Polyadic Constacyclic Codes Yun Fan Department of Mathematics, - - PowerPoint PPT Presentation

Polyadic Constacyclic Codes

Yun Fan

Department of Mathematics, Central China Normal University, Wuhan, 430079, CHINA Email: yfan@mail.ccnu.edu.cn Tel: (+86) 15002714631

A joint work with Bocong Chen, Hai Q. Dinh, San Ling

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Abstract

Necessary and sufficient conditions of the existence of polyadic constacyclic codes are reported in terms of p-valuations; the main ideas to solve the question are described. Some consequences are derived, and some examples are exhibited.

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Contents

1

Linear codes, Cyclic Codes

2

Constacyclic Codes

3

Polyadic Constacyclic Codes

4

Main Results

5

Applications

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Linear codes, Cyclic Codes Hamming Weight, Hamming Distance

Fq: finite field with q elements, q a prime power F∗

q: the multiplicative group of nonzero elements of Fq

Fn

q:

=

• (a0,a1,··· ,an−1)
• ai ∈ Fq
• ,

the string a = (a0,a1,··· ,an−1) is called a word over the alphabet Fq

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Linear codes, Cyclic Codes Hamming Weight, Hamming Distance

Fq: finite field with q elements, q a prime power F∗

q: the multiplicative group of nonzero elements of Fq

Fn

q:

=

• (a0,a1,··· ,an−1)
• ai ∈ Fq
• ,

the string a = (a0,a1,··· ,an−1) is called a word over the alphabet Fq

Definition

Hamming weight: w(a) := #{i |0 ≤ i < n, ai = 0}, ∀ a = (a0,a1,··· ,an−1) ∈ Fn

q

Hamming distance: d(a,a′) := w(a−a′), ∀ a,a′ ∈ Fn

q

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Linear Codes

Definition

Any given subspace C of Fn

q is said to be a linear code of length n

• ver the alphabet Fq; any c = (c0,c1,··· ,cn−1) ∈ C is said to be a

codeword.

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Linear Codes

Definition

Any given subspace C of Fn

q is said to be a linear code of length n

• ver the alphabet Fq; any c = (c0,c1,··· ,cn−1) ∈ C is said to be a

codeword. k := dimC, dimension of the linear code d = d(C) := min

c=c′∈C d(c,c′),

the minimum distance of the linear code w(C) := min

0=c∈C w(c),

the minimum weight of the linear code C is said to be an [n,k,d] code.

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Linear Codes

Definition

Any given subspace C of Fn

q is said to be a linear code of length n

• ver the alphabet Fq; any c = (c0,c1,··· ,cn−1) ∈ C is said to be a

codeword. k := dimC, dimension of the linear code d = d(C) := min

c=c′∈C d(c,c′),

the minimum distance of the linear code w(C) := min

0=c∈C w(c),

the minimum weight of the linear code C is said to be an [n,k,d] code.

• Remark. For linear codes, d(C) = w(C).

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What Codes Are Good Codes

Both the dimension and the minimum distance of the code are bigger. However, the dimension and the minimum distance of a code restrict each other. So the question is: how to trade off them.

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What Codes Are Good Codes

Both the dimension and the minimum distance of the code are bigger. However, the dimension and the minimum distance of a code restrict each other. So the question is: how to trade off them. The code has good mathematical structure

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Orthogonality, Self-duality

Euclidean inner product: a,a′ :=

n−1

∑

i=0

aia′

i,

∀ a = (a0,··· ,an−1), a′ = (a′

0,··· ,a′ n−1) ∈ Fn q

Definition

C ≤ Fn

q

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Orthogonality, Self-duality

Euclidean inner product: a,a′ :=

n−1

∑

i=0

aia′

i,

∀ a = (a0,··· ,an−1), a′ = (a′

0,··· ,a′ n−1) ∈ Fn q

Definition

C ≤ Fn

q

C ⊥ :=

• a ∈ Fn

q

• a,c = 0, ∀ c ∈ C
• , is called the orthogonal

code of C.

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Orthogonality, Self-duality

Euclidean inner product: a,a′ :=

n−1

∑

i=0

aia′

i,

∀ a = (a0,··· ,an−1), a′ = (a′

0,··· ,a′ n−1) ∈ Fn q

Definition

C ≤ Fn

q

C ⊥ :=

• a ∈ Fn

q

• a,c = 0, ∀ c ∈ C
• , is called the orthogonal

code of C. C is said to be self-orthogonal if C ⊆ C ⊥.

