Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir...
Transcript of Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir...
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Left Metacyclic Ideals
Samir Assuena
Centro Universiario da FEI
Joint work with Cesar Polcino Milies
CIMPA RESEARCH SCHOOL
ALGEBRAIC METHODS IN CODING THEORY
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Groups
Definition
A group G is metacyclic if G contains a cyclic normal subgroup H
such that the factor group G/H is also cyclic.
The dihedral groups and groups all whose Sylow subgroups are
cyclic are examples of such groups.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Groups
Let G be a metacyclic group, H = 〈a〉 its cyclic normal subgroup,
and set G/H = 〈bH〉. Then G has the following presentation
G =⟨a, b | am = 1, bn = as , bab−1 = ar
⟩and the integers m,n,s, r satisfy the relations
s | m, m | s(r − 1) , r < m, gcd(r ,m) = 1.
When s = m, we say G is split. In this case, G = 〈a〉o 〈b〉.
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Samir Assuena Left Metacyclic Ideals
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Group Codes
Definition
A group code over a field F is any ideal I of the group algebra FG
of a finite group G . A code is said to be metacyclic, abelian, or
dihedral in case the given group G is of that kind. If I is two-sided,
then it is called a central code. A minimal code is an ideal I
(left, two-sided) which is minimal in the set of all (left, two-sided)
ideals of FG .
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Samir Assuena Left Metacyclic Ideals
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Group Codes
The weight of an element α =∑
g∈G αgg is
w(α) =| {g | αg 6= 0, g ∈ G} |
that is, the number of elements of the support of α. The weight
of an ideal I is
w(I ) = min{w(α) | α 6= 0, α ∈ I}.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Definition
Let G1 and G2 be finite groups of the same order and let F be a
field. Let FG1 and FG2 be the corresponding group algebras. A
combinatorial equivalence is a linear isomorphism
φ : FG1 −→ FG2 induced by a bijection φ : G1 −→ G2. Codes
C1 ⊂ FG1 and C2 ⊂ FG2 are combinatorially equivalent if there
exists a combinatorial equivalence φ : FG1 −→ FG2 such that
φ(C1) = C2.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Theorem (Sabin and Lomonaco)
Metacyclic Central Codes are combinatorially equivalent to abelian
codes.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Let Fq be a finite field. For a subgroup S of a group G such that
gcd(q, | S |) = 1, we set
S = 1|S|
∑x∈S
x .
Then S is an idempotent of FqG and is central if and only if S is a
normal subgroup.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Let G be a split metaclyclic group of order pm`n with presentation
G =⟨a, b | apm = 1 = b`
n, bab−1 = ar
⟩and Fq be a finite field with q elements such that
gcd(q, pm`n) = 1. In this case, the group algebra FqG is
semisimple and it can be decomposed as direct sum of simple rings.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Set H = 〈a〉 and let
H = H0 ⊇ H1 ⊇ · · · ⊇ Hm = {1}
be the descending chain of all subgroups of H, i.e.,
Hj = 〈apj 〉, 0 ≤ j ≤ m. Consider the idempotents
e0 = H and ej = Hj − Hj−1, 1 ≤ j ≤ m,
which are central in FqG .
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Write H = a and 〈b〉 = b, so the elements bej , 1 ≤ j ≤ m, are non
central idempotents of FqG .
Proposition
Let e be a central idempotent. Then the left ideal FqG · be is
minimal if and only if the ideal FqG · e is minimal as a two-sided
ideal.
Proposition
The left codes FqG · bej and FqG · (1− b)ej are combinatorially
equivalent to cyclic codes.
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Lemma
For all j , 1 ≤ j ≤ m, the elements αj = ej + ba(1− b)ej are units
inside the ideals FqG · ej .
So, we can construct non central idempotents using the units αj as
follows αj
(bej
)α−1j and α−1
j
(bej
)αj .
