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Polyadic Constacyclic Codes

Yun Fan

Department of Mathematics, Central China Normal University, Wuhan, 430079,CHINA

Email: [email protected]

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 polyadicconstacyclic codes are reported in terms of p-valuations; the mainideas to solve the question are described. Some consequences arederived, 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 CodesHamming Weight, Hamming Distance

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

F∗q: the multiplicative group of nonzero elements of Fq

Fnq: =

{(a0,a1, · · · ,an−1)

∣∣ai ∈ Fq

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

Definition

Hamming weight:

w(a) := #{i |0≤ i < n, ai 6= 0}, ∀ a = (a0,a1, · · · ,an−1) ∈ Fnq

Hamming distance:

d(a,a′) := w(a−a′), ∀ a,a′ ∈ Fnq

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Linear codes, Cyclic CodesHamming Weight, Hamming Distance

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

F∗q: the multiplicative group of nonzero elements of Fq

Fnq: =

{(a0,a1, · · · ,an−1)

∣∣ai ∈ Fq

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

Definition

Hamming weight:

w(a) := #{i |0≤ i < n, ai 6= 0}, ∀ a = (a0,a1, · · · ,an−1) ∈ Fnq

Hamming distance:

d(a,a′) := w(a−a′), ∀ a,a′ ∈ Fnq

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Linear Codes

Definition

Any given subspace C of Fnq is said to be a linear code of length n

over the alphabet Fq; any c = (c0,c1, · · · ,cn−1) ∈ C is said to be acodeword.

k := dimC , dimension of the linear code

d = d(C ) := minc6=c′∈C

d(c,c′), the minimum distance of the

linear code

w(C ) := min06=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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Linear Codes

Definition

d = d(C ) := minc6=c′∈C

linear code

w(C ) := min06=c∈C

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Linear Codes

Definition

d = d(C ) := minc6=c′∈C

linear code

w(C ) := min06=c∈C

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What Codes Are Good Codes

Both the dimension and the minimum distance of the codeare bigger.

However, the dimension and the minimum distance of a coderestrict each other.

So the question is: how to trade off them.

The code has good mathematical structure

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What Codes Are Good Codes

Both the dimension and the minimum distance of the codeare bigger.

However, the dimension and the minimum distance of a coderestrict 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

Definition

C ≤ Fnq

C⊥ :={a ∈ Fn

∣∣〈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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〈a,a′〉 :=n−1

∑i=0

aia′i , ∀ a = (a0, · · · ,an−1), a′ = (a′0, · · · ,a′n−1) ∈ Fn

Definition

C ≤ Fnq

C⊥ :={a ∈ Fn

code of C .

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〈a,a′〉 :=n−1

∑i=0

aia′i , ∀ a = (a0, · · · ,an−1), a′ = (a′0, · · · ,a′n−1) ∈ Fn

Definition

C ≤ Fnq

C⊥ :={a ∈ Fn

code of C .

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〈a,a′〉 :=n−1

∑i=0

aia′i , ∀ a = (a0, · · · ,an−1), a′ = (a′0, · · · ,a′n−1) ∈ Fn

Definition

C ≤ Fnq

C⊥ :={a ∈ Fn

code of C .

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Cyclic Codes, Constacyclic Codes

C ≤ Fnq: 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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Cyclic Codes, Constacyclic Codes

C ≤ Fnq: 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

· · ·

Quadratic residue codes

Duadic codes

Polyadic codes

· · ·

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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 CodesHow to Determine Constacyclic Codes

λ ∈ F∗qFq[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.−→ Fnq

a0 +a1X + · · ·+an−1Xn−1 7−→ (a0,a1, · · · ,an−1)

C ≤ Fnq is a λ -constacyclic code if and only if the following set{

c0 + c1X + · · ·+ cn−1Xn−1

∣∣(c0,c1, · · · ,cn−1) ∈ C}

is an ideal of Fq[X ]/〈X n−λ 〉.

