decoding).

— Parametric Linear System Solving — Relating Error Correction to Parametric Linear System Solving — Early Termination for [Boyer and Kaltofen ’14] Algorithm. — Rational Vector Recovery.

2

Overview

— Error Correction:

— Reed-Solomon Codes (Gemmel/Sudan view of Welch/Berlekamp decoding).

— Parametric Linear System Solving — Relating Error Correction to Parametric Linear System Solving — Early Termination for [Boyer and Kaltofen ’14] Algorithm. — Rational Vector Recovery.

3

Overview Continued

3

Overview Continued

— A Polynomial Vector Recovery Model for a Burst Error Correcting Code

3

Overview Continued

— A Polynomial Vector Recovery Model for a Burst Error Correcting Code

— Description of the problem. — Description of the model. — A model to test our model. — Some experimental results.

3

Overview Continued

— A Polynomial Vector Recovery Model for a Burst Error Correcting Code

— Description of the problem. — Description of the model. — A model to test our model. — Some experimental results.

— Rank Deficient Parametric Linear Systems with Errors:

3

Overview Continued

— A Polynomial Vector Recovery Model for a Burst Error Correcting Code

— Description of the problem. — Description of the model. — A model to test our model. — Some experimental results.

— Rank Deficient Parametric Linear Systems with Errors:

— Description of the problem. — Two types of errors. — The Algorithm.

3

Overview Continued

— A Polynomial Vector Recovery Model for a Burst Error Correcting Code

— Description of the problem. — Description of the model. — A model to test our model. — Some experimental results.

— Rank Deficient Parametric Linear Systems with Errors:

— Description of the problem. — Two types of errors. — The Algorithm.

Possible Future Directions:

3

Overview Continued

— A Polynomial Vector Recovery Model for a Burst Error Correcting Code

— Description of the problem. — Description of the model. — A model to test our model. — Some experimental results.

— Rank Deficient Parametric Linear Systems with Errors:

— Description of the problem. — Two types of errors. — The Algorithm.

Possible Future Directions:

— Connect our Burst Error Model to work done in Information Theory — Error Correction as an Algorithmic Paradigm

4

Background

Error Correcting Codes

Definitions:

— Error Correcting Code: Adding redundant data to a transmitted message to aid in recovery if errors occur. — Linear Codes: Any linear combination of codewords (message plus redundant or parity data) is also a codeword. — Block Codes: Data is encoded in blocks of fixed length.

4

Background

Error Correcting Codes

Definitions:

— Error Correcting Code: Adding redundant data to a transmitted message to aid in recovery if errors occur. — Linear Codes: Any linear combination of codewords (message plus redundant or parity data) is also a codeword. — Block Codes: Data is encoded in blocks of fixed length.

General Idea:

Codeword Received word

(f(ξ0), f(ξ1),..., f(ξL−1)) (β0,β1,...,βL−1)

5

Background

Reed-Solomon Codes

Encoding: Example:

[Reed and Solomon 1960]

5

Background

Reed-Solomon Codes

Encoding:

— Construct a polynomial, f(u) ∈ K[u], with the message as its coefficients. — Let E ≥ |{ξℓ | f(ξℓ) = βℓ}|. — Pick L = deg(f)+2E +1 distinct elements ξℓ ∈ K, where K is a field. — The codeword is the evaluations of f(u) at the L distinct elements.

Example:

— Message : (1,2,1) — f(u) = u2 +2u+1 — We pick L distinct points ξ ← (0,1,2,3,4,5,6) — Codeword: f(ξℓ) → (1,4,9,16,25,36,49) — Received word: β ← (1,4,9,16,25,9,49) [Reed and Solomon 1960]

6

Background

Reed-Solomon Codes

Welch/Berlekamp Decoding:

[Reed and Solomon 1960] [Welch and Berlekamp 1986] [Gemmell and Sudan ’92] [Olshevsky and Shokrollahi ’03] [Kaltofen and Pernet ’13] [Kaltofen and Yang ’14]

6

Background

Reed-Solomon Codes

Welch/Berlekamp Decoding:

— Define Λ(u) such that Λ(ξλ) = 0 whenever f(ξλ) = βλ — Equation βℓΛ(ξℓ) = f(ξℓ)Λ(ξℓ) is true for all evaluation points ξℓ — Let Φ(u) and Ψ(u) be generic polynomial where, deg(Φ) = deg( f)+E, and deg(Ψ) = E — Solve the linear system βℓΨ(ξℓ) = Φ(ξℓ) for the coefficients of Φ and Ψ Φ(u)/Ψ(u) = f(u) [Reed and Solomon 1960] [Welch and Berlekamp 1986] [Gemmell and Sudan ’92] [Olshevsky and Shokrollahi ’03] [Kaltofen and Pernet ’13] [Kaltofen and Yang ’14]

7

Background

Parametric Linear Systems

Let A(u) ∈ K[u]m×n and b(u) ∈ K[u]m

7

Background

Parametric Linear Systems

Let A(u) ∈ K[u]m×n and b(u) ∈ K[u]m Assume rank(A(u)) = n and m ≥ n and that the system A(u)x = b(u) is consistent

7

Background

Parametric Linear Systems

Let A(u) ∈ K[u]m×n and b(u) ∈ K[u]m Assume rank(A(u)) = n and m ≥ n and that the system A(u)x = b(u) is consistent Find x(u) =     . . .

1 g(u) f [i](u)

. . .     ∈ K(u)n g = 0 is the monic least common denominator

8

Background

Systems with Errors

Consider A(u), b(u) ξℓ ˆ A[ℓ], ˆ b[ℓ] Note that ˆ A[ℓ] and ˆ b[ℓ] are not guaranteed to be A(ξℓ) and b(ξℓ) respectively

8

Background

Systems with Errors

Consider A(u), b(u) ξℓ ˆ A[ℓ], ˆ b[ℓ] Note that ˆ A[ℓ] and ˆ b[ℓ] are not guaranteed to be A(ξℓ) and b(ξℓ) respectively Assume that for kERR ≤ E evaluations ξλ we have ˆ A[λ] f(ξλ) = g(ξλ)ˆ b[λ] Our definition of errors rules out dealing with inconsistent systems Compute

1 g(u) f(u) such that A(u)f(u) = g(u)b(u) and Λ(u) such

that Λ(ξℓ) = 0

9

Evaluation Counts

Generalized Welch/Berlekamp Count LBK = d f +dg +R+2E +1 Cabay Count LCAB = max{dA +df ,db +dg}+R+2E +1

10

Early Termination

• 1. Initialize L
• 2. The algorithm tries all possible values for d∗

f and d∗ g such that

L∗

BK = max{df +d∗

g,dg +d∗ f }+R∗ +2E∗ +1 = L

• 3. If our algorithm determines that deg(f) > d∗

f and/or deg(g) > d∗ g

then the number of evaluations ,L, is incremented

• 4. The algorithm continues incrementing the number of evaluations

and then trying all possible pairs of d∗

f and d∗ g until the solution is

found

10

Early Termination

• 1. Initialize L
• 2. The algorithm tries all possible values for d∗

f and d∗ g such that

L∗

BK = max{df +d∗

g,dg +d∗ f }+R∗ +2E∗ +1 = L

• 3. If our algorithm determines that deg(f) > d∗

f and/or deg(g) > d∗ g

then the number of evaluations ,L, is incremented

• 4. The algorithm continues incrementing the number of evaluations

and then trying all possible pairs of d∗

f and d∗ g until the solution is

found The algorithm terminates when d∗

f ≥ deg(f) and d∗ g ≥ deg(g)

10

Early Termination

• 1. Initialize L
• 2. The algorithm tries all possible values for d∗

f and d∗ g such that

L∗

BK = max{df +d∗

g,dg +d∗ f }+R∗ +2E∗ +1 = L

• 3. If our algorithm determines that deg(f) > d∗

f and/or deg(g) > d∗ g

then the number of evaluations ,L, is incremented

• 4. The algorithm continues incrementing the number of evaluations

and then trying all possible pairs of d∗

f and d∗ g until the solution is

found The algorithm terminates when d∗

f ≥ deg(f) and d∗ g ≥ deg(g)

Since LET

BK is sufficient to solve our system and LET BK ≤ LBK,

We compute the solution with possibly fewer evaluations than the Generalized Welch/Berlekamp count

