On the quasi-cyclicity and linearity of the Gray image of a code over a Galois ring

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

([email protected] / [email protected]) XVIII Col´oquio Latino-Americano de ´Algebra

Sao Pedro, Brasil August, 2009

Abstract

IfR=GR(p²,m) is a Galois ring, Φ the Gray map on Rⁿ and C ⊆ Rⁿ is a λ-cyclic code of length n, it is shown that its Gray image Φ(C) is quasi-cyclic of indexnp over the residue field of R.

Also, ifC is linear, necessary and sufficient conditions for the code Φ(C) to be linear are given.

C O N T E N T

1. Introduction

2. Galois ring and Gray map 3. The λ-cyclicity

4. An example 5. Linearity

Introduction

In “Z_pk+1-Linear Codes´´ (Ling-blackford), IEEE Trans. Inf.

Theory, vol.48, pp. 2592-2605, (2002), it is shown that a code C ⊂Zⁿ_p2 is (1−p)-cyclic if and only if its Gray image is a quasi-cyclic code overZ_p. Also, in “The Z₄-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 aZ₄-linear code to be linear are given.

Since the ring ZZ_p², in particular if p = 2, is a special case of the Galois ringGR(p²,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.

Introduction

In “Z_pk+1-Linear Codes´´ (Ling-blackford), IEEE Trans. Inf.

Theory, vol.48, pp. 2592-2605, (2002), it is shown that a code C ⊂Zⁿ_p2 is (1−p)-cyclic if and only if its Gray image is a quasi-cyclic code overZ_p. Also, in “The Z₄-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 aZ₄-linear code to be linear are given.

Since the ring ZZ_p², in particular if p = 2, is a special case of the Galois ringGR(p²,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.

Introduction

In “Z_pk+1-Linear Codes´´ (Ling-blackford), IEEE Trans. Inf.

Theory, vol.48, pp. 2592-2605, (2002), it is shown that a code C ⊂Zⁿ_p2 is (1−p)-cyclic if and only if its Gray image is a quasi-cyclic code overZ_p. Also, in “The Z₄-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 aZ₄-linear code to be linear are given.

Since the ring ZZ_p², in particular if p = 2, is a special case of the Galois ringGR(p²,m), it is a natural question to ask if similar results as the mentioned above hold for this kind of Galois rings.

Galois ring and Gray map

Let ZZ/pⁿZZ be the ring of integers modulo pⁿ. An irreducible polynomialf(x)∈(ZZ/pⁿZZ)[x] is basicif its reduction modulop is irreducible. The Galois ring GR(pⁿ,m) is defined as:

R= GR(pⁿ,m) = (ZZ/pⁿZZ)[x]/hf(x)i

wheref(x)∈(ZZ/pⁿZZ)[x] is a monic, basic, primitive irreducible polynomial of degreem.

Galois ring and Gray map

Examples of Galois rings include:

1. GR(p,m) = GF(p,m) =IF_p^m, GR(pⁿ,1) = ZZ/pⁿZZ.

2. Let f(x) =x³+x+ 1∈(ZZ/4ZZ)[x] which is a monic, basic, irreducible polynomial over ZZ/4ZZ. Then

GR(2²,3) = (ZZ/4ZZ)[x]/hf(x)i.

3. Let g(x) =x³+ 2x²+x−1∈(ZZ/4ZZ)[x] which is also a monic, basic, irreducible polynomial over ZZ/4ZZ. Then GR(2²,3) = (ZZ/4ZZ)[x]/hg(x)i.

4. Let h(x) =x²+ 4x+ 8∈(ZZ/9ZZ)[x] which is a monic, basic, irreducible polynomial over ZZ/9ZZ. Then

GR(3²,2) = (ZZ/9ZZ)[x]/hh(x)i.

Galois ring and Gray map

Some properties:

1. (R,M=hpi) is local.

