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´
• pez-Andrade

U.A.Metropolitana-I / U.A. Puebla MEXICO (htr@xanum.uam.mx / calopez@cs.buap.mx) XVIII Col´

• quio Latino-Americano de ´

Algebra Sao Pedro, Brasil August, 2009

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(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 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´
• pez-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´
• pez-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 “Zpk+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 over Zp. 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 a Z4-linear code to be linear are given.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 “Zpk+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 over Zp. 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 a Z4-linear code to be linear are given. Since the ring Z Zp2, in particular if p = 2, is a special case of the Galois ring GR(p2, 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´
• pez-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 “Zpk+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 over Zp. 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 a Z4-linear code to be linear are given. Since the ring Z Zp2, in particular if p = 2, is a special case of the Galois ring GR(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.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 Z Z/pnZ Z be the ring of integers modulo pn. An irreducible polynomial f (x) ∈ (Z Z/pnZ Z)[x] is basic if its reduction modulo p is

• irreducible. The Galois ring GR(pn, m) is defined as:

R = GR(pn, m) = (Z Z/pnZ Z)[x]/f (x) where f (x) ∈ (Z Z/pnZ Z)[x] is a monic, basic, primitive irreducible polynomial of degree m.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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:

• 1. GR(p, m) = GF(p, m) = I

Fpm, GR(pn, 1) = Z Z/pnZ Z.

• 2. Let f (x) = x3 + x + 1 ∈ (Z

Z/4Z Z)[x] which is a monic, basic, irreducible polynomial over Z Z/4Z

• Z. Then

GR(22, 3) = (Z Z/4Z Z)[x]/f (x).

• 3. Let g(x) = x3 + 2x2 + x − 1 ∈ (Z

Z/4Z Z)[x] which is also a monic, basic, irreducible polynomial over Z Z/4Z

• Z. Then

GR(22, 3) = (Z Z/4Z Z)[x]/g(x).

• 4. Let h(x) = x2 + 4x + 8 ∈ (Z

Z/9Z Z)[x] which is a monic, basic, irreducible polynomial over Z Z/9Z

• Z. Then

GR(32, 2) = (Z Z/9Z Z)[x]/h(x).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 Fpm.

• 3. |R| = pnm and M = {zero − divisors} of R.
• 4. Ideals of R: {pi for 1 ≤ i ≤ n}.
• 5. R is a (finite) chain ring:

R = 1 ⊃ p ⊃ · · · ⊃ pn = {0}.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 = pm. For A ∈ R, A = r0(A) + pr1(A), with rj(A) ∈ T (the Teichm¨ uller set of representatives of R), let aj = µ(rj(A)) ∈ I F and let ω be a primitive element of the residue field I

• F. The Gray map
• n the Galois ring GR(p2, m) is defined as:

ϕ : R − → I Fq

q, ϕ(a) = (a1, a1 + a0, a1 + a0ω, ..., a1 + a0ωq−2)

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 = pm. For A ∈ R, A = r0(A) + pr1(A), with rj(A) ∈ T (the Teichm¨ uller set of representatives of R), let aj = µ(rj(A)) ∈ I F and let ω be a primitive element of the residue field I

• F. The Gray map
• n the Galois ring GR(p2, m) is defined as:

ϕ : R − → I Fq

q, ϕ(a) = (a1, a1 + a0, a1 + a0ω, ..., a1 + a0ωq−2)

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´
• pez-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 Rn coordinate-wise: Φ : Rn − → I Fqn, Φ(A) = (ϕ(A0), ..., ϕ(An−1)) where A = (A0, ..., An−1).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 Rn coordinate-wise: Φ : Rn − → I Fqn, Φ(A) = (ϕ(A0), ..., ϕ(An−1)) where A = (A0, ..., An−1). This map Φ has several properties of which one of the most important is that it is an isometry between (Rn, dh) and (I Fqn, dH) where dh is the homogeneous distance on Rn and dH is the Hamming distance on I Fqn.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 νλ : Rn − → Rn, νλ(A) = (λAn−1, A1, ..., An−2) where A = (A0, A1, ..., An−1).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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 νλ : Rn − → Rn, νλ(A) = (λAn−1, A1, ..., An−2) where A = (A0, A1, ..., An−1). With the notation as above we would like to show that Φ ◦ νλ = σ⊗np ◦ Φ

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2)

To see how things work, an example is given.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2)

To see how things work, an example is given. Let R = GR(32, 2) = Z Z9[x]/x2 + x + 8, {1, ω} be basis for the residue field I F = I F9 of R over I F3 and Ω = (1, ω).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2)

To see how things work, an example is given. Let R = GR(32, 2) = Z Z9[x]/x2 + x + 8, {1, ω} be basis for the residue field I F = I F9 of R over I F3 and Ω = (1, ω). For 0 ≤ k ≤ 8 let k = k0 + 3k1, 0 ≤ k0, k1 ≤ 2 be the 3-adic expression of k. Write: kΩ = ko + k1ω. The elements of I F are taken as: C0 = (0Ω, 1Ω, 2Ω, 3Ω, 4Ω, 5Ω, 6Ω, 7Ω, 8Ω) Let M be the matrix whose rows are C0 and the all one vector 1,

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2), cont.