11

Early Termination

• 1. Initialize L
• 2. The algorithm tries all possible values for d∗

f and d∗ g such that

L∗

CAB = max{dA +d∗

f ,db +d∗ g}+R∗ +2E∗ +1 = L

• 3. If our algorithm determines that deg(f) > d∗

f and/or deg(g) > d∗ g

then the number of evaluations ,L, is incremented

• 4. The algorithm continues incrementing the number of evaluations

and then trying all possible pairs of d∗

f and d∗ g until the solution is

found The algorithm terminates when d∗

f ≥ deg( f) and d∗ g ≥ deg(g)

Since LET

CAB is sufficient to find the solution the same Early

Termination strategy applies Since LET

CAB ≤ LCAB we compute the solution with possibly fewer

evaluations than the Cabay Count

12

Rational Vector Recovery

        

. . .

1 g(u) f [i](u)

. . .

         ξℓ     . . . γ[ℓ]

i.

. .     γ[ℓ]

i

=

• f [i](ξℓ)/g(ξℓ)

if g(ξℓ) = 0 ∞ if g(ξℓ) = 0 Note GCD(GCDi(f [i]),g) = 1 g(u) is the monic least common denominator

12

Rational Vector Recovery

        

. . .

1 g(u) f [i](u)

. . .

         ξℓ     . . . β [ℓ]

i

. . .     ∀ℓ / ∈ {λ1,...,λkERR}: β [ℓ] = γ[ℓ] ∀ℓ ∈ {λ1,...,λkERR}: β [ℓ] = γ[ℓ] kERR ≤ E

12

Rational Vector Recovery

        

. . .

1 g(u) f [i](u)

. . .

         ξℓ     . . . β [ℓ]

i

. . .     Let d f ≥ max

1≤i≤ndeg(f [i]) and dg ≥ deg(g)

Our algorithm can be specialized for rational vector recovery

12

Rational Vector Recovery

        

. . .

1 g(u) f [i](u)

. . .

         ξℓ     . . . β [ℓ]

i

. . .     Let d f ≥ max

1≤i≤ndeg(f [i]) and dg ≥ deg(g)

Our algorithm can be specialized for rational vector recovery

13

Rational Vector Recovery

Let ˆ A[ℓ] =     In 0...0 . . . ... . . . 0...0     and ˆ b[ℓ] =           β [ℓ]

1

. . . β [ℓ]

n

. . .           , at non-poles1 and ˆ A[ℓ] = 0m×n and ˆ b[ℓ] =     1 . . .    , at poles

1evaluations where the denominator is zero

14

Rational Vector Recovery

Lemma (Kaltofen, Pernet, Storjohann, and Waddell ’17)

If m = n = 1 and we take only L = deg(f)+deg(g)+2kERR then for a sufficiently large field we can always construct a second solution that has all the characteristics of the original solution.

14

Rational Vector Recovery

Theorem (Kaltofen, Pernet, Storjohann, and Waddell ’17)

If all degree bounds are exact then LCAB < LBK ⇐ ⇒ deg(A) < deg(g).

14

Rational Vector Recovery

Lemma (Kaltofen, Pernet, Storjohann, and Waddell ’17)

If A is full rank and f [i1] = f [i2] = 0 for 1 ≤ i1 < i2 ≤ n then deg(g) ≤ deg(A), thus in the exact case LBK ≤ LCAB.

15

Burst Errors

Consider f(ξℓ) =      f [1](ξℓ) f [2](ξℓ) . . . f [m](ξℓ)      where for all i,1 ≤ i ≤ m : f [i] ∈ K[u].

15

Burst Errors

Consider

15

Burst Errors

Consider       β [ℓ]

1 = f [1](ξℓ)

β [ℓ]

2 = f [2](ξℓ)

. . . β [ℓ]

m = f [m](ξℓ)

      where for all i,1 ≤ i ≤ m : f [i] ∈ K[u].