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Orthogonality, Self-duality

Euclidean inner product: a,a′ :=

n−1

∑

i=0

aia′

i,

∀ a = (a0,··· ,an−1), a′ = (a′

0,··· ,a′ n−1) ∈ Fn q

Definition

C ≤ Fn

q

C ⊥ :=

• a ∈ Fn

q

• a,c = 0, ∀ c ∈ C
• , is called the orthogonal

code of C. C is said to be self-orthogonal if C ⊆ C ⊥. C is said to be self-dual if C = C ⊥.

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Cyclic Codes, Constacyclic Codes

C ≤ Fn

q: a linear code

Definition

A linear code C is called a cyclic code if (cn−1,c0,··· ,cn−2) ∈ C, ∀ (c0,c1,··· ,cn−1) ∈ C

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Cyclic Codes, Constacyclic Codes

C ≤ Fn

q: a linear code

Definition

A linear code C is called a cyclic code if (cn−1,c0,··· ,cn−2) ∈ C, ∀ (c0,c1,··· ,cn−1) ∈ C λ ∈ F∗

q

Definition

A linear code C is called a λ-constacyclic code if (λcn−1,c0,··· ,cn−2) ∈ C, ∀ (c0,c1,··· ,cn−1) ∈ C λ = 1: λ-constacyclic code = cyclic code λ = −1: λ-constacyclic code is called negacyclic code

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Huge literatures on cyclic codes:

BCH codes RS codes Berlekamp Algorithm ···

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Huge literatures on cyclic codes:

BCH codes RS codes Berlekamp Algorithm ··· Quadratic residue codes Duadic codes Polyadic codes ···

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Constacyclic Codes How to Determine Constacyclic Codes

λ ∈ F∗

q

Fq[X]: the polynomial algebra over Fq X n −λ: the ideal of Fq[X] generated by X n −λ Fq[X]/X n −λ: the quotient algebra Fq[X]/X n −λ

linear iso.

− → Fn

q

a0 +a1X +···+an−1X n−1 − → (a0,a1,··· ,an−1)

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Constacyclic Codes How to Determine Constacyclic Codes

λ ∈ F∗

q

Fq[X]: the polynomial algebra over Fq X n −λ: the ideal of Fq[X] generated by X n −λ Fq[X]/X n −λ: the quotient algebra Fq[X]/X n −λ

linear iso.

− → Fn

q

a0 +a1X +···+an−1X n−1 − → (a0,a1,··· ,an−1) C ≤ Fn

q is a λ-constacyclic code if and only if the following set

• c0 +c1X +···+cn−1X n−1

(c0,c1,··· ,cn−1) ∈ C

• is an ideal of Fq[X]/X n −λ.

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Since Fq[X] is a principal ideal ring, one gets that

Proposition

λ-constacyclic codes C over Fq of length n are one to one corresponding to the monomial divisors g(X) of X n −λ such that C =

• f (X)g(X) (mod X n −λ)
• f (X) ∈ Fq[X]
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Since Fq[X] is a principal ideal ring, one gets that

Proposition

λ-constacyclic codes C over Fq of length n are one to one corresponding to the monomial divisors g(X) of X n −λ such that C =

• f (X)g(X) (mod X n −λ)
• f (X) ∈ Fq[X]
• Definition

Let X n −λ = g(X)h(X) and C =

• f (X)g(X) (mod X n −λ)
• f (X) ∈ Fq[X]
• .

Then g(X) is called a generator polynomial of C, while h(X) is called a check polynomial of C. C =

• c(X) ∈ Fq[X]/X n −λ
• c(X)h(X) ≡ 0 (mod X n −λ)
• .

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How to Determine the Divisors of X n −λ

From now on we always assume that gcd(q,n) = 1

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How to Determine the Divisors of X n −λ

From now on we always assume that gcd(q,n) = 1 Zn: the residue ring of the integer ring Z modulo n Z∗

n: the multiplicative group of units of Zn

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How to Determine the Divisors of X n −λ

From now on we always assume that gcd(q,n) = 1 Zn: the residue ring of the integer ring Z modulo n Z∗

n: the multiplicative group of units of Zn

r := ordF∗

q(λ), the order of λ in the group F∗

q, hence r | (q −1)

and q ∈ Z∗

nr

e := ordZ∗

nr (q), the order of q in the group Z∗

nr, hence nr|(qe −1)

ω ∈ Fqe: a primitive nr-th root of unity

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How to Determine the Divisors of X n −λ

From now on we always assume that gcd(q,n) = 1 Zn: the residue ring of the integer ring Z modulo n Z∗

n: the multiplicative group of units of Zn

r := ordF∗

q(λ), the order of λ in the group F∗

q, hence r | (q −1)

and q ∈ Z∗

nr

e := ordZ∗

nr (q), the order of q in the group Z∗

nr, hence nr|(qe −1)

ω ∈ Fqe: a primitive nr-th root of unity 1+rZnr = {1+rk | k = 0,1,··· ,n −1} ⊆ Znr X n −λ =

∏

i∈1+rZnr

(X −ωi)

• Proof. (ωi)n = λ if and only if i ≡ 1 (mod r).