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
The non central idempotents are
(b ± ba(1− b))ej , 1 ≤ j ≤ m.
The dimension of FqG · bej over Fq is pj − pj−1 = ϕ(pj), where ϕ
denotes Euler’s totient function. Hence the dimension of
FqG · (b ± ba(1− b))ej over Fq is also ϕ(pj).
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Samir Assuena Left Metacyclic Ideals
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Metacyclic Codes
Proposition
Write f = (b + ba(1− b))ej . If ej is a central primitive idempotent
of Fq 〈a〉, then the set
B = {f , af , a2f , · · · , aϕ(pj )−1f }
is a basis of the left ideal FqG · f over Fq.
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Samir Assuena Left Metacyclic Ideals
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Examples
Example 1: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F5:
f1 = ba, f2 = (1− b)a, e1 = 1− a;
F5G ∼= F5 ⊕ F25 ⊕M3(F25).
f = (b + ba(1− b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F5G · f .
We have found a minimal [21,6,10] left code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 16: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/16.jpg)
Examples
Example 1: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F5:
f1 = ba, f2 = (1− b)a, e1 = 1− a;
F5G ∼= F5 ⊕ F25 ⊕M3(F25).
f = (b + ba(1− b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F5G · f .
We have found a minimal [21,6,10] left code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 17: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/17.jpg)
Examples
Example 1: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F5:
f1 = ba, f2 = (1− b)a, e1 = 1− a;
F5G ∼= F5 ⊕ F25 ⊕M3(F25).
f = (b + ba(1− b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F5G · f .
We have found a minimal [21,6,10] left code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 18: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/18.jpg)
Examples
Example 1: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F5:
f1 = ba, f2 = (1− b)a, e1 = 1− a;
F5G ∼= F5 ⊕ F25 ⊕M3(F25).
f = (b + ba(1− b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F5G · f .
We have found a minimal [21,6,10] left code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 19: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/19.jpg)
Examples
Example 1: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F5:
f1 = ba, f2 = (1− b)a, e1 = 1− a;
F5G ∼= F5 ⊕ F25 ⊕M3(F25).
f = (b + ba(1− b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F5G · f .
We have found a minimal [21,6,10] left code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 20: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/20.jpg)
Examples
Example 2: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F2:
f1 = ba, f2 = (1− b)a,
f3 =1
7
(3 + (ξ + ξ2 + ξ4)Γa + (ξ3 + ξ5 + ξ6)Γa3
),
f4 =1
7
(3 + (ξ3 + ξ5 + ξ6)Γa + (ξ + ξ2 + ξ4)Γa3
), where ξ is a
primitive 7th root of unity;
F2G ∼= F2 ⊕ F4 ⊕M3(F2)⊕M3(F2).
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Samir Assuena Left Metacyclic Ideals
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Examples
Example 2: Set G =⟨a, b | a7 = 1 = b3, bab−1 = a2
⟩.
Central primitive idempotents over F2:
f1 = ba, f2 = (1− b)a,
f3 =1
7
(3 + (ξ + ξ2 + ξ4)Γa + (ξ3 + ξ5 + ξ6)Γa3
),
f4 =1
7
(3 + (ξ3 + ξ5 + ξ6)Γa + (ξ + ξ2 + ξ4)Γa3
), where ξ is a
primitive 7th root of unity;
F2G ∼= F2 ⊕ F4 ⊕M3(F2)⊕M3(F2).
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Samir Assuena Left Metacyclic Ideals
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Examples
Take e1 = 1 + a which is not a central primitive idempotent and
f = (b + ba(1 + b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F2G · f .
This is a [21,6,8]-code, which is not minimal and it has the same
weight of the best known [21,6]-code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 23: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/23.jpg)
Examples
Take e1 = 1 + a which is not a central primitive idempotent and
f = (b + ba(1 + b))e1;
B = {f , af , a2f , a3f , a4f , a5f } basis of the left ideal F2G · f .