2. Its residue field IF=R/Mis isomorphic toIF_p^m. 3. |R|=p^nm andM={zero−divisors} of R.

4. Ideals of R:{hpⁱi for 1≤i ≤n}.

5. R is a (finite) chain ring:

R=h1i ⊃ hpi ⊃ · · · ⊃ hpⁿi={0}.

Galois ring and Gray map

Letq =p^m. For A∈ R,A=r0(A) +pr1(A), withrj(A)∈ T (the Teichm¨uller set of representatives of R), leta_j =µ(r_j(A))∈IFand letω be a primitive element of the residue field IF. The Gray map on the Galois ringGR(p²,m) is defined as:

ϕ:R −→IF^q_q, ϕ(a) = (a₁,a₁+a₀,a₁+a₀ω, ...,a₁+a₀ω^q−2)

A direct consequence of the definition of the Gray map is: ifA is any element ofR, then:

ϕ(pA) = (A,A, ...,A)

Galois ring and Gray map

Letq =p^m. For A∈ R,A=r0(A) +pr1(A), withrj(A)∈ T (the Teichm¨uller set of representatives of R), leta_j =µ(r_j(A))∈IFand letω be a primitive element of the residue field IF. The Gray map on the Galois ringGR(p²,m) is defined as:

ϕ:R −→IF^q_q, ϕ(a) = (a₁,a₁+a₀,a₁+a₀ω, ...,a₁+a₀ω^q−2) A direct consequence of the definition of the Gray map is: ifA is any element ofR, then:

ϕ(pA) = (A,A, ...,A)

The Gray map

Ifn is a natural number, the Gray map can be extended toRⁿ coordinate-wise:

Φ :Rⁿ−→IF^qn, Φ(A) = (ϕ(A0), ..., ϕ(An−1)) whereA= (A₀, ...,An−1).

This map Φ has several properties of which one of the most important is that it is an isometry between (Rⁿ,d_h) and (IF^qn,d_H) whered_h is the homogeneous distance onRⁿ andd_H is the Hamming distance onIF^qn.

The Gray map

Ifn is a natural number, the Gray map can be extended toRⁿ coordinate-wise:

Φ :Rⁿ−→IF^qn, Φ(A) = (ϕ(A0), ..., ϕ(An−1)) whereA= (A₀, ...,An−1).

This map Φ has several properties of which one of the most important is that it is an isometry between (Rⁿ,d_h) and (IF^qn,d_H) whered_h is the homogeneous distance onRⁿ andd_H is the Hamming distance onIF^qn.

The λ-cyclicity

Letλ= 1−p∈U(R) and let

ν_λ :Rⁿ−→ Rⁿ, ν_λ(A) = (λAn−1,A₁, ...,An−2) whereA= (A₀,A₁, ...,An−1).

With the notation as above we would like to show that Φ◦ν_λ =σ^⊗np◦Φ

The λ-cyclicity

Letλ= 1−p∈U(R) and let

ν_λ :Rⁿ−→ Rⁿ, ν_λ(A) = (λAn−1,A₁, ...,An−2) whereA= (A₀,A₁, ...,An−1).

With the notation as above we would like to show that Φ◦ν_λ =σ^⊗np◦Φ

The case GR (3

, 2)

To see how things work, an example is given.

LetR =GR(3²,2) = ZZ₉[x]/hx²+x+ 8i,{1, ω}be basis for the residue fieldIF=IF₉ of R overIF₃ and Ω = (1, ω).

For 0≤k ≤8 let k=k0+ 3k1, 0≤k0,k1 ≤2 be the 3-adic expression ofk. Write:kΩ =k_o +k₁ω. The elements of IFare taken as:

C₀ = (0Ω,1Ω,2Ω,3Ω,4Ω,5Ω,6Ω,7Ω,8Ω)

LetM be the matrix whose rows are C0 and the all one vector 1,

The case GR (3

, 2)

To see how things work, an example is given.

LetR=GR(3²,2) = ZZ₉[x]/hx²+x+ 8i,{1, ω}be basis for the residue fieldIF=IF₉ of R overIF₃ and Ω = (1, ω).