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X n −λ = ∏

i∈1+rZnr

(X −ωi) is not a decomposition in Fq[X] in general !

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X n −λ = ∏

i∈1+rZnr

(X −ωi) is not a decomposition in Fq[X] in general ! A polynomial f (X) ∈ Fqe[X] belongs to Fq[X] if and only if it is invariant by the Galois group Gal(Fqe/Fq), which is a cyclic group generated by γq : Fqe − → Fqe, α − → αq

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X n −λ = ∏

i∈1+rZnr

(X −ωi) is not a decomposition in Fq[X] in general ! A polynomial f (X) ∈ Fqe[X] belongs to Fq[X] if and only if it is invariant by the Galois group Gal(Fqe/Fq), which is a cyclic group generated by γq : Fqe − → Fqe, α − → αq ωi − → ωqi {ωi | i ∈ 1+rZnr} − →

γq

{ωi | i ∈ 1+rZnr}

• 
• 
• 1+rZnr

µq

− → 1+rZnr i − → qi µq : 1+rZnr − → 1+rZnr, i − → qi

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q-Costs

Definition

µq-orbits on 1+rZnr are called q-cosets. The set of q-cosets

• n 1+rZnr is denoted by (1+rZnr)/µq.

For any Q ∈ (1+rZnr)/µq, let MQ(X) = ∏

i∈Q

(X −ωi) For any Q ∈ (1+rZnr)/µq, let CQ be the constacyclic code with MQ(X) as a check polynomial.

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q-Costs

Definition

µq-orbits on 1+rZnr are called q-cosets. The set of q-cosets

• n 1+rZnr is denoted by (1+rZnr)/µq.

For any Q ∈ (1+rZnr)/µq, let MQ(X) = ∏

i∈Q

(X −ωi) For any Q ∈ (1+rZnr)/µq, let CQ be the constacyclic code with MQ(X) as a check polynomial.

Proposition

(i) Every MQ(X) is an irreducible polynomial in Fq[X] with degMQ(X) = #Q (in particular, dimCQ = #Q). (ii) X n −λ = ∏

Q∈(1+rZnr)/µq

MQ(X) is a decomposition of X n −λ in Fq[X]. (iii) Fq[X]/X n −λ =

• Q∈(1+rZnr)/µq

CQ.

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Polyadic Constacyclic Codes Multiplier µs

Assume: s is an integer such that s ∈ Z∗

nr ∩(1+rZnr)

Define a permutation of 1+rZnr: µs : 1+rZnr − → 1+rZnr, x − → sx;

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Polyadic Constacyclic Codes Multiplier µs

Assume: s is an integer such that s ∈ Z∗

nr ∩(1+rZnr)

Define a permutation of 1+rZnr: µs : 1+rZnr − → 1+rZnr, x − → sx; Define an algebra automorphism (also denoted by µs) µs : Fq[X]/X n −λ − → Fq[X]/X n −λ ∑n−1

i=0 aiX i

− → ∑n−1

i=0 aiX is

(mod X n −λ), then µs also keeps the Hamming-weight structure. µs is called a mutiplier

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Polyadic Constacyclic Codes

Definition

If 1+rZnr has a partition 1+rZnr = X0 ∪X1 ···∪Xm−1 such that every Xj is µq-invariant and µs(Xj) = Xj+1 for j = 0,1,··· ,m −1 (the subscripts are taken modulo m), then

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Polyadic Constacyclic Codes

Definition

If 1+rZnr has a partition 1+rZnr = X0 ∪X1 ···∪Xm−1 such that every Xj is µq-invariant and µs(Xj) = Xj+1 for j = 0,1,··· ,m −1 (the subscripts are taken modulo m), then the system X0,X1,··· ,Xm−1 is said to be a Type I m-adic splitting of 1+rZnr given by the multiplier µs;

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Polyadic Constacyclic Codes

Definition

If 1+rZnr has a partition 1+rZnr = X0 ∪X1 ···∪Xm−1 such that every Xj is µq-invariant and µs(Xj) = Xj+1 for j = 0,1,··· ,m −1 (the subscripts are taken modulo m), then the system X0,X1,··· ,Xm−1 is said to be a Type I m-adic splitting of 1+rZnr given by the multiplier µs; the constacyclic codes CXj =

• Q∈Xj/µq

CQ are called Type I m-adic constacyclic codes given by the multiplier µs.