This is a [21,6,8]-code, which is not minimal and it has the same
weight of the best known [21,6]-code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 24: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/24.jpg)
Dihedral Codes
D =⟨a, b | apm = 1 = b2, bab = a−1
⟩.
Suppose that U(Zpm) = 〈q〉. The elements
e11 =(
1+b2
)e, e12 =
(1+b
2
)a(
1−b2
)e,
e21 = 4((a− a−1)e)−2(
1−b2
)a(
1+b2
)e, e22 =
(1−b
2
)e.
form a set of matrix units for (FD)e.
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Samir Assuena Left Metacyclic Ideals
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Dihedral Codes
D =⟨a, b | apm = 1 = b2, bab = a−1
⟩.
Suppose that U(Zpm) = 〈q〉. The elements
e11 =(
1+b2
)e, e12 =
(1+b
2
)a(
1−b2
)e,
e21 = 4((a− a−1)e)−2(
1−b2
)a(
1+b2
)e, e22 =
(1−b
2
)e.
form a set of matrix units for (FD)e.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 26: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/26.jpg)
Dihedral Codes
Example 3: Let D9 be dihedral group of order 18, set
e = e1 = H1 − H0, f = e11 − e22.
The set {f , af } is a basis of the minimal left ideal I = FqD9 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 15 and it is the same as that of the best known code
of same dimension and this code is not equivalent to any abelian
code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 27: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/27.jpg)
Dihedral Codes
Example 3: Let D9 be dihedral group of order 18, set
e = e1 = H1 − H0, f = e11 − e22.
The set {f , af } is a basis of the minimal left ideal I = FqD9 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 15 and it is the same as that of the best known code
of same dimension and this code is not equivalent to any abelian
code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 28: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/28.jpg)
Dihedral Codes
Example 3: Let D9 be dihedral group of order 18, set
e = e1 = H1 − H0, f = e11 − e22.
The set {f , af } is a basis of the minimal left ideal I = FqD9 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 15 and it is the same as that of the best known code
of same dimension and this code is not equivalent to any abelian
code.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 29: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/29.jpg)
Dihedral Codes
Example 4: Let D6 be dihedral group of order 6.
Set e = 1− a and set f = e11 − e12.
The set {f , af } is a basis of the minimal left ideal I = FqD6 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 5 and it is the same as that of the best known code
of same dimension.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 30: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/30.jpg)
Dihedral Codes
Example 4: Let D6 be dihedral group of order 6.
Set e = 1− a and set f = e11 − e12.
The set {f , af } is a basis of the minimal left ideal I = FqD6 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 5 and it is the same as that of the best known code
of same dimension.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 31: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/31.jpg)
Dihedral Codes
Example 4: Let D6 be dihedral group of order 6.
Set e = 1− a and set f = e11 − e12.
The set {f , af } is a basis of the minimal left ideal I = FqD6 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 5 and it is the same as that of the best known code
of same dimension.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 32: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/32.jpg)
Dihedral Codes
Example 4: Let D6 be dihedral group of order 6.
Set e = 1− a and set f = e11 − e12.
The set {f , af } is a basis of the minimal left ideal I = FqD6 · f .
If the characteristic of Fq is different from 2,3,5 and 7, the weight
of I of weight 5 and it is the same as that of the best known code
of same dimension.
www.fei.edu.br
Samir Assuena Left Metacyclic Ideals
![Page 33: Left Metacyclic Ideals - IME-USPcimpars/schedule/assuena.pdf · Left Metacyclic Ideals Samir Assuena Centro Universi ario da FEI Joint work with C esar Polcino Milies samir.assuena@fei.edu.br](https://reader031.fdocuments.in/reader031/viewer/2022021909/5bf8da0a09d3f2ac7c8c9c2c/html5/thumbnails/33.jpg)
Thank you!!!!
Samir Assuena Left Metacyclic Ideals