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Constacyclic CodesHow to Determine Constacyclic Codes

λ ∈ F∗qFq[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.−→ Fnq

a0 +a1X + · · ·+an−1Xn−1 7−→ (a0,a1, · · · ,an−1)

C ≤ Fnq is a λ -constacyclic code if and only if the following set{

c0 + c1X + · · ·+ cn−1Xn−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 onecorresponding 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

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

Proposition

λ -constacyclic codes C over Fq of length n are one to onecorresponding 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 ) iscalled 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

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∗

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. (ω i )n = λ if and only if i ≡ 1 (mod r).

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1+ rZnr = {1+ rk | k = 0,1, · · · ,n−1} ⊆ Znr

(X −ωi )

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1+ rZnr = {1+ rk | k = 0,1, · · · ,n−1} ⊆ Znr

(X −ωi )

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1+ rZnr = {1+ rk | k = 0,1, · · · ,n−1} ⊆ Znr

(X −ωi )

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(X −ω i ) is not a decomposition in Fq[X ] in

general !

A polynomial f (X ) ∈ Fqe [X ] belongs to Fq[X ] if and only if it isinvariant by the Galois group Gal(Fqe/Fq), which is a cyclic groupgenerated by

γq : Fqe −→ Fqe , α 7−→ αq

ω i 7−→ ωqi

{ω i | i ∈ 1+ rZnr} −→γq

{ω i | i ∈ 1+ rZnr}xy xy1+ rZnr

µq−→ 1+ rZnr

i 7−→ qi

µq : 1+ rZnr −→ 1+ rZnr , i 7−→ qi

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general !

ω i 7−→ ωqi

{ω i | i ∈ 1+ rZnr} −→γq

µq−→ 1+ rZnr

i 7−→ qi

µq : 1+ rZnr −→ 1+ rZnr , i 7−→ qi

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general !

ω i 7−→ ωqi

{ω i | i ∈ 1+ rZnr} −→γq

µq−→ 1+ rZnr

i 7−→ qi

µq : 1+ rZnr −→ 1+ rZnr , i 7−→ qi

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q-Costs

Definition

µq-orbits on 1+ rZnr are called q-cosets. The set of q-cosetson 1+ rZnr is denoted by (1+ rZnr )/µq.

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

(X −ω i )

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

Proposition

(i) Every MQ(X ) is an irreducible polynomial in Fq[X ] withdegMQ(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

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q-Costs

Definition

µq-orbits on 1+ rZnr are called q-cosets. The set of q-cosetson 1+ rZnr is denoted by (1+ rZnr )/µq.

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

(X −ω i )

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

Proposition

(i) Every MQ(X ) is an irreducible polynomial in Fq[X ] withdegMQ(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

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Polyadic Constacyclic CodesMultiplier 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 7−→ sx ;

Define an algebra automorphism (also denoted by µs)

µs : Fq[X ]/〈X n−λ 〉 −→ Fq[X ]/〈X n−λ 〉∑

n−1i=0 aiX

i 7−→ ∑n−1i=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 CodesMultiplier 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 7−→ sx ;

Define an algebra automorphism (also denoted by µs)

µs : Fq[X ]/〈X n−λ 〉 −→ Fq[X ]/〈X n−λ 〉∑

n−1i=0 aiX

i 7−→ ∑n−1i=0 aiX

is (mod X n−λ ),

then µs also keeps the Hamming-weight structure.