For 0≤k ≤8 let k=k0+ 3k1, 0≤k0,k1 ≤2 be the 3-adic expression ofk. Write:kΩ =k_o +k₁ω. The elements of IFare taken as:

C₀ = (0Ω,1Ω,2Ω,3Ω,4Ω,5Ω,6Ω,7Ω,8Ω)

LetM be the matrix whose rows are C0 and the all one vector 1,

The case GR (3

, 2)

To see how things work, an example is given.

LetR=GR(3²,2) = ZZ₉[x]/hx²+x+ 8i,{1, ω}be basis for the residue fieldIF=IF₉ of R overIF₃ and Ω = (1, ω).

For 0≤k ≤8 let k=k0+ 3k1, 0≤k0,k1 ≤2 be the 3-adic expression ofk. Write:kΩ =k_o +k₁ω. The elements of IFare taken as:

C₀ = (0Ω,1Ω,2Ω,3Ω,4Ω,5Ω,6Ω,7Ω,8Ω)

LetM be the matrix whose rows are C0 and the all one vector1,

The case GR (3

, 2), cont.

In this exampleλ= 1−p =−2 = 7 and letn = 4. If

Ai =r0(Ai) + 3r1(Ai)∈ Rwith rj(Ai)∈ T, the Teichmuller set, let a_ij =µ(r_j(A_i))∈IF. Then Φ(A) = (Φ(A₀),Φ(A₁),Φ(A₂),Φ(A₃)) can be taken as the concatenation of the rows of following array:

The case GR (3

, 2), cont.

Φ(A) =

a01+ 0Ωa00 a11+ 0Ωa10 a21+ 0Ωa20 a31+ 0Ωa30

a₀₁+ 1Ωa₀₀ a₁₁+ 1Ωa₁₀ a₂₁+ 1Ωa₂₀ a₃₁+ 1Ωa₃₀ a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ a₃₁+ 2Ωa₃₀ a₀₁+ 3Ωa₀₀ a₁₁+ 3Ωa₁₀ a₂₁+ 3Ωa₂₀ a₃₁+ 3Ωa₃₀ a01+ 4Ωa00 a11+ 4Ωa10 a21+ 4Ωa20 a31+ 4Ωa30

a₀₁+ 5Ωa₀₀ a₁₁+ 5Ωa₁₀ a₂₁+ 5Ωa₂₀ a₃₁+ 5Ωa₃₀ a01+ 6Ωa00 a11+ 6Ωa10 a21+ 6Ωa20 a31+ 6Ωa30

a₀₁+ 7Ωa₀₀ a₁₁+ 7Ωa₁₀ a₂₁+ 7Ωa₂₀ a₃₁+ 7Ωa₃₀ a01+ 8Ωa00 a11+ 8Ωa10 a21+ 8Ωa20 a31+ 8Ωa30

The case GR (3

, 2), cont.

In this caseλA₃=a₃₁+ 2a₃₀. For 0≤j,k ≤8 letj = (j₀,j₁), k= (k₀,k₁) (the coefficients of the 3-adic representation). Then jΩ +kΩ = (j0+k0,j1+k1)Ω (each coordinate is reduced modulo 3). A direct calculation shows that

Φ(ν_λ(A)) = (Φ(λA₃),Φ(A₀),Φ(A₁),Φ(A₂)) is the concatenation of the rows of the following array:

The case GR (3

, 2), cont.

Φ(νλ(A)) =

a31+ 2Ωa30 a01+ 0Ωa00 a11+ 0Ωa10 a21+ 1Ωa20

a₃₁+ 0Ωa₃₀ a₀₁+ 1Ωa₀₀ a₁₁+ 1Ωa₁₀ a₂₁+ 1Ωa₂₀ a₃₁+ 1Ωa₃₀ a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ a₃₁+ 5Ωa₃₀ a₀₁+ 3Ωa₀₀ a₁₁+ 3Ωa₁₀ a₂₁+ 3Ωa₂₀ a31+ 3Ωa30 a01+ 4Ωa00 a11+ 4Ωa10 a21+ 4Ωa20