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Polyadic Constacyclic Codes

Definition

If 1+rZnr has a partition 1+rZnr = X0 ∪X1 ···∪Xm−1 such that every Xj is µq-invariant and µs(Xj) = Xj+1 for j = 0,1,··· ,m −1 (the subscripts are taken modulo m), then the system X0,X1,··· ,Xm−1 is said to be a Type I m-adic splitting of 1+rZnr given by the multiplier µs; the constacyclic codes CXj =

• Q∈Xj/µq

CQ are called Type I m-adic constacyclic codes given by the multiplier µs. At that case, µs(CX0) = CX1, µs(CX1) = CX2, ··· , µs(CXm−1) = CX0, Fq[X]/X n −λ = CX0 ⊕CX1 ⊕···⊕CXm−1

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Polyadic Constacyclic Codes

Definition

If 1+rZnr has a partition 1+rZnr = X0 ∪X1 ···∪Xm−1 such that every Xj is µq-invariant and µs(Xj) = Xj+1 for j = 0,1,··· ,m −1 (the subscripts are taken modulo m), then the system X0,X1,··· ,Xm−1 is said to be a Type I m-adic splitting of 1+rZnr given by the multiplier µs; the constacyclic codes CXj =

• Q∈Xj/µq

CQ are called Type I m-adic constacyclic codes given by the multiplier µs. At that case, µs(CX0) = CX1, µs(CX1) = CX2, ··· , µs(CXm−1) = CX0, Fq[X]/X n −λ = CX0 ⊕CX1 ⊕···⊕CXm−1 If m = 2, “2-adic” also said to be “duadic”

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An Example

q = 3, r = 2, λ = −1, n = 4, m = 2, s = −1 1+rZnr = 1+2Z8 = {1,3,5,7}

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An Example

q = 3, r = 2, λ = −1, n = 4, m = 2, s = −1 1+rZnr = 1+2Z8 = {1,3,5,7} X0 = {1,3}, X1 = {5,7} Then X0, X1 are µ3-invariant partition of {1,3,5,7}, and µ−1(X0) = X1, µ−1(X1) = X0 X n −λ = X 4 +1 = (X 2 −X −1)(X 2 +X −1)

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An Example

q = 3, r = 2, λ = −1, n = 4, m = 2, s = −1 1+rZnr = 1+2Z8 = {1,3,5,7} X0 = {1,3}, X1 = {5,7} Then X0, X1 are µ3-invariant partition of {1,3,5,7}, and µ−1(X0) = X1, µ−1(X1) = X0 X n −λ = X 4 +1 = (X 2 −X −1)(X 2 +X −1) CX0 =

• f (X)(X 2 +X −1) (mod X 4 +1)
• f (X) ∈ Fq[X]
• =

Span

• (1,1,−1,0), (0,1,1,−1)
• CX1

=

• f (X)(X 2 −X −1) (mod X 4 +1)
• f (X) ∈ Fq[X]
• =

Span

• (1,−1,−1,0), (0,1,−1,−1)
• µ−1(CX0) = CX1,

µ−1(CX1) = CX0 Both CX0, CX1 are self-dual [4,2,3] codes.

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References

• C. J. Lim, Consta-Abelian polyadic codes, IEEE Trans. Inform.

Theory, 51(2005), 2198-2206. Polyadic cyclic codes were generalized to polyadic consta-abelian codes, some sufficient conditions for the existence were established.

• T. Blackford, Negacyclic duadic codes, Finite Fields Appl.,

14(2008), 930-943. Duadic negacyclic codes were considered: on existence conditions and on good codes.

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• T. Blackford, Isodual constacyclic codes, Finite Fields Appl.,

24(2013), 29-44. Conditions for the existence of Type I duadic constacyclic codes were obtained, conditions for the existence of Type I duadic constacyclic codes given by a multiplier were shown.

• B. Chen, H. Q. Dinh, A note on isodual constacyclic codes, Finite

Fields Appl.(accepted, 2014). A correction for the above reference are made.

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Main Results Two General Questions

As far as we know, there are no solutions to the general questions: For any given positive integer m, do the Type I m-adic λ-constacyclic codes of length n exist? For any given integer s, can s be a multiplier of a Type I m-adic λ-constacyclic code of length n?

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p-Adic Valuations

Let t be a non-zero integer. For any prime p there is a unique non-negative integer νp(t) such that pνp(t)t. The function νp is called the p-adic valuation of t.

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p-Adic Valuations

Let t be a non-zero integer. For any prime p there is a unique non-negative integer νp(t) such that pνp(t)t. The function νp is called the p-adic valuation of t. Of course, t = ±∏

p pνp(t) where p runs over all primes, but

νp(t) = 0 for almost all primes p except for finite many ones.