µs is called a mutiplier

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Definition

If 1+ rZnr has a partition 1+ rZnr = X0∪X1 · · ·∪Xm−1 suchthat every Xj is µq-invariant and µs(Xj) = Xj+1 forj = 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-adicsplitting 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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Definition

⊕Q∈Xj/µq

At that case,

Fq[X ]/〈X n−λ 〉= CX0 ⊕CX1 ⊕·· ·⊕CXm−1

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Definition

⊕Q∈Xj/µq

At that case,

Fq[X ]/〈X n−λ 〉= CX0 ⊕CX1 ⊕·· ·⊕CXm−1

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Definition

⊕Q∈Xj/µq

At that case,

Fq[X ]/〈X n−λ 〉= CX0 ⊕CX1 ⊕·· ·⊕CXm−1

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Definition

⊕Q∈Xj/µq

At that case,

Fq[X ]/〈X n−λ 〉= CX0 ⊕CX1 ⊕·· ·⊕CXm−1

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An Exampleq = 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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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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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.17 / 38

References

C. J. Lim, Consta-Abelian polyadic codes, IEEE Trans. Inform.Theory, 51(2005), 2198-2206.

Polyadic cyclic codes were generalized to polyadicconsta-abelian codes, some sufficient conditions for theexistence were established.

T. Blackford, Negacyclic duadic codes, Finite Fields Appl.,14(2008), 930-943.

Duadic negacyclic codes were considered: on existenceconditions 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 codeswere obtained, conditions for the existence of Type I duadicconstacyclic codes given by a multiplier were shown.

B. Chen, H. Q. Dinh, A note on isodual constacyclic codes, FiniteFields Appl.(accepted, 2014).

A correction for the above reference are made.

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Main ResultsTwo 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-adicλ -constacyclic codes of length n exist?

For any given integer s, can s be a multiplier of a Type Im-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 uniquenon-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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p-Adic Valuations

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p-Adic Valuations

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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, thenνp(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 thenν2(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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Theorem

(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:

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Theorem

(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:

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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 mis 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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Theorem

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

otherwise:

νp(Ms) = max{

min{νp(q−1), νp(nr)}− |νp(s−1)|, 0};

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Theorem

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

νp(Ms) = max{

min{νp(q−1), νp(nr)}− |νp(s−1)|, 0};

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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) 6= ν2(q +1);

0, if ν2(s +1) = ν2(q +1).

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Ideas of the Proof: Elementary Group Theory

Let µ be a permutation of a finite set X and m be a positiveinteger. Then the following statements are equivalent:

(i) There is partition X = X0∪X1∪·· ·∪Xm−1 such thatµ(Xi ) = Xi+1 for i = 0,1, · · · ,m−1 (the subscripts are takenmodulo m).

(ii) The length of every µ-orbit on X is divided by m.

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Let G , H be finite groups, and X , Y be finite G -set and H-setrespectively. Then X ×Y is a finite G ×H-set with natural actionof G ×H, and the following hold:

(i) For g ∈ G and h ∈ H, the order of (g ,h) ∈ G ×H is equal tothe least common multiple of the order of g in G and theorder of h in H, i.e., lcm

(ordG (g),ordH(h)

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

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Let a finite group G act on a finite set X freely, and N be anormal 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, thelength 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 = pα11 · · ·pαk

k p′1α ′

1 · · ·p′k ′α ′

k ′ , αj > 0, α′j > 0;

r = pβ1

1 · · ·pβk

k p′′1β ′′

1 · · ·p′′k ′′β ′′

k ′′ , βj > 0, β′′j > 0.

Thennr = p

α1+β1

1 · · ·pαk+βk

k n′r ′′

with n′ = p′1α ′

1 · · ·p′k ′α ′

k ′ and r ′′ = p′′1β ′′

1 · · ·p′′k ′′β ′′

k ′′ .

Applying the Chinese Remainder Theorem, we rewrite Znr asfollows:

ZnrCRT= Z

pα1+β11

×·· ·×Zp

αk+βkk

×Zn′×Zr ′′ .

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n = pα11 · · ·pαk

k p′1α ′

1 · · ·p′k ′α ′

k ′ , αj > 0, α′j > 0;

r = pβ1

1 · · ·pβk

k p′′1β ′′

1 · · ·p′′k ′′β ′′

k ′′ , βj > 0, β′′j > 0.

Thennr = p

α1+β1

1 · · ·pαk+βk

k n′r ′′

1 · · ·p′k ′α ′

k ′ and r ′′ = p′′1β ′′

1 · · ·p′′k ′′β ′′

k ′′ .

