Parametric Linear System Solving with Error Correction. Cleveland Waddell S YMBOLIC -N UMERIC C OMPUTING S EMINAR 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 ( ξ 0 ) , f ( ξ 1 ) ,..., f ( ξ L − 1 )) ( β 0 , β 1 ,..., β L − 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 ≥ |{ ξ ℓ | f ( ξ ℓ ) � = β ℓ }| . — Pick L = deg ( f )+ 2 E + 1 distinct elements ξ ℓ ∈ 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 ) = u 2 + 2 u + 1 — We pick L distinct points ξ ← ( 0 , 1 , 2 , 3 , 4 , 5 , 6 ) — Codeword: f ( ξ ℓ ) → ( 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 ( ξ λ ) = 0 whenever f ( ξ λ ) � = β λ — Equation β ℓ Λ ( ξ ℓ ) = f ( ξ ℓ ) Λ ( ξ ℓ ) is true for all evaluation points ξ ℓ — Let Φ ( u ) and Ψ ( u ) be generic polynomial where, deg ( Φ ) = deg ( f )+ E , and deg ( Ψ ) = E — Solve the linear system β ℓ Ψ ( ξ ℓ ) = Φ ( ξ ℓ ) 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 . . . g ( u ) f [ i ] ( u ) 1 ∈ K ( u ) n x ( u ) = . . . g � = 0 is the monic least common denominator
8 Background Systems with Errors Consider A [ ℓ ] , ˆ b [ ℓ ] ξ ℓ ˆ A(u), b(u) A [ ℓ ] and ˆ b [ ℓ ] are not guaranteed to be A ( ξ ℓ ) and b ( ξ ℓ ) Note that ˆ respectively
8 Background Systems with Errors Consider A [ ℓ ] , ˆ b [ ℓ ] ξ ℓ ˆ A(u), b(u) A [ ℓ ] and ˆ b [ ℓ ] are not guaranteed to be A ( ξ ℓ ) and b ( ξ ℓ ) Note that ˆ respectively Assume that for k ERR ≤ E evaluations ξ λ we have A [ λ ] f ( ξ λ ) � = g ( ξ λ ) ˆ b [ λ ] ˆ Our definition of errors rules out dealing with inconsistent systems 1 g ( u ) f ( u ) such that A ( u ) f ( u ) = g ( u ) b ( u ) and Λ ( u ) such Compute that Λ ( ξ ℓ ) = 0
9 Evaluation Counts Generalized Welch/Berlekamp Count L BK = d f + d g + R + 2 E + 1 Cabay Count L CAB = max { d A + d f , d b + d g } + R + 2 E + 1
10 Early Termination 1. Initialize L 2. The algorithm tries all possible values for d ∗ f and d ∗ g such that f } + R ∗ + 2 E ∗ + 1 = L L ∗ BK = max { d f + d ∗ g , d g + d ∗ 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 f } + R ∗ + 2 E ∗ + 1 = L L ∗ BK = max { d f + d ∗ g , d g + d ∗ 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 f } + R ∗ + 2 E ∗ + 1 = L L ∗ BK = max { d f + d ∗ g , d g + d ∗ 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 L ET BK is sufficient to solve our system and L ET BK ≤ L BK , We compute the solution with possibly fewer evaluations than the Generalized Welch/Berlekamp count
Recommend
More recommend