a₃₁+ 4Ωa₃₀ a₀₁+ 5Ωa₀₀ a₁₁+ 5Ωa₁₀ a₂₁+ 5Ωa₂₀ a31+ 8Ωa30 a01+ 6Ωa00 a11+ 6Ωa10 a21+ 6Ωa20

a₃₁+ 6Ωa₃₀ a₀₁+ 7Ωa₀₀ a₁₁+ 7Ωa₁₀ a₂₁+ 7Ωa₂₀ a31+ 7Ωa30 a01+ 8Ωa00 a11+ 8Ωa10 a21+ 8Ωa20

The case GR (3

, 2), cont.

We look at the first three rows of the array corresponding to Φ(A):

a₀₁+ 0Ωa₀₀ a₁₁+ 0Ωa₁₀ a₂₁+ 0Ωa₂₀ a₃₁+ 0Ωa₃₀ a01+ 1Ωa00 a11+ 1Ωa10 a21+ 1Ωa20 a31+ 1Ωa30

a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ a₃₁+ 2Ωa₃₀

and the first three rows of the array corresponding to Φ(ν_λ(A)) = a₃₁+ 2Ωa₃₀ a₀₁+ 0Ωa₀₀ a₁₁+ 0Ωa₁₀ a₂₁+ 1Ωa₂₀ a31+ 0Ωa30 a01+ 1Ωa00 a11+ 1Ωa10 a21+ 1Ωa20

a₃₁+ 1Ωa₃₀ a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ Observe that if the usual shiftσ is applied to the first array the second one is obtained.

The case GR (3

, 2), cont.

We look at the first three rows of the array corresponding to Φ(A):

a₀₁+ 0Ωa₀₀ a₁₁+ 0Ωa₁₀ a₂₁+ 0Ωa₂₀ a₃₁+ 0Ωa₃₀ a01+ 1Ωa00 a11+ 1Ωa10 a21+ 1Ωa20 a31+ 1Ωa30

a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ a₃₁+ 2Ωa₃₀ and the first three rows of the array corresponding to Φ(ν_λ(A)) =

a₃₁+ 2Ωa₃₀ a₀₁+ 0Ωa₀₀ a₁₁+ 0Ωa₁₀ a₂₁+ 1Ωa₂₀ a31+ 0Ωa30 a01+ 1Ωa00 a11+ 1Ωa10 a21+ 1Ωa20

a₃₁+ 1Ωa₃₀ a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀

Observe that if the usual shiftσ is applied to the first array the second one is obtained.

The case GR (3

, 2), cont.

We look at the first three rows of the array corresponding to Φ(A):

a₀₁+ 0Ωa₀₀ a₁₁+ 0Ωa₁₀ a₂₁+ 0Ωa₂₀ a₃₁+ 0Ωa₃₀ a01+ 1Ωa00 a11+ 1Ωa10 a21+ 1Ωa20 a31+ 1Ωa30

a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ a₃₁+ 2Ωa₃₀ and the first three rows of the array corresponding to Φ(ν_λ(A)) =

a₃₁+ 2Ωa₃₀ a₀₁+ 0Ωa₀₀ a₁₁+ 0Ωa₁₀ a₂₁+ 1Ωa₂₀ a31+ 0Ωa30 a01+ 1Ωa00 a11+ 1Ωa10 a21+ 1Ωa20

a₃₁+ 1Ωa₃₀ a₀₁+ 2Ωa₀₀ a₁₁+ 2Ωa₁₀ a₂₁+ 2Ωa₂₀ Observe that if the usual shiftσ is applied to the first array the

The case GR (3

, 2), cont.

Doing the same for the next two blocks consisting of 3 rows each of the array corresponding to Φ(A) and Φ(νλ(A)) it follows that,

Φ(νλ(A)) =σ^⊗12(Φ(A))

We have the following, Proposition

With the notation as above,

Φ◦ν_λ =σ^⊗np◦Φ

The case GR (3

, 2), cont.

