Projected Subcodes of the Second Order Binary Reed-Muller Code - PowerPoint PPT Presentation

Motivation and principle Recalls Results Conclusion and further works Projected Subcodes of the Second Order Binary Reed-Muller Code Matthieu Legeay IRMAR, University of Rennes 1, France CBC 2012

Motivation and principle Recalls Results Conclusion and further works Plan Motivation and principle 1 Recalls 2 Results 3 Conclusion and further works 4

Motivation and principle Recalls Results Conclusion and further works Motivation Reed-Muller codes have efficient decoding algorithms

Motivation and principle Recalls Results Conclusion and further works Motivation Reed-Muller codes have efficient decoding algorithms ⇒ No algorithm reaches the lower bound on the minimum distance decoding capability

Motivation and principle Recalls Results Conclusion and further works Motivation Reed-Muller codes have efficient decoding algorithms ⇒ No algorithm reaches the lower bound on the minimum distance decoding capability Other algorithms using algebraic properties practically correct more errors

Motivation and principle Recalls Results Conclusion and further works Motivation Reed-Muller codes have efficient decoding algorithms ⇒ No algorithm reaches the lower bound on the minimum distance decoding capability Other algorithms using algebraic properties practically correct more errors ⇒ The complexity of the decoder is quadratic in the code length

Motivation and principle Recalls Results Conclusion and further works Principle Take y = c + e and compute : � � � λ i σ i ( y ) = λ i σ i ( c ) + λ i σ i ( e ) i i i where ( σ i ) i ∈ Perm ( C ) and ( λ i ) i ∈ F 2 .

Motivation and principle Recalls Results Conclusion and further works Principle Take y = c + e and compute : � � � λ i σ i ( y ) = λ i σ i ( c ) + λ i σ i ( e ) i i i where ( σ i ) i ∈ Perm ( C ) and ( λ i ) i ∈ F 2 . ⇒ c ′ = � i λ i σ i ( c ) lives in a subcode C ad of C , with k ad ≤ k .

Motivation and principle Recalls Results Conclusion and further works Principle Take y = c + e and compute : � � � λ i σ i ( y ) = λ i σ i ( c ) + λ i σ i ( e ) i i i where ( σ i ) i ∈ Perm ( C ) and ( λ i ) i ∈ F 2 . ⇒ c ′ = � i λ i σ i ( c ) lives in a subcode C ad of C , with k ad ≤ k . ⇒ e ′ = � i λ i σ i ( e ) is an error vector, wt ( e ′ ) ≤ λ t .

Motivation and principle Recalls Results Conclusion and further works Recalls r -order Reed-Muller codes Let 0 ≤ r ≤ m , n = 2 m and ( α 1 , . . . , α n ) ∈ ( F m 2 ) n . R ( r , m ) = { ( f ( α 1 ) , . . . , f ( α n )) ∈ F n 2 } with f ( x 1 , . . . , x m ) a binary multivariate polynomial of degree ≤ r .

Motivation and principle Recalls Results Conclusion and further works Recalls r -order Reed-Muller codes Let 0 ≤ r ≤ m , n = 2 m and ( α 1 , . . . , α n ) ∈ ( F m 2 ) n . R ( r , m ) = { ( f ( α 1 ) , . . . , f ( α n )) ∈ F n 2 } with f ( x 1 , . . . , x m ) a binary multivariate polynomial of degree ≤ r . r R ( r , m ) is a [ n = 2 m , k = � m , d = 2 m − r ] code. � � i i =0

Motivation and principle Recalls Results Conclusion and further works Recalls r -order Reed-Muller codes Let 0 ≤ r ≤ m , n = 2 m and ( α 1 , . . . , α n ) ∈ ( F m 2 ) n . R ( r , m ) = { ( f ( α 1 ) , . . . , f ( α n )) ∈ F n 2 } with f ( x 1 , . . . , x m ) a binary multivariate polynomial of degree ≤ r . r R ( r , m ) is a [ n = 2 m , k = � m , d = 2 m − r ] code. � � i i =0 R (0 , m ) is the repetition code.

Motivation and principle Recalls Results Conclusion and further works Recalls r -order Reed-Muller codes Let 0 ≤ r ≤ m , n = 2 m and ( α 1 , . . . , α n ) ∈ ( F m 2 ) n . R ( r , m ) = { ( f ( α 1 ) , . . . , f ( α n )) ∈ F n 2 } with f ( x 1 , . . . , x m ) a binary multivariate polynomial of degree ≤ r . r R ( r , m ) is a [ n = 2 m , k = � m , d = 2 m − r ] code. � � i i =0 R (0 , m ) is the repetition code. R ( m , m ) is all the space F n 2 .

Motivation and principle Recalls Results Conclusion and further works Permutation group Theorem Perm ( R ( r , m )) = GA m ( F 2 ) = T ⋊ GL m ( F 2 )

Motivation and principle Recalls Results Conclusion and further works Permutation group Theorem Perm ( R ( r , m )) = GA m ( F 2 ) = T ⋊ GL m ( F 2 ) � T α : F m F m � → 2 2 , α ∈ F m T = 2 x �→ x + α T α · f ( x ) def = f ( T α ( x )) = f ( x + α )

Motivation and principle Recalls Results Conclusion and further works Permutation group Theorem Perm ( R ( r , m )) = GA m ( F 2 ) = T ⋊ GL m ( F 2 ) � T α : F m F m � → 2 2 , α ∈ F m T = 2 x �→ x + α T α · f ( x ) def = f ( T α ( x )) = f ( x + α ) GL m ( F 2 ) = { non-singular binary matrices G of size m × m } G · f ( x ) def = f ( G . x )

Motivation and principle Recalls Results Conclusion and further works With T Proposition 1 ( Id + T α ) · R (2 , m ) def = { f + T α · f | f ∈ R (2 , m ) } is a subcode of R (2 , m ).