16

Burst Errors

Recall that the system we solve is overdetermined Theorem [Bleichenbacher, Kiayias, and Yung ’03] Assume that the errors are random field elements and they occur in bursts. The matrix of the system, β [ℓ]

i Ψ(ξℓ)−Φ[i](ξℓ) = 0, has a non-zero

minor of dimension m(d f +E +1)+E with probability 1− E

|K|

Theorem [Kaltofen, and Waddell 2019] Assume the errors occur in bursts The matrix of the system, β [ℓ]

i Ψ(ξℓ)−Φ[i](ξℓ) = 0, always has a

non-zero minor of dimension m(df +E +1)+E We remove the requirement that the errors be random field elements

17

Burst Errors

β [ℓ]

i Ψ(ξℓ)−Φ[i](ξℓ) = 0

d f +2E +1 df +E +1 E +1

17

Burst Errors

β [ℓ]

i Ψ(ξℓ)−Φ[i](ξℓ) = 0

d f +E +1 E +1 d f +E +1 E

17

Burst Errors

β [ℓ]

i Ψ(ξℓ)−Φ[i](ξℓ) = 0

df +E +1 E +1 df +E +1

18

Data Transmission

Consider f(u) =      f [1](u) f [2](u) . . . f [m](u)      where for all i,1 ≤ i ≤ m : f [i] ∈ K[u].

[Bleichenbacher, Kiayias, and Yung ’03], [Schmidt, Sidorenko, and Bossert ’06]

18

Data Transmission

Consider f(u) =      f [1](u) f [2](u) . . . f [m](u)      where for all i,1 ≤ i ≤ m : f [i] ∈ K[u]. Recall that in traditional Reed-Solomon decoding codewords are: [ f [1](ξ0), f [1](ξ1),..., f [1](ξL−1)], [ f [2](ξ0), f [2](ξ1),..., f [2](ξL−1)], ...,[ f [m](ξ0), f [m](ξ1),..., f [m](ξL−1)]

[Bleichenbacher, Kiayias, and Yung ’03], [Schmidt, Sidorenko, and Bossert ’06]

18

Data Transmission

Consider f(u) =      f [1](u) f [2](u) . . . f [m](u)      where for all i,1 ≤ i ≤ m : f [i] ∈ K[u]. In the vector model, if we assume L = m the codewords are: [ f [1](ξ0), f [2](ξ0),..., f [m](ξ0)], [ f [1](ξ1), f [2](ξ1),..., f [m](ξ1)], ...,[ f [1](ξL−1), f [2](ξL−1),..., f [m](ξL−1)]

[Bleichenbacher, Kiayias, and Yung ’03], [Schmidt, Sidorenko, and Bossert ’06]

18

Data Transmission

Consider f(u) =      f [1](u) f [2](u) . . . f [m](u)      where for all i,1 ≤ i ≤ m : f [i] ∈ K[u]. In the vector model, if we assume L = m the codewords are: [ f [1](ξ0), f [2](ξ0),..., f [m](ξ0)], [ f [1](ξ1), f [2](ξ1),..., f [m](ξ1)], ...,[ f [1](ξL−1), f [2](ξL−1),..., f [m](ξL−1)] So the vector model can be thought of as an interleaved Reed-Solomon code.

[Bleichenbacher, Kiayias, and Yung ’03], [Schmidt, Sidorenko, and Bossert ’06]

19

Data Transmission

— Simplified Gilbert Channel

19

Data Transmission

— Simplified Gilbert Channel — Channel Burst Length CBL = 1 1−P

BB

. — Symbol Error Rate ρ = 1−P

GG

1−P

BB +1−P GG

. — Mean Number of Bursts CBN = ρML/CBL.

20

Data Transmission

Let d f = 3 and E = 4 then L = df +2E +1 = 12 ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4]

20

Data Transmission

Let d f = 3 and E = 4 then L = df +2E +1 = 12 ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4]

20

Data Transmission

Let d f = 3 and E = 4 then L = df +2E +1 = 12 ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1](ξ8) f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [1](ξ9) f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [1](ξ10) f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [1](ξ11)

21

Data Transmission

ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [3] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4]

21

Data Transmission

ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 X f [1] f [1] X f [1] f [1] X f [1] f [1] f [1] f [1] f [1] X f [2] X f [2] f [2] f [2] X f [2] f [3] f [3] f [3] f [3] X f [3] X f [3] f [4] X f [4] f [4] f [4] f [4] f [4] f [4]

21

Data Transmission

ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 X f [2] f [2] X f [2] f [2] X f [2] f [3] f [3] f [3] f [3] X f [1] X f [1] f [1] f [1] X f [1] f [1] f [1] f [1] f [1] X f [3] X f [3] f [4] X f [4] f [4] f [4] f [4] f [4] f [4]