Doing the same for the next two blocks consisting of 3 rows each of the array corresponding to Φ(A) and Φ(νλ(A)) it follows that,

Φ(νλ(A)) =σ^⊗12(Φ(A)) We have the following,

Proposition

With the notation as above,

Φ◦ν_λ =σ^⊗np◦Φ

Quasi-cyclicity

Definition.A code C ⊆ Rⁿ is called λ-cyclic if νλ(C) =C. A codeD⊆IF^m is called cyclic of indexs ifσ^⊗s(D) =D.

We have the following. Theorem

A code C ⊆Rⁿ is λ-cyclic if and only if its Gray image Φ(C)is cyclic of index np.

Idea of Proof:

It follows directly from the above relation and the fact that the Gray map is injective.

Quasi-cyclicity

Definition.A code C ⊆ Rⁿ is called λ-cyclic if νλ(C) =C. A codeD⊆IF^m is called cyclic of indexs ifσ^⊗s(D) =D.

We have the following.

Theorem

A code C ⊆Rⁿ is λ-cyclic if and only if its Gray image Φ(C)is cyclic of index np.

Idea of Proof:

It follows directly from the above relation and the fact that the Gray map is injective.

Quasi-cyclicity

Definition.A code C ⊆ Rⁿ is called λ-cyclic if νλ(C) =C. A codeD⊆IF^m is called cyclic of indexs ifσ^⊗s(D) =D.

We have the following.

Theorem

A code C ⊆Rⁿ is λ-cyclic if and only if its Gray image Φ(C)is cyclic of index np.

Idea of Proof:

It follows directly from the above relation and the fact that the Gray map is injective.

Linearity

The Witt ring W₂(IF).

Let (IF,+,∗) = (IF_p^m,+,∗): finite field with p^m elements.

As a set,W₂(IF) =IF×IF. The operations “+w”, “∗_w” are: (x0,x1) +w(y0,y1) = (S0(x0,x1,y0,y1),S1(x0,x1,y0,x1)) where

S0(x0,x1,y0,y1) =x0+y0

S₁(x₀,x₁,y₀,y₁) =

(x₁+y₁)−¹_p((x₀+y₀)^p−x₀^p−y₀^p) and

(x₀,x₁)∗_w(y₀,y₁) = (x₀y₀,x₀^py₁+y₀^px₁) (fora,b∈IFwe write a∗b=ab).

Linearity

The Witt ring W₂(IF).

Let (IF,+,∗) = (IF_p^m,+,∗): finite field with p^m elements.

As a set,W₂(IF) =IF×IF.

The operations “+w”, “∗_w” are: (x0,x1) +w(y0,y1) = (S0(x0,x1,y0,y1),S1(x0,x1,y0,x1)) where

S0(x0,x1,y0,y1) =x0+y0

S₁(x₀,x₁,y₀,y₁) =

(x₁+y₁)−¹_p((x₀+y₀)^p−x₀^p−y₀^p) and

(x₀,x₁)∗_w(y₀,y₁) = (x₀y₀,x₀^py₁+y₀^px₁) (fora,b∈IFwe write a∗b=ab).

Linearity

The Witt ring W₂(IF).

Let (IF,+,∗) = (IF_p^m,+,∗): finite field with p^m elements.

As a set,W₂(IF) =IF×IF. The operations “+w”, “∗_w” are:

(x0,x1) +w(y0,y1) = (S0(x0,x1,y0,y1),S1(x0,x1,y0,x1)) where

S0(x0,x1,y0,y1) =x0+y0

S₁(x₀,x₁,y₀,y₁) =

(x₁+y₁)−¹_p((x₀+y₀)^p−x₀^p−y₀^p) and

(x₀,x₁)∗_w(y₀,y₁) = (x₀y₀,x₀^py₁+y₀^px₁) (fora,b∈IFwe write a∗b=ab).

Linearity

The Witt ring W₂(IF).

Let (IF,+,∗) = (IF_p^m,+,∗): finite field with p^m elements.