ZnrCRT= Z

pα1+β11

×·· ·×Zp

αk+βkk

×Zn′×Zr ′′ .

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n = pα11 · · ·pαk

k p′1α ′

1 · · ·p′k ′α ′

k ′ , αj > 0, α′j > 0;

r = pβ1

1 · · ·pβk

k p′′1β ′′

1 · · ·p′′k ′′β ′′

k ′′ , βj > 0, β′′j > 0.

Thennr = p

α1+β1

1 · · ·pαk+βk

k n′r ′′

1 · · ·p′k ′α ′

k ′ and r ′′ = p′′1β ′′

1 · · ·p′′k ′′β ′′

k ′′ .

ZnrCRT= Z

pα1+β11

×·· ·×Zp

αk+βkk

×Zn′×Zr ′′ .

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Since 1+ rZnr = Ker(Znr → Zr

), we get the key decomposition:

1+ rZrnCRT=

α1+β11

)×·· ·×

αk+βkk

)×Zn′×{1}.

Z∗rn∩(1+rZrn)

α1+β11

)×·· ·×

αk+βkk

)×Z∗

n′×{1}.

Correspondingly, we can write

(1+pτ1

1 q1, · · · , 1+pτkk qk , q′, 1

(1+pσ1

1 s1, · · · , 1+pσkk sk , s ′, 1

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

structure of(1+p

α1+β11

)×·· ·×

αk+βkk

), though the

analyzing is very delicate.

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1+ rZrnCRT=

α1+β11

)×·· ·×

αk+βkk

)×Zn′×{1}.

Z∗rn∩(1+rZrn)

α1+β11

)×·· ·×

αk+βkk

)×Z∗

n′×{1}.

(1+pτ1

1 q1, · · · , 1+pτkk qk , q′, 1

(1+pσ1

1 s1, · · · , 1+pσkk sk , s ′, 1

Then both the theorems could be proved by analyzing the group

structure of(1+p

α1+β11

)×·· ·×

αk+βkk

), though the

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1+ rZrnCRT=

α1+β11

)×·· ·×

αk+βkk

)×Zn′×{1}.

Z∗rn∩(1+rZrn)

α1+β11

)×·· ·×

αk+βkk

)×Z∗

n′×{1}.

(1+pτ1

1 q1, · · · , 1+pτkk qk , q′, 1

(1+pσ1

1 s1, · · · , 1+pσkk sk , s ′, 1

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

structure of(1+p

α1+β11

)×·· ·×

αk+βkk

), though the

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Applicationsp-Adic Constacyclic Codes

Corollary

Let m = p be a prime. The p-adic (q,n, r)-constacyclic codes ofType 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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Applicationsp-Adic Constacyclic Codes

Corollary

Let m = p be a prime. The p-adic (q,n, r)-constacyclic codes ofType 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 CodesProposition

Assume that m = p is a prime, q is a prime power withνp(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.

1+ ri∣∣∣ jn

p ≤ i < (j+1)np

}, j = 0,1, · · · ,p−1,

(i) CXjfor j = 0,1, · · · ,p−1 are Type I p-adic λ -constacyclic

codes given by µ1+ nrp;

(ii) for any 0 < k < p, the constacyclic code

C = CX0 ⊕CX1 ⊕·· ·⊕CXk−1is an [n, kn

p , (p−k)np +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 ], deg f (X ) < k}

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Assume that m = p is a prime, q is a prime power withνp(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

∣∣∣ jnp ≤ i < (j+1)n

}, j = 0,1, · · · ,p−1,

p , (p−k)np +1]

C ={(f (1),ω−1f (ω−2), · · · ,ω−(n−1)f (ω−2(n−1))

)∣∣∣ f (X ) ∈ Fq[X ], deg f (X ) < k}

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Assume that m = p is a prime, q is a prime power withνp(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

