On the quasi-cyclicity and linearity of the Gray image of a code - PowerPoint PPT Presentation

On the quasi-cyclicity and linearity of the Gray image of a code over a Galois ring H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) oquio Latino-Americano de ´ XVIII Col´ Algebra Sao Pedro, Brasil August, 2009 H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Abstract If R = GR ( p 2 , m ) is a Galois ring, Φ the Gray map on R n and C ⊆ R n is a λ -cyclic code of length n , it is shown that its Gray image Φ( C ) is quasi-cyclic of index np over the residue field of R . Also, if C is linear, necessary and sufficient conditions for the code Φ( C ) to be linear are given. H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

C O N T E N T 1. Introduction 2. Galois ring and Gray map 3. The λ -cyclicity 4. An example 5. Linearity H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Introduction In “ Z p k +1 -Linear Codes´´ (Ling-blackford), IEEE Trans. Inf. Theory, vol.48, pp. 2592-2605, (2002), it is shown that a code C ⊂ Z n p 2 is (1 − p )-cyclic if and only if its Gray image is a quasi-cyclic code over Z p . Also, in “The Z 4 -linearity of Kerdock, Preparata, Goethals and related codes´´ (Hammond et al.), IEEE Trans. Inf. Theory, vol.40, pp. 301-319, (1994), necessary and sufficient conditions for the Gray image of a Z 4 -linear code to be linear are given. H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Introduction In “ Z p k +1 -Linear Codes´´ (Ling-blackford), IEEE Trans. Inf. Theory, vol.48, pp. 2592-2605, (2002), it is shown that a code C ⊂ Z n p 2 is (1 − p )-cyclic if and only if its Gray image is a quasi-cyclic code over Z p . Also, in “The Z 4 -linearity of Kerdock, Preparata, Goethals and related codes´´ (Hammond et al.), IEEE Trans. Inf. Theory, vol.40, pp. 301-319, (1994), necessary and sufficient conditions for the Gray image of a Z 4 -linear code to be linear are given. Since the ring Z Z p 2 , in particular if p = 2, is a special case of the Galois ring GR ( p 2 , m ), it is a natural question to ask if similar results as the mentioned above hold for this kind of Galois rings. H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Introduction In “ Z p k +1 -Linear Codes´´ (Ling-blackford), IEEE Trans. Inf. Theory, vol.48, pp. 2592-2605, (2002), it is shown that a code C ⊂ Z n p 2 is (1 − p )-cyclic if and only if its Gray image is a quasi-cyclic code over Z p . Also, in “The Z 4 -linearity of Kerdock, Preparata, Goethals and related codes´´ (Hammond et al.), IEEE Trans. Inf. Theory, vol.40, pp. 301-319, (1994), necessary and sufficient conditions for the Gray image of a Z 4 -linear code to be linear are given. Since the ring Z Z p 2 , in particular if p = 2, is a special case of the Galois ring GR ( p 2 , m ), it is a natural question to ask if similar results as the mentioned above hold for this kind of Galois rings. In this talk an answer to these question is given. H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Galois ring and Gray map Z / p n Z Z be the ring of integers modulo p n . An irreducible Let Z Z / p n Z polynomial f ( x ) ∈ (Z Z)[ x ] is basic if its reduction modulo p is irreducible. The Galois ring GR( p n , m ) is defined as: R = GR( p n , m ) = (Z Z / p n Z Z)[ x ] / � f ( x ) � Z / p n Z where f ( x ) ∈ (Z Z)[ x ] is a monic, basic, primitive irreducible polynomial of degree m . H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Galois ring and Gray map Examples of Galois rings include: F p m , GR( p n , 1) = Z Z / p n Z 1. GR( p , m ) = GF( p , m ) = I Z. 2. Let f ( x ) = x 3 + x + 1 ∈ (Z Z / 4Z Z)[ x ] which is a monic, basic, irreducible polynomial over Z Z / 4Z Z. Then GR(2 2 , 3) = (Z Z / 4Z Z)[ x ] / � f ( x ) � . 3. Let g ( x ) = x 3 + 2 x 2 + x − 1 ∈ (Z Z / 4Z Z)[ x ] which is also a monic, basic, irreducible polynomial over Z Z / 4Z Z. Then GR(2 2 , 3) = (Z Z / 4Z Z)[ x ] / � g ( x ) � . 4. Let h ( x ) = x 2 + 4 x + 8 ∈ (Z Z / 9Z Z)[ x ] which is a monic, basic, irreducible polynomial over Z Z / 9Z Z. Then GR(3 2 , 2) = (Z Z / 9Z Z)[ x ] / � h ( x ) � . H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Galois ring and Gray map Some properties: 1. ( R , M = � p � ) is local. 2. Its residue field I F = R / M is isomorphic to I F p m . 3. |R| = p nm and M = { zero − divisors } of R . 4. Ideals of R : {� p i � for 1 ≤ i ≤ n } . 5. R is a (finite) chain ring: R = � 1 � ⊃ � p � ⊃ · · · ⊃ � p n � = { 0 } . H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Galois ring and Gray map Let q = p m . For A ∈ R , A = r 0 ( A ) + pr 1 ( A ), with r j ( A ) ∈ T (the Teichm¨ uller set of representatives of R ), let a j = µ ( r j ( A )) ∈ I F and let ω be a primitive element of the residue field I F . The Gray map on the Galois ring GR ( p 2 , m ) is defined as: F q q , ϕ ( a ) = ( a 1 , a 1 + a 0 , a 1 + a 0 ω, ..., a 1 + a 0 ω q − 2 ) ϕ : R − → I H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