Motivation and principle Recalls Results Conclusion and further works With T Proposition 1 ( Id + T α ) · R (2 , m ) def = { f + T α · f | f ∈ R (2 , m ) } is a subcode of R (2 , m ). Proposition 2 ( Id + T α ) · R (2 , m ) is isomorphic to R (1 , m − 1).

Motivation and principle Recalls Results Conclusion and further works With T Proposition 1 ( Id + T α ) · R (2 , m ) def = { f + T α · f | f ∈ R (2 , m ) } is a subcode of R (2 , m ). Proposition 2 ( Id + T α ) · R (2 , m ) is isomorphic to R (1 , m − 1). Idea for proof... 1 ( f + T α · f ) is an affine function x ⇒ r ′ = 1 2 ( f + T α · f )( x + α ) = ( f + T α · f )( x ) ⇒ m ′ = m − 1

Motivation and principle Recalls Results Conclusion and further works With GL m ( F 2 ) Proposition 1 ( Id + G ) · R (2 , m ) def = { f + G · f | f ∈ R (2 , m ) } is a subcode of R (2 , m ).

Motivation and principle Recalls Results Conclusion and further works With GL m ( F 2 ) Proposition 1 ( Id + G ) · R (2 , m ) def = { f + G · f | f ∈ R (2 , m ) } is a subcode of R (2 , m ). What are the properties of this subcode ? Length ? Dimension ? Minimum Distance ?

Motivation and principle Recalls Results Conclusion and further works With GL m ( F 2 ) Proposition 1 ( Id + G ) · R (2 , m ) def = { f + G · f | f ∈ R (2 , m ) } is a subcode of R (2 , m ). What are the properties of this subcode ? Length ? Dimension ? Minimum Distance ? ⇒ Hard to answer in the general case.

Motivation and principle Recalls Results Conclusion and further works With GL m ( F 2 ) • By writing f ( x ) = x t Fx + a f , with F upper triangular, ( f + G · f )( x ) = x t ( F + G t FG ) x � P G : M m ( F 2 ) → M m ( F 2 ) does not keep F + G t FG F �→ upper-triangularity.

Motivation and principle Recalls Results Conclusion and further works With GL m ( F 2 ) • By writing f ( x ) = x t Fx + a f , with F upper triangular, ( f + G · f )( x ) = x t ( F + G t FG ) x � P G : M m ( F 2 ) → M m ( F 2 ) does not keep F + G t FG F �→ upper-triangularity. • Rewrite G = Id + E , hence ( f + G · f )( x ) = x t ( E t F + FE + E t FE ) x � P E : M m ( F 2 ) → M m ( F 2 ) E t F + FE + E t FE �→ F

Motivation and principle Recalls Results Conclusion and further works With GL m ( F 2 ) • By writing f ( x ) = x t Fx + a f , with F upper triangular, ( f + G · f )( x ) = x t ( F + G t FG ) x � P G : M m ( F 2 ) → M m ( F 2 ) does not keep F + G t FG F �→ upper-triangularity. • Rewrite G = Id + E , hence ( f + G · f )( x ) = x t ( E t F + FE + E t FE ) x � P E : M m ( F 2 ) → M m ( F 2 ) E t F + FE + E t FE �→ F ⇒ Rank of E

Motivation and principle Recalls Results Conclusion and further works Result on length Proposition 2 ( Id + G ) · R (2 , m ) is isomorphic to a subcode of length n − 2 m − r

Motivation and principle Recalls Results Conclusion and further works Result on length Proposition 2 ( Id + G ) · R (2 , m ) is isomorphic to a subcode of length n − 2 m − r If r = 1, n ′ = 2 m − 1 we find again that the subcode is isomorphic to R (1 , m − 1). If r = 2, n ′ = 2 m − 2 m − 2 ...

Motivation and principle Recalls Results Conclusion and further works Result on length Proposition 2 ( Id + G ) · R (2 , m ) is isomorphic to a subcode of length n − 2 m − r If r = 1, n ′ = 2 m − 1 we find again that the subcode is isomorphic to R (1 , m − 1). If r = 2, n ′ = 2 m − 2 m − 2 ... ⇒ We can do better... Some columns are equal in practice.

Motivation and principle Recalls Results Conclusion and further works Result on dimension Proposition 3 ( Id + G ) · R (2 , m ) has dimension k ′ ≤ 4 r ( m − r ) + 1 Idea for proof... 1 Rank( E t F + FE + E t FE ) ≤ 2 r j − 1 r (2 m − 2 i )(2 m − 2 i ) 2 N ( m , r ) = ≤ 2 (2 m − r ) r +1) � � 2 j − 2 i j =0 i =0

Motivation and principle Recalls Results Conclusion and further works Result on dimension Proposition 3 ( Id + G ) · R (2 , m ) has dimension k ′ ≤ 4 r ( m − r ) + 1 Idea for proof... 1 Rank( E t F + FE + E t FE ) ≤ 2 r j − 1 r (2 m − 2 i )(2 m − 2 i ) 2 N ( m , r ) = ≤ 2 (2 m − r ) r +1) � � 2 j − 2 i j =0 i =0 If r = 1, k ′ ≤ 4( m − 1) + 1 If r = 2, k ′ ≤ 8( m − 2) + 1...