∣∣∣ jnp ≤ i < (j+1)n

}, j = 0,1, · · · ,p−1,

p , (p−k)np +1]

C ={(f (1),ω−1f (ω−2), · · · ,ω−(n−1)f (ω−2(n−1))

)∣∣∣ f (X ) ∈ Fq[X ], deg f (X ) < k}

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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 ], deg f (X ) < 4

}CX0 is a self-dual generalized Reed-Solomon code with parameters[8,4,5]

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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 ], deg f (X ) < 4

CX0 is a self-dual generalized Reed-Solomon code with parameters[8,4,5]

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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 ], deg f (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 6= 1.

Then Type I p-adic splittings of 1+ rZnr given by µs exist if andonly if both the two conditions are satisfied: (1) p divides both nand 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 andone 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)| andν2(nr) > |ν2(s−1)|;

(iii) ν2(q−1) = ν2(s−1) = 1, |ν2(s +1)|> ν2(q +1) andν2(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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Corollary

(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)| andν2(nr) > |ν2(s−1)|;

(iii) ν2(q−1) = ν2(s−1) = 1, |ν2(s +1)|> ν2(q +1) andν2(nr) > ν2(q +1);

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Corollary

(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);

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Before the corollary, there is no necessary and sufficient conditionfor µs being a multiplier of a duadic constacyclic codes appeared inliteratures. Though a paper of Blackford and a joint paper of Chenand Dihn considered the question, their results didn’t provide acomplete 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 onlyif 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).

35 / 38

Before the corollary, there is no necessary and sufficient conditionfor µs being a multiplier of a duadic constacyclic codes appeared inliteratures. Though a paper of Blackford and a joint paper of Chenand Dihn considered the question, their results didn’t provide acomplete 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 onlyif 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` with `

being an odd divisor of q +1, and

X0 = {1, 3, · · · , n−1}, X1 = {n+1, n+3, · · · , 2n−1}.

(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 duadicnegacyclic MDS [n, n

2 , n2 +1] code;

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

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Proposition

X0 = {1, 3, · · · , n−1}, X1 = {n+1, n+3, · · · , 2n−1}.

given by µ−1;

2 , n2 +1] code;

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Proposition

X0 = {1, 3, · · · , n−1}, X1 = {n+1, n+3, · · · , 2n−1}.

given by µ−1;

2 , n2 +1] code;

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Proposition

X0 = {1, 3, · · · , n−1}, X1 = {n+1, n+3, · · · , 2n−1}.

given by µ−1;

2 , n2 +1] code;

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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}

C̃X0 ={(f (1),ω−1f (ω−2), · · · ,ω−7f (ω−14)

)∣∣∣ f (X ) ∈ F72 [X ],deg f (X ) < 4}

CX0 = C̃X0 |Fq

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

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An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s =−1

1+2Z16 = X0∪X1

X0 = {1,3,5,7}, X1 = {9,11,13,15}

C̃X0 ={(f (1),ω−1f (ω−2), · · · ,ω−7f (ω−14)

)∣∣∣ f (X ) ∈ F72 [X ],deg f (X ) < 4}

CX0 = C̃X0 |Fq

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An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s =−1

1+2Z16 = X0∪X1

X0 = {1,3,5,7}, X1 = {9,11,13,15}

C̃X0 ={(f (1),ω−1f (ω−2), · · · ,ω−7f (ω−14)

)∣∣∣ f (X ) ∈ F72 [X ],deg f (X ) < 4}

CX0 = C̃X0 |Fq

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An Example

q = 7, m = p = r = 2, n = q +1 = 8, and s =−1

1+2Z16 = X0∪X1

X0 = {1,3,5,7}, X1 = {9,11,13,15}

C̃X0 ={(f (1),ω−1f (ω−2), · · · ,ω−7f (ω−14)

)∣∣∣ f (X ) ∈ F72 [X ],deg f (X ) < 4}

CX0 = C̃X0 |Fq

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THANK YOU

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