As a set,W₂(IF) =IF×IF. The operations “+w”, “∗_w” are:

(x0,x1) +w(y0,y1) = (S0(x0,x1,y0,y1),S1(x0,x1,y0,x1)) where

S0(x0,x1,y0,y1) =x0+y0

S₁(x₀,x₁,y₀,y₁) =

(x₁+y₁)−¹_p((x₀+y₀)^p−x₀^p−y₀^p)

and

(x₀,x₁)∗_w(y₀,y₁) = (x₀y₀,x₀^py₁+y₀^px₁) (fora,b∈IFwe write a∗b=ab).

Linearity

The Witt ring W₂(IF).

Let (IF,+,∗) = (IF_p^m,+,∗): finite field with p^m elements.

As a set,W₂(IF) =IF×IF. The operations “+w”, “∗_w” are:

(x0,x1) +w(y0,y1) = (S0(x0,x1,y0,y1),S1(x0,x1,y0,x1)) where

S0(x0,x1,y0,y1) =x0+y0

S₁(x₀,x₁,y₀,y₁) =

(x₁+y₁)−¹_p((x₀+y₀)^p−x₀^p−y₀^p) and

Linearity

An isomorphism between the Galois and Witt rings LetIF=R/M: residue field of the Galois ring R.

The mapping

ψ:R −→ W₂(IF), ˆa=ψ(a) = (a₀,a^p₁) (1) wherea=a₀+a₁p∈ R,a₀,a₁∈ T, is a ring isomorphism.

Its inverse is:

ψ⁻¹ :W₂(IF)−→ R, ψ⁻¹ b₀,b₁

=B₀+pB₁^1/p (2) whereB₀,B₁ ∈ T are such thatB_i =b_i (the bar means the image under the canonical mappingµ).

Linearity

An isomorphism between the Galois and Witt rings LetIF=R/M: residue field of the Galois ring R.

The mapping

ψ:R −→ W₂(IF), ˆa=ψ(a) = (a₀,a^p₁) (1) wherea=a₀+a₁p∈ R,a₀,a₁∈ T, is a ring isomorphism.

Its inverse is:

ψ⁻¹ :W₂(IF)−→ R, ψ⁻¹ b₀,b₁

=B₀+pB₁^1/p (2) whereB₀,B₁ ∈ T are such thatB_i =b_i (the bar means the image under the canonical mappingµ).

Linearity

An isomorphism between the Galois and Witt rings LetIF=R/M: residue field of the Galois ring R.

The mapping

ψ:R −→ W₂(IF), ˆa=ψ(a) = (a₀,a^p₁) (1) wherea=a₀+a₁p∈ R,a₀,a₁∈ T, is a ring isomorphism.

Its inverse is:

ψ⁻¹ :W₂(IF)−→ R, ψ⁻¹ b₀,b₁

=B₀+pB₁^1/p (2) whereB ,B ∈ T are such thatB =b (the bar means the image

Linearity

Let

h(x,y) = 1

p((x+y)^p−x^p−y^p).

Ifa,b∈ R with a=r₀(a) +pr₁(a),b =r₀(b) +pr₁(b), r_i(x)∈ T, by means of the previous isomorphism it can be seen that

r₀(a+b) =r₀(a) +r₀(b) and

(r1(a+b))^p=r1(a) +r1(b)−h(r0(a),r0(b))

Linearity

Let

h(x,y) = 1

p((x+y)^p−x^p−y^p).

Ifa,b∈ R with a=r₀(a) +pr₁(a),b =r₀(b) +pr₁(b), r_i(x)∈ T, by means of the previous isomorphism it can be seen that

r₀(a+b) =r₀(a) +r₀(b) and

(r1(a+b))^p=r1(a) +r1(b)−h(r0(a),r0(b))

Linearity

For any two elementsA=r₀(A) +r₁(A)p,B =r₀(B) +r₁(B)p of Rletaj =µ(rj(A)),bj =µ(rj(B)),j = 0,1 and let

Θ(A,B) = [a1+b1−h(a0,b0]^1p, whereh(x,y) is the polynomial introduced above.