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p-Adic Valuations

Let t be a non-zero integer. For any prime p there is a unique non-negative integer νp(t) such that pνp(t)t. The function νp is called the p-adic valuation of t. Of course, t = ±∏

p pνp(t) where p runs over all primes, but

νp(t) = 0 for almost all primes p except for finite many ones. We adopt a convention that νp(0) = −∞ and |νp(0)| = ∞.

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Our Main Results: For a Given m

Theorem

There is a unique integer M = ∏

p pνp(M) such that Type I m-adic

(q,n,r)-constacyclic codes exist if and only if m is a divisor of M, where νp(M) is determined as follows: if p ∤ r or p ∤ n, then νp(M) = 0;

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Our Main Results: For a Given m

Theorem

There is a unique integer M = ∏

p pνp(M) such that Type I m-adic

(q,n,r)-constacyclic codes exist if and only if m is a divisor of M, where νp(M) is determined as follows: if p ∤ r or p ∤ n, then νp(M) = 0; otherwise: (i) if p is odd or νp(r) ≥ 2, then νp(M) = min{νp(q −1)−νp(r), νp(n)};

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Our Main Results: For a Given m

Theorem

There is a unique integer M = ∏

p pνp(M) such that Type I m-adic

(q,n,r)-constacyclic codes exist if and only if m is a divisor of M, where νp(M) is determined as follows: if p ∤ r or p ∤ n, then νp(M) = 0; otherwise: (i) if p is odd or νp(r) ≥ 2, then νp(M) = min{νp(q −1)−νp(r), νp(n)}; (ii) if p = 2 and ν2(r) = 1, there are two subcases:

(ii.1) if ν2(q −1) ≥ 2 then ν2(M) = max{min{ν2(q −1)−2, ν2(n)−1}, 1}; (ii.2) if ν2(q −1) = 1 then ν2(M) = min{ν2(q +1)−1, ν2(n)−1}.

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Our Main Results: For a Given Multiplier µs

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Our Main Results: For a Given Multiplier µs

Theorem

There is a unique integer Ms = ∏

p pνp(Ms) such that µs is a

multiplier of a Type I m-adic splitting for 1+rZnr if and only if m is a divisor of Ms, where νp(Ms) is determined as follows: if p ∤ r or p ∤ n, then νp(Ms) = 0;

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Our Main Results: For a Given Multiplier µs

Theorem

There is a unique integer Ms = ∏

p pνp(Ms) such that µs is a

multiplier of a Type I m-adic splitting for 1+rZnr if and only if m is a divisor of Ms, where νp(Ms) is determined as follows: if p ∤ r or p ∤ n, then νp(Ms) = 0; otherwise: (i) if p is odd or both νp(q −1) ≥ 2 and νp(s −1) ≥ 2, then νp(Ms) = max

• min{νp(q −1), νp(nr)}−|νp(s −1)|, 0
• ;

(to continue next page)

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(Continue: For a Given Multiplier µs)

(ii) if p = 2, ν2(q −1) = 1 and s ≡ 1 (mod 4), then ν2(Ms) = max

• min{ν2(q +1)+1, ν2(nr)}−|ν2(s −1)|, 0
• ;

(iii) if p = 2, ν2(q −1) ≥ 2 and ν2(s −1) = 1, then ν2(Ms) = max

• min{ν2(q −1), ν2(nr)}−|ν2(s +1)|, 1
• ;

(iv) if p = 2, ν2(q −1) = 1 and ν2(s −1) = 1, then ν2(Ms) =      max

• min{ν2(q +1)+1,ν2(nr)}−min{|ν2(s +1)|,ν2(q +1)},0
• ,

if ν2(s +1) = ν2(q +1); 0, if ν2(s +1) = ν2(q +1).

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Ideas of the Proof: Elementary Group Theory

Lemma

Let µ be a permutation of a finite set X and m be a positive

• integer. Then the following statements are equivalent:

(i) There is partition X = X0 ∪X1 ∪···∪Xm−1 such that µ(Xi) = Xi+1 for i = 0,1,··· ,m −1 (the subscripts are taken modulo m). (ii) The length of every µ-orbit on X is divided by m.

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Lemma

Let G, H be finite groups, and X , Y be finite G-set and H-set

• respectively. Then X ×Y is a finite G ×H-set with natural action
• f G ×H, and the following hold:

(i) For g ∈ G and h ∈ H, the order of (g,h) ∈ G ×H is equal to the least common multiple of the order of g in G and the

• rder of h in H, i.e., lcm
• rdG(g),ordH(h)
• .