22

Data Transmission

ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [3] f [3] f [3] f [3] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [3] f [3] f [3] f [3] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [6] f [6] f [6] f [6] f [6] f [6] f [6] f [6] f [7] f [7] f [7] f [7] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [7] f [7] f [7] f [7] f [8] f [8] f [8] f [8] f [8] f [8] f [8] f [8] ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [10] f [10] f [10] f [10] f [10] f [10] f [10] f [10] f [11] f [11] f [11] f [11] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [11] f [11] f [11] f [11] f [12] f [12] f [12] f [12] f [12] f [12] f [12] f [12]

22

Data Transmission

ξ0 ξ1 ξ2 ξ3 ξ4 ξ5 ξ6 ξ7 ξ8 ξ9 ξ10 ξ11 f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [2] f [3] f [3] f [3] f [3] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [1] f [3] f [3] f [3] f [3] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [4] f [6] f [6] f [6] f [6] f [6] f [6] f [6] f [6] f [7] f [7] f [7] f [7] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [5] f [7] f [7] f [7] f [7] f [8] f [8] f [8] f [8] f [8] f [8] f [8] f [8] f [10] f [10] f [10] f [10] f [10] f [10] f [10] f [10] f [11] f [11] f [11] f [11] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [9] f [11] f [11] f [11] f [11] f [12] f [12] f [12] f [12] f [12] f [12] f [12] f [12]

23

Some Results

Table: Stack + Not Burst Only + Not Minimal Interleave c = ˆ c;EBRST = ENBRST = 2 Message Code Pct of Total Total Num Max Max Burst P

GG

P

BB df c Length Length Correct

Num Bursts Burst Length Decoding Bursts Corrected Length Corrected 0.9999 0.9 1110 660 1000 100.00 18.00 18.00 48.00 48.00 0.85 5 7 672 1372 100.00 19.00 19.00 11.00 11.00 0.999 0.9 1110 660 1000 100.00 125.00 125.00 45.00 45.00 0.85 5 7 672 1372 98.00 156.00 146.00 25.00 25.00 0.99 0.8535 7 576 1176 98.00 1151.00 1109.00 45.00 45.00 Table: Not Stack + Not Burst Only + Not Minimal Interleave c = ˆ c;EBRST = ENBRST = 2 Message Code Pct of Total Total Num Max Max Burst P

GG

P

BB df c Length Length Correct

Num Bursts Burst Length Decoding Bursts Corrected Length Corrected 0.9999 0.9 1110 660 1000 98.00 17.00 13.00 48.00 26.00 0.85 5 7 672 1372 100.00 19.00 19.00 11.00 11.00 0.999 0.9 1110 660 1000 85.00 125.00 86.00 45.00 34.00 0.85 5 7 672 1372 85.00 156.00 104.00 25.00 24.00 0.99 0.8535 7 576 1176 87.00 1151.00 937.00 45.00 45.00

24

Rank Deficient Systems

The Matrix is fine it’s the Right Side Vector that is off

25

Rank Deficient Systems

Let Wℓ = {w ∈ Kn | A(ξℓ)w = b(ξℓ)} and Wℓ = { ˆ w ∈ Kn | ˆ A[ℓ] ˆ w = ˆ b[ℓ]}. Error at ξλ : There exists w ∈ Wλ such that w / ∈ Wλ. Two types of Errors:

• 1. Matrix Error: There exists w ∈ Wλ such that given ˆ

A[λ] there is no right side vector such that that w ∈ Wλ.

• 2. Right Side Vector Error: There exists w ∈ Wλ such that w /

∈ Wλ for the given ˆ A[λ] and ˆ b[λ], but there exists a scalar right side vector such that there is no error.

Let R ≥ |{ℓ | Wℓ ⊂ Wℓ for 0 ≤ ℓ ≤ L−1}|.

26

Rank Deficient Systems

Theorem There are no Matrix Errors if and only if N(A(ξℓ)) ⊆ N( ˆ A[ℓ]).

26

Rank Deficient Systems

Theorem There are no Matrix Errors if and only if N(A(ξℓ)) ⊆ N( ˆ A[ℓ]). Fact If N(A(ξℓ)) ⊆ N( ˆ A[ℓ]) then all column dependencies that exist in A(ξℓ) also exist in ˆ A[ℓ].

