Quaternary and binary codes as Gray images of constacyclic codes over - PowerPoint PPT Presentation

Quaternary and binary codes as Gray images of constacyclic codes over Z 2 k +1 Henry Chimal Dzul Depto. de Matem´ aticas, UAM-Iztapalapa Noncommutative rings and their applications IV University of Artois, Lens, France 8-11 June 2015 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 1 / 27

Outline Preliminaries 1 Formulation of the problem 2 Some contributions 3 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 2 / 27

Outline Preliminaries 1 Formulation of the problem 2 Some contributions 3 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 3 / 27

Constacyclic codes Let R be a finite commutative ring with 1, γ ∈ U ( R ) and n ≥ N . C ⊆ R n is a constacyclic code or a γ -cyclic code if ν γ ( C ) = C , where ν γ : ( a 0 , a 1 , . . . , a n − 1 ) �→ ( γa n − 1 , a 0 , . . . , a n − 2 ) . C ⊆ R n is a cyclic code if σ ( C ) = C , where σ = ν 1 . C ⊆ R n is a negacyclic code if ν ( C ) = C , donde ν = ν − 1 . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 4 / 27

Constacyclic codes Let R be a finite commutative ring with 1, γ ∈ U ( R ) and n ≥ N . C ⊆ R n is a constacyclic code or a γ -cyclic code if ν γ ( C ) = C , where ν γ : ( a 0 , a 1 , . . . , a n − 1 ) �→ ( γa n − 1 , a 0 , . . . , a n − 2 ) . C ⊆ R n is a cyclic code if σ ( C ) = C , where σ = ν 1 . C ⊆ R n is a negacyclic code if ν ( C ) = C , donde ν = ν − 1 . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 4 / 27

γ -quasi-cyclic codes Let m be a positive integer C ⊆ ( R n ) m ia a γ -quasi-cyclic code of index m and length mn if ν ⊗ m ( C ) = C , where γ � A (0) | · · · | A ( m − 1) � � � A (0) � � � A ( m − 1) �� � � · · · � ν γ ν ⊗ m : �→ ν γ , γ with A ( i ) ∈ R n , 0 ≤ i ≤ m − 1 . C ⊆ ( R n ) m is quasi-cyclic if σ ⊗ m ( C ) = C . C ⊆ ( R n ) m es quasi-negacyclic if ν ⊗ m ( C ) = C . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 5 / 27

γ -quasi-cyclic codes Let m be a positive integer C ⊆ ( R n ) m ia a γ -quasi-cyclic code of index m and length mn if ν ⊗ m ( C ) = C , where γ � A (0) | · · · | A ( m − 1) � � � A (0) � � � A ( m − 1) �� � � · · · � ν γ ν ⊗ m : �→ ν γ , γ with A ( i ) ∈ R n , 0 ≤ i ≤ m − 1 . C ⊆ ( R n ) m is quasi-cyclic if σ ⊗ m ( C ) = C . C ⊆ ( R n ) m es quasi-negacyclic if ν ⊗ m ( C ) = C . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 5 / 27

Beginings of the linear codes over rings The history of linear codes over rings backs to the 70’s with the works of I. F. Blake, Codes over certain rings 20 (1972), Inf. and Control E. Spiegel, Codes over the ring Z m 35 (1977), Inf. and Control However the community did not pay a lot of attention. H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 6 / 27

The theory of codes over rings was really initiated A. A. Nechaev, Kerdock code in a cyclic form , Discrete Math. and Appl. 1 (1991) A. R. Hammons, et. al, The Z 4 -Linearity of Kerdock, Preparata, Goethals, and Related Codes , IEEE Trans. Inf. Theory 40 (1994) The classical Gray Map dual ✲ K ⊥ = P ⊂ Z n K ⊂ Z n 4 4 φ : → F 2 × F 2 Z 4 0 �→ (0 , 0) φ φ 1 �→ (0 , 1) 2 �→ (1 , 1) ❄ ❄ 3 �→ (1 , 0) K = φ ( K ) ⊂ F 2 n P = φ ( P ) ⊂ F 2 n 2 2 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 7 / 27

Analysis of the cyclic properties J. Wolfman, Negacyclic and cyclic codes over Z 4 . IEEE Trans. Inf. Theory. 45 (1999) µ ✲ D ⊂ Z n C ⊂ Z n Linear Cyclic code Linear Negacyclic code 4 4 φ φ � ❄ ❄ N φ ( C ) ⊂ F 2 n ✲ φ ( D ) ⊂ F 2 n quasi-cyclic Cyclic Code 2 2 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 8 / 27

Some generalizations S. Ling, T. Blackford, Z p k +1 -Linear Codes . IEEE Tans. Info. Theory. 48 (2002) (1 − p k ) -cyclic codes over Z p k +1 H. Tapia-Recillas, G. Vega, Some Constacyclic Codes over Z 2 k +1 and Binary Quasi-Cyclic Codes . Disc. App. Math. 128 (2003) (1 + 2 k ) -cyclic codes over Z 2 k +1 S. Jitman, P. Udomkavanich. The Gray Image of Cyclic Codes over Finite Chaing Rings . Inter. J. of Contemporary Mathematics 5 (2010). (1 − θ k ) -cyclic codes over a finite chaing ring R with maximal ideal � θ � , θ k +1 = 0 . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 9 / 27