(ii) For x ∈ X and y ∈ Y , the length of the (g,h)-orbit on X ×Y containing (x,y) is equal to the least common multiple of the length of the g-orbit on X containing x and the length of the h-orbit on Y containing y.

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Lemma

Let a finite group G act on a finite set X freely, and N be a normal subgroup of G. Let X /N be the set of N-orbits on X . Then the quotient G/N acts on X /N freely; specifically, the length of any G/N-orbit on X /N is equal to the index |G : N|.

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Ideas of the Proof: Elementary Number Theory

Let p1,··· ,pk, p′

1,··· ,p′ k′, p′′ 1,··· ,p′′ k′′ be distinct primes such that

n = pα1

1 ···pαk k p′ 1 α′

1 ···p′

k′α′

k′,

αj > 0, α′

j > 0;

r = pβ1

1 ···pβk k p′′ 1 β ′′

1 ···p′′

k′′β ′′

k′′,

βj > 0, β ′′

j > 0.

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Ideas of the Proof: Elementary Number Theory

Let p1,··· ,pk, p′

1,··· ,p′ k′, p′′ 1,··· ,p′′ k′′ be distinct primes such that

n = pα1

1 ···pαk k p′ 1 α′

1 ···p′

k′α′

k′,

αj > 0, α′

j > 0;

r = pβ1

1 ···pβk k p′′ 1 β ′′

1 ···p′′

k′′β ′′

k′′,

βj > 0, β ′′

j > 0.

Then nr = pα1+β1

1

···pαk+βk

k

n′r′′ with n′ = p′

1 α′

1 ···p′

k′α′

k′ and r′′ = p′′

1 β ′′

1 ···p′′

k′′β ′′

k′′. 28 / 38

Ideas of the Proof: Elementary Number Theory

Let p1,··· ,pk, p′

1,··· ,p′ k′, p′′ 1,··· ,p′′ k′′ be distinct primes such that

n = pα1

1 ···pαk k p′ 1 α′

1 ···p′

k′α′

k′,

αj > 0, α′

j > 0;

r = pβ1

1 ···pβk k p′′ 1 β ′′

1 ···p′′

k′′β ′′

k′′,

βj > 0, β ′′

j > 0.

Then nr = pα1+β1

1

···pαk+βk

k

n′r′′ with n′ = p′

1 α′

1 ···p′

k′α′

k′ and r′′ = p′′

1 β ′′

1 ···p′′

k′′β ′′

k′′.

Applying the Chinese Remainder Theorem, we rewrite Znr as follows: Znr

CRT

= Zpα1+β1

1

×···×Zp

αk +βk k

×Zn′ ×Zr′′.

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Since 1+rZnr = Ker

• Znr → Zr
• , we get the key decomposition:

1+rZrn

CRT

=

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• ×Zn′ ×{1}.

Z∗

rn∩(1+rZrn) CRT

=

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• ×Z∗

n′ ×{1}.

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Since 1+rZnr = Ker

• Znr → Zr
• , we get the key decomposition:

1+rZrn

CRT

=

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• ×Zn′ ×{1}.

Z∗

rn∩(1+rZrn) CRT

=

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• ×Z∗

n′ ×{1}.

Correspondingly, we can write q CRT =

• 1+pτ1

1 q1, ··· , 1+pτk k qk, q′, 1

• s CRT

=

• 1+pσ1

1 s1, ··· , 1+pσk k sk, s′, 1

• 29 / 38

Since 1+rZnr = Ker

• Znr → Zr
• , we get the key decomposition:

1+rZrn

CRT

=

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• ×Zn′ ×{1}.

Z∗

rn∩(1+rZrn) CRT

=

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• ×Z∗

n′ ×{1}.

Correspondingly, we can write q CRT =

• 1+pτ1

1 q1, ··· , 1+pτk k qk, q′, 1

• s CRT

=

• 1+pσ1

1 s1, ··· , 1+pσk k sk, s′, 1

• Then both the theorems could be proved by analyzing the group

structure of

• 1+pβ1

1 Zpα1+β1

1

• ×···×
• 1+pβk

k Zp

αk +βk k

• , though the

analyzing is very delicate.

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Applications p-Adic Constacyclic Codes

Corollary

Let m = p be a prime. The p-adic (q,n,r)-constacyclic codes of Type I exist if and only if one of the following two holds: (i) νp(n) ≥ 1 and νp(q −1) > νp(r) ≥ 1 (it is allowed that p = 2); (ii) p = 2, ν2(r) = 1 and min{ν2(q −1),ν2(n)} ≥ 2.

