Multidimensional Quasi-Cyclic and Convolutional Codes Buket - PowerPoint PPT Presentation

Quasi-Cyclic and Convolutional Codes Multidimensional QC Codes Links to Multidimensional Convolutional Codes Multidimensional Quasi-Cyclic and Convolutional Codes Buket ¨ Ozkaya joint work with Cem G¨ uneri 7 May 2015 Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes For m , ℓ integers with ( m , q ) = 1, a QC code of length m ℓ and index ℓ over F q is a linear code C ⊆ F m ℓ q , if it is invariant under shift of codewords by ℓ positions. Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes For m , ℓ integers with ( m , q ) = 1, a QC code of length m ℓ and index ℓ over F q is a linear code C ⊆ F m ℓ q , if it is invariant under shift of codewords by ℓ positions.   c 00 . . . c 0 ,ℓ − 1   . . . .  ∈ F m × ℓ ≃ F m ℓ c =  . . q q c m − 1 , 0 . . . c m − 1 ,ℓ − 1 Invariance under shift by ℓ units is equivalent to being closed under row shift. In particular, a QC code of index ℓ = 1 is a cyclic code. Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes For m , ℓ integers with ( m , q ) = 1, a QC code of length m ℓ and index ℓ over F q is a linear code C ⊆ F m ℓ q , if it is invariant under shift of codewords by ℓ positions.   c 00 . . . c 0 ,ℓ − 1   . . . .  ∈ F m × ℓ ≃ F m ℓ c =  . . q q c m − 1 , 0 . . . c m − 1 ,ℓ − 1 Invariance under shift by ℓ units is equivalent to being closed under row shift. In particular, a QC code of index ℓ = 1 is a cyclic code. If C is also closed under column shift, then it’s called a 2D cyclic code. Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Algebraic Structure The codewords of a cyclic code can be viewed as polynomials via the identification: F q [ x ] / � x m − 1 � = R F m − → q   c 0 m − 1 �  .  . c i x i �→ c ( x ) =   . c m − 1 i =0 The shift by 1 unit corresponds to x . c ( x ) ⇒ a cyclic code is an ideal in R . Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Algebraic Structure Similarly one can define a QC code in R ℓ : F m × ℓ R ℓ − → q   c 00 c 01 . . . c 0 ,ℓ − 1  . . .  . . . �→ � c ( x ) = ( c 0 ( x ) , . . . , c ℓ − 1 ( x ))   . . . c m − 1 , 0 c m − 1 , 1 . . . c m − 1 ,ℓ − 1 m − 1 � c ij x i , ∀ 0 ≤ j ≤ ℓ − 1. where c j ( x ) = i =0 Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Algebraic Structure Similarly one can define a QC code in R ℓ : F m × ℓ R ℓ − → q   c 00 c 01 . . . c 0 ,ℓ − 1  . . .  . . . �→ � c ( x ) = ( c 0 ( x ) , . . . , c ℓ − 1 ( x ))   . . . c m − 1 , 0 c m − 1 , 1 . . . c m − 1 ,ℓ − 1 m − 1 � c ij x i , ∀ 0 ≤ j ≤ ℓ − 1. where c j ( x ) = i =0 Row shift in F m × ℓ corresponds to coordinatewise multiplication by x in R ℓ . q Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Algebraic Structure Similarly one can define a QC code in R ℓ : F m × ℓ R ℓ − → q   c 00 c 01 . . . c 0 ,ℓ − 1  . . .  . . . �→ � c ( x ) = ( c 0 ( x ) , . . . , c ℓ − 1 ( x ))   . . . c m − 1 , 0 c m − 1 , 1 . . . c m − 1 ,ℓ − 1 m − 1 � c ij x i , ∀ 0 ≤ j ≤ ℓ − 1. where c j ( x ) = i =0 Row shift in F m × ℓ corresponds to coordinatewise multiplication by x in R ℓ . q ⇒ C ⊆ R ℓ is an R -submodule. Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Algebraic Structure Let S := F q [ x , y ] / � x m − 1 , y ℓ − 1 � and view a codeword c = ( c ij ) ∈ C as a 2-variate polynomial in S : m − 1 ℓ − 1 � � c ij x i y j c ( x , y ) = i =0 j =0 Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Algebraic Structure Let S := F q [ x , y ] / � x m − 1 , y ℓ − 1 � and view a codeword c = ( c ij ) ∈ C as a 2-variate polynomial in S : m − 1 ℓ − 1 � � c ij x i y j c ( x , y ) = i =0 j =0 Then C is QC ⇔ C is an R -submodule in S . C is 2D-cyclic ⇔ C is an ideal in S . Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Constituents (Ling-Sol´ e, 2001) Consider the factorization of x m − 1 into irreducibles in F q [ x ]: x m − 1 = f 1 ( x ) . . . f s ( x ) Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Constituents (Ling-Sol´ e, 2001) Consider the factorization of x m − 1 into irreducibles in F q [ x ]: x m − 1 = f 1 ( x ) . . . f s ( x ) Since ( m , q ) = 1, there are no repeating factors. By CRT we have: R = F q [ x ] / � x m − 1 � ≃ F q [ x ] / � f 1 ( x ) � ⊕ . . . ⊕ F q [ x ] / � f s ( x ) � Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Constituents (Ling-Sol´ e, 2001) Consider the factorization of x m − 1 into irreducibles in F q [ x ]: x m − 1 = f 1 ( x ) . . . f s ( x ) Since ( m , q ) = 1, there are no repeating factors. By CRT we have: R = F q [ x ] / � x m − 1 � ≃ F q [ x ] / � f 1 ( x ) � ⊕ . . . ⊕ F q [ x ] / � f s ( x ) � s � R ≃ E 1 ⊕ . . . ⊕ E s ⇒ R ℓ ≃ E ℓ i i =1 Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Constituents (Ling-Sol´ e, 2001) Consider the factorization of x m − 1 into irreducibles in F q [ x ]: x m − 1 = f 1 ( x ) . . . f s ( x ) Since ( m , q ) = 1, there are no repeating factors. By CRT we have: R = F q [ x ] / � x m − 1 � ≃ F q [ x ] / � f 1 ( x ) � ⊕ . . . ⊕ F q [ x ] / � f s ( x ) � s � R ≃ E 1 ⊕ . . . ⊕ E s ⇒ R ℓ ≃ E ℓ i i =1 s � C i where C i ⊆ E ℓ Hence, C = i is a length ℓ code over E i for each i =1 1 ≤ i ≤ s . C i ’s are said to be the constituents of C . Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Concatenated Form Let � θ i � be the minimal cyclic code of length m over F q with the check polynomial f i ( x ) and the primitive idempotent generator θ i . Note that � θ i � is isomorphic to E i = F q deg fi . Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes

Quasi-Cyclic and Convolutional Codes Quasi-cyclic codes Multidimensional QC Codes Concatenated structure of QC codes Links to Multidimensional Convolutional Codes Convolutional codes Concatenated Form Let � θ i � be the minimal cyclic code of length m over F q with the check polynomial f i ( x ) and the primitive idempotent generator θ i . Note that � θ i � is isomorphic to E i = F q deg fi . Theorem (Jensen, 1985) Let C be a QC code. For some subset I of { 1 , . . . , s } , we have � C = ( � θ i � � C i ) , i ∈I where C i is a linear code over E i of length ℓ . Converse also holds. Buket ¨ Ozkaya Multidimensional Quasi-Cyclic and Convolutional Codes