All the works aforementioned analyze the gray images of γ -cyclic codes where γ is γ = 1 − θ k , k is the index of nilpotence of R In terms of the chain of ideals R � � θ � � � θ 2 � � · · · � � θ k − 1 � � � θ k � � � 0 � unit ❄ γ = 1 − θ k H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 10 / 27

All the works aforementioned analyze the gray images of γ -cyclic codes where γ is γ = 1 − θ k , k is the index of nilpotence of R In terms of the chain of ideals R � � θ � � � θ 2 � � · · · � � θ k − 1 � � � θ k � � � 0 � unit ❄ γ = 1 − θ k H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 10 / 27

Outline Preliminaries 1 Formulation of the problem 2 Some contributions 3 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 11 / 27

Formulation of the problem... Take R = Z 2 k +1 Z 2 k +1 � � 2 � � � 2 2 � � · · · � � 2 k − 1 � � 2 k � � � 0 � � units unit ❄ ❄ δ 1 = 1 + 2 k − 1 1 − 2 k , 1 δ 2 = 1 + 2 k − 1 + 2 k γ = 1 + 2 k , 1 We will analyze the Gray image of (1 + 2 k − 1 ) , (1 + 2 k − 1 + 2 k ) -cyclic codes, and the Gray image of quasi-cyclic codes and (1 + 2 k ) -quasi-cyclic codes. H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 12 / 27

The 2-adic representation of z ∈ Z 2 k +1 is: z = r 0 ( z ) + 2 r 1 ( z ) + 2 2 r 2 ( z ) + · · · + 2 k r k ( z ) , r i ( z ) ∈ F 2 . The 2-adic representation of Z = ( z 0 , . . . , z n − 1 ) ∈ Z n 2 k +1 is: Z = r 0 ( Z ) + 2 r 1 ( Z ) + 2 2 r 2 ( Z ) + · · · + 2 k r k ( Z ) , where r i ( Z ) = ( r i ( z 0 ) , . . . , r i ( z n − 1 )) ∈ F n 2 . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 13 / 27

The homogeneous weight The homogeneous weight ω h : Z 2 k +1 → Z is ω h (2 k ) = 2 k ω h ( a ) = 2 k − 1 , a � = 0 , 2 k ω h (0) = 0 Extension to Z n 2 k +1 → Z ω h ( a 0 , . . . , a n − 1 ) = ω h ( a 0 ) + · · · + ω h ( a n − 1 ) The homogeneous distance δ H : Z n 2 k +1 × Z n 2 k +1 → Z δ h ( A, B ) = ω h ( A − B ) H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 14 / 27

The Gray isometry M. Greferath, S. Schmidt, Gray Isometries over Finite Chaing Rings and a Nonlinear Ternary (36 , 3 12 , 15) code . IEEE Trans. Inf. Theory. 45 (1999) 2 k +1 → F 2 k n Definition of Φ : Z n 2 � � � � � � c k c k c k 0 ⊗ r 0 ( Z ) ⊕ 1 ⊗ r 1 ( Z ) ⊕ · · · ⊕ k ⊗ r k ( Z ) Φ( Z ) = Theorem → ( F 2 k n Φ : ( Z n 2 k +1 , δ h ) − , δ H ) is an inyective isometry. 2 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 15 / 27

Outline Preliminaries 1 Formulation of the problem 2 Some contributions 3 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 16 / 27

An step isometry Definition of the step Gray isometry isometry Z n ϕ ✲ Z 2 k − 1 n 2 k +1 Z n 4 2 k +1 Φ Φ φ ❄ ✲ F 2 k n ❄ 2 F 2 k n 2 H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 17 / 27

Image of quasi-ciclic codes Theorem The following statements are equivalents: (1) C ⊆ Z mn 2 k +1 is a quasi-cyclic code of index m . (2) ϕ ( C ) is a quaternary quasi-cyclic code of index 2 k − 1 m and of length 2 k − 1 mn . (3) Φ( C ) is a binary quasi-cyclic code of index 2 k m and of length 2 k mn . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 18 / 27

Image of (1 + 2 k ) -cyclic codes Theorem The following statements are equivalent 1 C ⊆ Z mn 2 k +1 is a λ -quasi-cyclic code of index m . 2 ϕ ( C ) is a quaternary quasi-negacyclic code of index 2 k − 1 m and of length 2 k − 1 mn . 3 Φ( C ) is permutation equivalent to a binary quasi-cyclic code of index 2 k − 1 m and of length 2 k mn . H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 19 / 27

Images of the new constacyclic codes: A permutation π the permutation on Z 2 k − 1 n Let � induced by the permutation 4 l + 2 n ) · · · ((2 k − 2 − 1) n l + (2 k − 2 − 1) n ) , π = (0 l )( n l + n )(2 n donde l = 2 k − 2 n .   l l � � �� � � �� � �   ∗ | ⊛ | · · · | ⋄ � ∗ | ⊛ | · · · | ⋄ � �� � � �� � � �� � � �� � � �� � � �� � n n n n n n � �� � 2 k − 1 n H. Chimal-Dzul (hcdzul@xanum.uam.mx) Quaternary and binary codes NCRA IV 20 / 27