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Applications p-Adic Constacyclic Codes

Corollary

Let m = p be a prime. The p-adic (q,n,r)-constacyclic codes of Type I exist if and only if one of the following two holds: (i) νp(n) ≥ 1 and νp(q −1) > νp(r) ≥ 1 (it is allowed that p = 2); (ii) p = 2, ν2(r) = 1 and min{ν2(q −1),ν2(n)} ≥ 2. One of Blackford’s results is the case of the corollary when p = 2

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Generalized Reed-Solomon Codes

Proposition

Assume that m = p is a prime, q is a prime power with νp(q −1) ≥ 2, and nr | (q −1) such that νp(r) ≥ 1 and νp(n) ≥ 1. Let ω ∈ Fq be a primitive nr-th root of unity and λ = ωn.

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Generalized Reed-Solomon Codes

Proposition

Assume that m = p is a prime, q is a prime power with νp(q −1) ≥ 2, and nr | (q −1) such that νp(r) ≥ 1 and νp(n) ≥ 1. Let ω ∈ Fq be a primitive nr-th root of unity and λ = ωn. Set Xj =

• 1+ri
• jn

p ≤ i < (j+1)n p

• ,

j = 0,1,··· ,p −1,

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Generalized Reed-Solomon Codes

Proposition

Assume that m = p is a prime, q is a prime power with νp(q −1) ≥ 2, and nr | (q −1) such that νp(r) ≥ 1 and νp(n) ≥ 1. Let ω ∈ Fq be a primitive nr-th root of unity and λ = ωn. Set Xj =

• 1+ri
• jn

p ≤ i < (j+1)n p

• ,

j = 0,1,··· ,p −1, Then (i) CXj for j = 0,1,··· ,p −1 are Type I p-adic λ-constacyclic codes given by µ1+ nr

p ;

(ii) for any 0 < k < p, the constacyclic code C = CX0 ⊕CX1 ⊕···⊕CXk−1 is an [n, kn

p , (p−k)n p

+1] generalized RS-code as follows:

C =

• f (1),ω−1f (ω−2),··· ,ω−(n−1)f (ω−2(n−1))
• f (X) ∈ Fq[X], degf (X) < k
• 31 / 38

An Example

q = 17, m = p = r = 2, n = 8, and s = −1 ω = 6, λ = 68 = −1, ω−1 = 3, ω−2 = 9

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An Example

q = 17, m = p = r = 2, n = 8, and s = −1 ω = 6, λ = 68 = −1, ω−1 = 3, ω−2 = 9 1+2Z16 = X0 ∪X1 X0 = {1,3,5,7}, X1 = {9,11,13,15} CX0 =

• f (1),3f (9),··· ,37f (97)
• f (X) ∈ F17[X], degf (X) < 4
• 32 / 38

An Example

q = 17, m = p = r = 2, n = 8, and s = −1 ω = 6, λ = 68 = −1, ω−1 = 3, ω−2 = 9 1+2Z16 = X0 ∪X1 X0 = {1,3,5,7}, X1 = {9,11,13,15} CX0 =

• f (1),3f (9),··· ,37f (97)
• f (X) ∈ F17[X], degf (X) < 4
• CX0 is a self-dual generalized Reed-Solomon code with parameters

[8,4,5]

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p-Adic Constacyclic Codes given by µs

The statement is rather easy when p is odd.

Corollary

Assume that m = p is an odd prime, s ∈ Z∗

nr ∩1+rZnr and s = 1.

Then Type I p-adic splittings of 1+rZnr given by µs exist if and

• nly if both the two conditions are satisfied: (1) p divides both n

and r; (2) νp(s −1) is less than both νp(q −1) and νp(nr).

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Corollary

Assume that s ∈ Z∗

nr ∩(1+rZnr). Then Type I duadic splittings for

1+rZnr given by µs exist if and only if both n and r are even and

• ne of the following four holds:

(i) ν2(q −1) > |ν2(s −1)| and ν2(nr) > |ν2(s −1)|;

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Corollary

Assume that s ∈ Z∗

nr ∩(1+rZnr). Then Type I duadic splittings for

1+rZnr given by µs exist if and only if both n and r are even and

• ne of the following four holds:

(i) ν2(q −1) > |ν2(s −1)| and ν2(nr) > |ν2(s −1)|; Note: |ν2(s −1)| ≥ 1 when s ∈ Z∗

rn ∩(1+rZrn) and r is even; so (i)

implies that ν2(q −1) > 1, i.e. q ≡ 1 (mod 4).