26

Rank Deficient Systems

Theorem There are no Matrix Errors if and only if N(A(ξℓ)) ⊆ N( ˆ A[ℓ]). Fact If N(A(ξℓ)) ⊆ N( ˆ A[ℓ]) then all column dependencies that exist in A(ξℓ) also exist in ˆ A[ℓ]. Non Essential Column: A∗,j is a Non Essential Column of A if and only if Rg(A) = Rg(↓ A∗, j).

26

Rank Deficient Systems

Theorem There are no Matrix Errors if and only if N(A(ξℓ)) ⊆ N( ˆ A[ℓ]). Fact If N(A(ξℓ)) ⊆ N( ˆ A[ℓ]) then all column dependencies that exist in A(ξℓ) also exist in ˆ A[ℓ]. Non Essential Column: A∗,j is a Non Essential Column of A if and only if Rg(A) = Rg(↓ A∗, j). Minimal Linearly Dependent Set: S is a Minimal Linearly Dependent Set of vectors if and only if S is a linearly dependent set and no proper subset of S is a linearly dependent set.

26

Rank Deficient Systems

Theorem There are no Matrix Errors if and only if N(A(ξℓ)) ⊆ N( ˆ A[ℓ]). Fact If N(A(ξℓ)) ⊆ N( ˆ A[ℓ]) then all column dependencies that exist in A(ξℓ) also exist in ˆ A[ℓ]. Non Essential Column: A∗,j is a Non Essential Column of A if and only if Rg(A) = Rg(↓ A∗, j). Minimal Linearly Dependent Set: S is a Minimal Linearly Dependent Set of vectors if and only if S is a linearly dependent set and no proper subset of S is a linearly dependent set. Lemma The non zero entries in the last column of a canonical basis for the N(A) corresponds to the Right Most Minimal Linearly Dependent Set of columns in A.

26

Rank Deficient Systems

Theorem There are no Matrix Errors if and only if N(A(ξℓ)) ⊆ N( ˆ A[ℓ]). Fact If N(A(ξℓ)) ⊆ N( ˆ A[ℓ]) then all column dependencies that exist in A(ξℓ) also exist in ˆ A[ℓ]. Non Essential Column: A∗,j is a Non Essential Column of A if and only if Rg(A) = Rg(↓ A∗, j). Minimal Linearly Dependent Set: S is a Minimal Linearly Dependent Set of vectors if and only if S is a linearly dependent set and no proper subset of S is a linearly dependent set. Lemma The non zero entries in the last column of a canonical basis for the N(A) corresponds to the Right Most Minimal Linearly Dependent Set of columns in A. Lemma Let A∗, j be a Non Essential Column of A. There exists a permutation matrix P such that A∗, j is in the Right Most Minimal Linearly Dependent Set of Columns in AP.

27

Rank Deficient Systems

Assume there are M different Minimal Linearly Dependent Sets of columns on A(u).

27

Rank Deficient Systems

Assume there are M different Minimal Linearly Dependent Sets of columns on A(u). Let A(u)P

1,A(u)P 2,...,A(u)P M be such that each Minimal

Linearly Dependent Set of columns in A(u) is the Right Most Minimal Linearly Dependent Set of columns in some A(u)P

η.

27

Rank Deficient Systems

Assume there are M different Minimal Linearly Dependent Sets of columns on A(u). Let A(u)P

1,A(u)P 2,...,A(u)P M be such that each Minimal

Linearly Dependent Set of columns in A(u) is the Right Most Minimal Linearly Dependent Set of columns in some A(u)P

η.

Let dΩη be an upper bound on the minimum degree of the vectors in the N(A(u)P

η) with the property that only the rows that

correspond to the Right Most Minimal Linearly Dependent Set of columns in A(u)P

η are non zero.

27

Rank Deficient Systems

Assume there are M different Minimal Linearly Dependent Sets of columns on A(u). Let A(u)P

1,A(u)P 2,...,A(u)P M be such that each Minimal

Linearly Dependent Set of columns in A(u) is the Right Most Minimal Linearly Dependent Set of columns in some A(u)P

η.