In this example λ = 1 − p = −2 = 7 and let n = 4. If Ai = r0(Ai) + 3r1(Ai) ∈ R with rj(Ai) ∈ T , the Teichmuller set, let aij = µ(rj(Ai)) ∈ I

• F. Then Φ(A) = (Φ(A0), Φ(A1), Φ(A2), Φ(A3))

can be taken as the concatenation of the rows of following array:

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 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

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2), cont.

In this case λA3 = a31 + 2a30. For 0 ≤ j, k ≤ 8 let j = (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(νλ(A)) = (Φ(λA3), Φ(A0), Φ(A1), Φ(A2)) is the concatenation

• f the rows of the following array:
• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 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

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 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

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 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(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

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 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(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 Observe that if the usual shift σ is applied to the first array the second one is obtained.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2), cont.

Doing the same for the next two blocks consisting of 3 rows each

• f the array corresponding to Φ(A) and Φ(νλ(A)) it follows that,

Φ(νλ(A)) = σ⊗12(Φ(A))

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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(32, 2), cont.

Doing the same for the next two blocks consisting of 3 rows each

• f 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 ◦ νλ = σ⊗np ◦ Φ

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Quasi-cyclicity

• Definition. A code C ⊆ Rn is called λ-cyclic if νλ(C) = C. A

code D ⊆ I Fm is called cyclic of index s if σ⊗s(D) = D.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Quasi-cyclicity

• Definition. A code C ⊆ Rn is called λ-cyclic if νλ(C) = C. A

code D ⊆ I Fm is called cyclic of index s if σ⊗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.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Quasi-cyclicity

• Definition. A code C ⊆ Rn is called λ-cyclic if νλ(C) = C. A

code D ⊆ I Fm is called cyclic of index s if σ⊗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.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

The Witt ring W2(I F). Let (I F, +, ∗) = (I Fpm, +, ∗): finite field with pm elements.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

The Witt ring W2(I F). Let (I F, +, ∗) = (I Fpm, +, ∗): finite field with pm elements. As a set, W2(I F) = I F × I F.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

The Witt ring W2(I F). Let (I F, +, ∗) = (I Fpm, +, ∗): finite field with pm elements. As a set, W2(I F) = I F × I

• F. The operations “+w”, “∗w” are:
• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

The Witt ring W2(I F). Let (I F, +, ∗) = (I Fpm, +, ∗): finite field with pm elements. As a set, W2(I F) = I F × I

• F. 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 − xp 0 − yp 0 )

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

The Witt ring W2(I F). Let (I F, +, ∗) = (I Fpm, +, ∗): finite field with pm elements. As a set, W2(I F) = I F × I

• F. 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 − xp 0 − yp 0 )

• and

(x0, x1) ∗w (y0, y1) = (x0y0, xp

0 y1 + yp 0 x1)

(for a, b ∈ I F we write a ∗ b = ab).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

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

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

An isomorphism between the Galois and Witt rings Let I F = R/M: residue field of the Galois ring R. The mapping ψ : R − → W2 (I F) , ˆ a = ψ(a) = (a0, ap

1)

(1) where a = a0 + a1p ∈ R, a0, a1 ∈ T , is a ring isomorphism.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

An isomorphism between the Galois and Witt rings Let I F = R/M: residue field of the Galois ring R. The mapping ψ : R − → W2 (I F) , ˆ a = ψ(a) = (a0, ap

1)

(1) where a = a0 + a1p ∈ R, a0, a1 ∈ T , is a ring isomorphism. Its inverse is: ψ−1 : W2 (I F) − → R, ψ−1 b0, b1

• = B0 + pB1/p

1

(2) where B0, B1 ∈ T are such that Bi = bi (the bar means the image under the canonical mapping µ).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Let h(x, y) = 1 p((x + y)p − xp − yp).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Let h(x, y) = 1 p((x + y)p − xp − yp). If a, b ∈ R with a = r0(a) + pr1(a), b = r0(b) + pr1(b), ri(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))

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

For any two elements A = r0(A) + r1(A)p, B = r0(B) + r1(B)p of R let aj = µ(rj(A)), bj = µ(rj(B)), j = 0, 1 and let Θ(A, B) = [a1 + b1 − h(a0, b0]

1 p ,

where h(x, y) is the polynomial introduced above. We have the following,

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Proposition

Let ϕ be 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).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

A direct consequence of the previous Proposition is the following,

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

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 − → I Fnq 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))).

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Now we have the following,

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Now we have the following,

Theorem

Let C ⊂ Rn be a R-linear code of length n and let Φ be the Gray map on Rn. Then the Gray image Φ(C) is a I F-linear code if and

• nly if for all A, B ∈ C, pΩ(r0(A), r0(B)) ∈ C.
• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

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

I F2, then, Ω(a, b) = a0b0 ϕ(a) + ϕ(b) − ϕ(a + b) = ϕ(2a0b0)

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

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

I F2, 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.

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

If the residue field I F of the Galois ring R is I Fp then, Ω(a, b) = h(a0, b0) = p−1

i=1 ai 0bp−i

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))

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Now we have the following,

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Linearity

Now we have the following,

Theorem

Let C ⊂ Rn be a R-linear code of length n and let Φ be the Gray map on Rn. Then the Gray image Φ(C) is a I F-linear code if and

• nly if for all A, B ∈ C, pΩ(r0(A), r0(B)) ∈ C.
• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

T H A N K Y O U ! !

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

T H A N K Y O U ! !

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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

Monte Alban, Oaxaca, M´ exico

• H. Tapia-Recillas (*)/ C.A. L´
• pez-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