We have the following,

Linearity

Proposition

Letϕbe the Gray map on R. With the notation as above, for any two elements A,B ∈ R:

ϕ(A) +ϕ(B)−ϕ(A+B) =ϕ(pΩ(r₀(A),r₀(B))) where

Ω(A,B) =a₁+b₁−Θ(A,B).

Linearity

A direct consequence of the previous Proposition is the following,

Corollary

Let A= (A0,A1, ...,An−1),B = (B0,B1, ...,Bn−1)∈ Rⁿ and let Φ :Rⁿ−→IF^nq be the Gray map onRⁿ. Then,

Φ(A) + Φ(B)−Φ(A+B) = Φ(pΩ(r0(A),r0(B)) where

Ω(r₀(A),r₀(B)) = (Ω(r₀(A₀),r₀(B₀)), ...,Ω(r₀(An−1),r₀(Bn−1))).

Linearity

A direct consequence of the previous Proposition is the following, Corollary

Let A= (A0,A1, ...,An−1),B = (B0,B1, ...,Bn−1)∈ Rⁿ and let Φ :Rⁿ−→IF^nq be the Gray map on Rⁿ. Then,

Φ(A) + Φ(B)−Φ(A+B) = Φ(pΩ(r0(A),r0(B)) where

Ω(r₀(A),r₀(B)) = (Ω(r₀(A₀),r₀(B₀)), ...,Ω(r₀(An−1),r₀(Bn−1))).

Linearity

Now we have the following,

Theorem

LetC ⊂ Rⁿ be a R-linear code of length n and let Φbe the Gray map onRⁿ. Then the Gray imageΦ(C) is a IF-linear code if and only if for all A,B∈ C, pΩ(r₀(A),r0(B))∈ C.

Linearity

Now we have the following, Theorem

LetC ⊂ Rⁿ be a R-linear code of length n and let Φbe the Gray map onRⁿ. Then the Gray imageΦ(C) is a IF-linear code if and only if for all A,B∈ C, pΩ(r₀(A),r0(B))∈ C.

Linearity

Observation.In the classical case, i.e., p= 2, the residue field is IF₂, then,

Ω(a,b) =a₀b₀

ϕ(a) +ϕ(b)−ϕ(a+b) =ϕ(2a₀b₀)

and the relation

Φ(A) + Φ(B) + Φ(A+B) = Φ(2r₀(A)∗r₀(B)) given in Hammond et al. is recovered.

Linearity

Observation.In the classical case, i.e., p= 2, the residue field is IF₂, then,

Ω(a,b) =a₀b₀

ϕ(a) +ϕ(b)−ϕ(a+b) =ϕ(2a₀b₀) and the relation

Φ(A) + Φ(B) + Φ(A+B) = Φ(2r₀(A)∗r₀(B)) given in Hammond et al. is recovered.

Linearity

If the residue fieldIFof the Galois ring R isIFp then, Ω(a,b) =h(a₀,b₀) =Pp−1

i=1 aⁱ₀b^p−i₀ ϕ(a) +ϕ(b)−ϕ(a+b) =ϕ(ph(a0,b0)) and we have the relation

Φ(A) + Φ(B) + Φ(A+B) = Φ(ph(r0(A),r0(B))

Linearity

Now we have the following,

Theorem

LetC ⊂ Rⁿ be a R-linear code of length n and let Φbe the Gray map onRⁿ. Then the Gray imageΦ(C) is a IF-linear code if and only if for all A,B∈ C, pΩ(r₀(A),r0(B))∈ C.

Linearity

Now we have the following, Theorem

LetC ⊂ Rⁿ be a R-linear code of length n and let Φbe the Gray map onRⁿ. Then the Gray imageΦ(C) is a IF-linear code if and only if for all A,B∈ C, pΩ(r₀(A),r0(B))∈ C.

T H A N K Y O U ! !

T H A N K Y O U ! !

Monte Alban, Oaxaca, M´exico