Let dΩη be an upper bound on the minimum degree of the vectors in the N(A(u)P

η) with the property that only the rows that

correspond to the Right Most Minimal Linearly Dependent Set of columns in A(u)P

η are non zero.

Let dΩ ≥ max

1≤η≤M{dΩη,df }.

27

Rank Deficient Systems

Assume there are M different Minimal Linearly Dependent Sets of columns on A(u). Let A(u)P

1,A(u)P 2,...,A(u)P M be such that each Minimal

Linearly Dependent Set of columns in A(u) is the Right Most Minimal Linearly Dependent Set of columns in some A(u)P

η.

Let dΩη be an upper bound on the minimum degree of the vectors in the N(A(u)P

η) with the property that only the rows that

correspond to the Right Most Minimal Linearly Dependent Set of columns in A(u)P

η are non zero.

Let dΩ ≥ max

1≤η≤M{dΩη,df }.

Let LKW = max{dA +dΩ,db +dg}+R+2E +1.

28

Rank Deficient Systems

Solve ˆ A[ℓ]     . . . Ω[i](ξℓ) . . .     = 0m (1) deg(Ω[i]) ≤ dΩ +E 0 ≤ ℓ ≤ LKW −1

28

Rank Deficient Systems

Solve ˆ A[ℓ]     . . . Ω[i](ξℓ) . . .     = 0m (1) deg(Ω[i]) ≤ dΩ +E 0 ≤ ℓ ≤ LKW −1 Let Ωlast be the last column in the Column Echelon Form for a basis of the solution of equation (1). Theorem There is a Matrix Error at ξλ due to a difference in the column dependency in the Right Most Minimal Dependent Set of columns in A(ξℓ)P

η and ˆ

A[λ]P

η if and only if (u−ξλ) is a factor of

Ωlast(u).

29

Rank Deficient Systems

Solve simultaneously: Ψ(ξℓ)ˆ b[ℓ] − ˆ A[ℓ]Φ(ξℓ) = 0 and Φ[j](ξℓ) = 0 where ˆ A[ℓ]

∗,j has no pivot in the Row Echelon Form of ˆ

A[ℓ] deg(Φ) ≤ df +E, deg(Ψ) ≤ dg +E, and 0 ≤ ℓ ≤ LCAB −1.

29

Rank Deficient Systems

Solve simultaneously: Ψ(ξℓ)ˆ b[ℓ] − ˆ A[ℓ]Φ(ξℓ) = 0 and Φ[j](ξℓ) = 0 where ˆ A[ℓ]

∗,j has no pivot in the Row Echelon Form of ˆ

A[ℓ] deg(Φ) ≤ df +E, deg(Ψ) ≤ dg +E, and 0 ≤ ℓ ≤ LCAB −1. Then (Φmin,Ψmin) = ( ¯ Λ ¯ f, ¯ Λ ¯ g) and ¯ Λ | Λ. Warning: The solution 1

¯ g ¯

f may not satisfy the input degree bounds.

30

Rank Deficient Systems

{Matrix Errors} ∪ {Right Side Vector Errors} We can remove all errors as well as compute an error locator polynomial Λ.

30

Rank Deficient Systems

{Matrix Errors} ∪ {Right Side Vector Errors} We can remove all errors as well as compute an error locator polynomial Λ. Solve Ψ(ξℓ)ˆ b[ℓ] − ˆ A[ℓ]     . . . Φ[i](ξℓ) . . .     = 0, (2) deg(Φ[i]) ≤ df ,deg(Ψ) ≤ dg,0 ≤ ℓ ≤ LCAB −(E +1). The solution to equation (2) satisfies the input degree bounds.

31

Future Work

Prove conjectured polynomial time algorithm for rank deficient case correct

31

Future Work

Prove conjectured polynomial time algorithm for rank deficient case correct Continue work on burst errors:

— Refine model — Hamming distance — List decoding

31

Future Work

Prove conjectured polynomial time algorithm for rank deficient case correct Continue work on burst errors:

— Refine model — Hamming distance — List decoding

Investigate error correction as an algorithmic paradigm

— [Roche 2018]

31

Future Work

Prove conjectured polynomial time algorithm for rank deficient case correct Continue work on burst errors:

— Refine model — Hamming distance — List decoding

Investigate error correction as an algorithmic paradigm

— [Roche 2018]

32

Thank you!