Galois ring and Gray map Let q = p m . For A ∈ R , A = r 0 ( A ) + pr 1 ( A ), with r j ( A ) ∈ T (the Teichm¨ uller set of representatives of R ), let a j = µ ( r j ( A )) ∈ I F and let ω be a primitive element of the residue field I F . The Gray map on the Galois ring GR ( p 2 , m ) is defined as: F q q , ϕ ( a ) = ( a 1 , a 1 + a 0 , a 1 + a 0 ω, ..., a 1 + a 0 ω q − 2 ) ϕ : R − → I A direct consequence of the definition of the Gray map is: if A is any element of R , then: ϕ ( pA ) = ( A , A , ..., A ) H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

The Gray map If n is a natural number, the Gray map can be extended to R n coordinate-wise: Φ : R n − F qn , Φ( A ) = ( ϕ ( A 0 ) , ..., ϕ ( A n − 1 )) → I where A = ( A 0 , ..., A n − 1 ). H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

The Gray map If n is a natural number, the Gray map can be extended to R n coordinate-wise: Φ : R n − F qn , Φ( A ) = ( ϕ ( A 0 ) , ..., ϕ ( A n − 1 )) → I where A = ( A 0 , ..., A n − 1 ). This map Φ has several properties of which one of the most important is that it is an isometry between ( R n , d h ) and ( I F qn , d H ) where d h is the homogeneous distance on R n and d H is the F qn . Hamming distance on I H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

The λ -cyclicity Let λ = 1 − p ∈ U ( R ) and let ν λ : R n − → R n , ν λ ( A ) = ( λ A n − 1 , A 1 , ..., A n − 2 ) where A = ( A 0 , A 1 , ..., A n − 1 ). H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

The λ -cyclicity Let λ = 1 − p ∈ U ( R ) and let ν λ : R n − → R n , ν λ ( A ) = ( λ A n − 1 , A 1 , ..., A n − 2 ) where A = ( A 0 , A 1 , ..., A n − 1 ). With the notation as above we would like to show that Φ ◦ ν λ = σ ⊗ np ◦ Φ H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

The case GR (3 2 , 2) To see how things work, an example is given. H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code

The case GR (3 2 , 2) To see how things work, an example is given. Z 9 [ x ] / � x 2 + x + 8 � , { 1 , ω } be basis for the Let R = GR (3 2 , 2) = Z residue field I F = I F 9 of R over I F 3 and Ω = (1 , ω ). H. Tapia-Recillas (*)/ C.A. L´ opez-Andrade U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) On the quasi-cyclicity and linearity of the Gray image of a code