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Corollary

Assume that s ∈ Z∗

nr ∩(1+rZnr). Then Type I duadic splittings for

1+rZnr given by µs exist if and only if both n and r are even and

• ne of the following four holds:

(i) ν2(q −1) > |ν2(s −1)| and ν2(nr) > |ν2(s −1)|; (ii) ν2(q −1) = 1, ν2(s −1) > 1, ν2(q +1)+1 > |ν2(s −1)| and ν2(nr) > |ν2(s −1)|; (iii) ν2(q −1) = ν2(s −1) = 1, |ν2(s +1)| > ν2(q +1) and ν2(nr) > ν2(q +1); (iv) ν2(q −1) = ν2(s −1) = 1, |ν2(s +1)| < ν2(q +1) and |ν2(s +1)| < ν2(nr). Note: |ν2(s −1)| ≥ 1 when s ∈ Z∗

rn ∩(1+rZrn) and r is even; so (i)

implies that ν2(q −1) > 1, i.e. q ≡ 1 (mod 4).

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Before the corollary, there is no necessary and sufficient condition for µs being a multiplier of a duadic constacyclic codes appeared in

• literatures. Though a paper of Blackford and a joint paper of Chen

and Dihn considered the question, their results didn’t provide a complete answer.

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Before the corollary, there is no necessary and sufficient condition for µs being a multiplier of a duadic constacyclic codes appeared in

• literatures. Though a paper of Blackford and a joint paper of Chen

and Dihn considered the question, their results didn’t provide a complete answer. The special case “s = −1” has a special interests.

Corollary

Type I duadic splittings for 1+rZrn given by µ−1 exist if and only if both n and r are even and one of the following two holds: (i) ν2(q −1) ≥ 2 (i.e. q ≡ 1 (mod 4)); (ii) ν2(q −1) = 1 (i.e. q ≡ 3 (mod 4)) and ν2(q +1) < ν2(nr).

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Alternant self-dual Constacyclic MDS Codes

Proposition

Assume that q is a power of any odd prime. Let n = q+1

ℓ

with ℓ being an odd divisor of q +1, and X0 = {1, 3, ··· , n −1}, X1 = {n +1, n +3, ··· , 2n −1}. Then

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Alternant self-dual Constacyclic MDS Codes

Proposition

Assume that q is a power of any odd prime. Let n = q+1

ℓ

with ℓ being an odd divisor of q +1, and X0 = {1, 3, ··· , n −1}, X1 = {n +1, n +3, ··· , 2n −1}. Then (i) X0 and X1 form a Type I duadic splitting of 1+2Z2n over Fq given by µ−1;

36 / 38

Alternant self-dual Constacyclic MDS Codes

Proposition

Assume that q is a power of any odd prime. Let n = q+1

ℓ

with ℓ being an odd divisor of q +1, and X0 = {1, 3, ··· , n −1}, X1 = {n +1, n +3, ··· , 2n −1}. Then (i) X0 and X1 form a Type I duadic splitting of 1+2Z2n over Fq given by µ−1; (ii) the duadic negacyclic code CX0 over Fq is a self-dual duadic negacyclic MDS [n, n

2, n 2 +1] code;

36 / 38

Alternant self-dual Constacyclic MDS Codes

Proposition

Assume that q is a power of any odd prime. Let n = q+1

ℓ

with ℓ being an odd divisor of q +1, and X0 = {1, 3, ··· , n −1}, X1 = {n +1, n +3, ··· , 2n −1}. Then (i) X0 and X1 form a Type I duadic splitting of 1+2Z2n over Fq given by µ−1; (ii) the duadic negacyclic code CX0 over Fq is a self-dual duadic negacyclic MDS [n, n

2, n 2 +1] code;

(iii) CX0 is alternant code, i.e. the restriction code of a generalized RS-code over Fq2 restricted to Fq.

36 / 38

An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s = −1 ω ∈ F72 is primitive 16-th root of unity, λ = ω8 = −1

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An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s = −1 ω ∈ F72 is primitive 16-th root of unity, λ = ω8 = −1 1+2Z16 = X0 ∪X1 X0 = {1,3,5,7}, X1 = {9,11,13,15}

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An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s = −1 ω ∈ F72 is primitive 16-th root of unity, λ = ω8 = −1 1+2Z16 = X0 ∪X1 X0 = {1,3,5,7}, X1 = {9,11,13,15}

• CX0 =
• f (1),ω−1f (ω−2),··· ,ω−7f (ω−14)
• f (X) ∈ F72[X],degf (X) < 4
• 37 / 38

An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s = −1 ω ∈ F72 is primitive 16-th root of unity, λ = ω8 = −1 1+2Z16 = X0 ∪X1 X0 = {1,3,5,7}, X1 = {9,11,13,15}

• CX0 =
• f (1),ω−1f (ω−2),··· ,ω−7f (ω−14)
• f (X) ∈ F72[X],degf (X) < 4
• CX0 =

CX0|Fq is an alternant self-dual negacyclic code with parameters [8,4,5].

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THANK YOU

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