Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Finite dihedral group algebras and coding theory
César Polcino Milies
Universidade de São Paulo
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Basic FActs
The basic elements to build a code are the following:
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Basic FActs
The basic elements to build a code are the following:
A finite set, A called the alphabet. We shall denote by
q = |A| the number of elements in A.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Basic FActs
The basic elements to build a code are the following:
A finite set, A called the alphabet. We shall denote by
q = |A| the number of elements in A.
Finite sequences of elements of the alphabet, that are called
words. The number of elements in a word is called its length.
We shall only consider codes in which all the words have the
same length n.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Basic FActs
The basic elements to build a code are the following:
A finite set, A called the alphabet. We shall denote by
q = |A| the number of elements in A.
Finite sequences of elements of the alphabet, that are called
words. The number of elements in a word is called its length.
We shall only consider codes in which all the words have the
same length n.
A q-ary block code of length n is any subset of the set of
all words of length n, i.e., the code C is a subset:
C ⊂ An = A
| × A ×{z · · · × A} .
n veces
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
A classical scheme due to Shannon
Information
−→
codification
signal
−→
channel
↑
noise
signal
−→
decodification
−→
receiver
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
A classical scheme due to Shannon
Information
−→
codification
signal
−→
channel
signal
−→
decodification
−→
↑
noise
The basic idea in coding theory, is to add information to the
message, called redundancy, in such a way that it will turn
possible to detect errors and correct them.
receiver
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Definition
Given two elements x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) in
An , the number of coordinates in which the two elements differ is
called the Hamming distance from x to y ; i.e.:
d(x, y ) = | {i | xi 6= yi , 1 ≤ i ≤ n} |
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Definition
Given two elements x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) in
An , the number of coordinates in which the two elements differ is
called the Hamming distance from x to y ; i.e.:
d(x, y ) = | {i | xi 6= yi , 1 ≤ i ≤ n} |
Definición
Given a code C ⊂ An the minimum distance of C is the number:
d = min{d(x, y ) | x, y ∈ C, x 6= y }.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem
Let C be a code with minimum distance d and set
d −1
κ=
2
where [x] denotes the integral part of the real number x; i.e., the
greatest integer smaller than or equal to x.
Then C is capable of detecting n − 1 errors and correcting κ errores.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem
Let C be a code with minimum distance d and set
d −1
κ=
2
where [x] denotes the integral part of the real number x; i.e., the
greatest integer smaller than or equal to x.
Then C is capable of detecting n − 1 errors and correcting κ errores.
Definition
The number κ is called the capacity of the code C.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Linear Codes
We shall take, as an alphabet A, a finite field F.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Linear Codes
We shall take, as an alphabet A, a finite field F.
In this case, Fn is an n-dimensional vector space over F.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Linear Codes
We shall take, as an alphabet A, a finite field F.
In this case, Fn is an n-dimensional vector space over F.
We shall take, as codes, subespaces of Fn of dimensión m < n.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Linear Codes
We shall take, as an alphabet A, a finite field F.
In this case, Fn is an n-dimensional vector space over F.
We shall take, as codes, subespaces of Fn of dimensión m < n.
Definition
A code C as above is called a linear code over F.
If d the minimum distance of C, we shall call it a (n,m,d)-code.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Definition
A linear code C ⊂ Fn is called a cyclic code if for every vector
(a0 , a1 , . . . , an−2 , an−1 ) in the code, we have that also the vector
(an−1 , a0 , a1 , . . . , an−2 ) is in the code.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Definition
A linear code C ⊂ Fn is called a cyclic code if for every vector
(a0 , a1 , . . . , an−2 , an−1 ) in the code, we have that also the vector
(an−1 , a0 , a1 , . . . , an−2 ) is in the code.
Notice that the definition implies that if (a0 , a1 , . . . , an−2 , an−1 ) is in the
code, then all the vectors obtained from this one by a cyclic permutation
of its coordinates are also in the code.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let
Rn =
F[X ]
;
hX n − 1i
We shall denote by [f ] the class of the polynomial f ∈ F[X ] in Rn .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let
Rn =
F[X ]
;
hX n − 1i
We shall denote by [f ] the class of the polynomial f ∈ F[X ] in Rn .
The mapping:
ϕ : Fn →
(a0 , a1 , . . . , an−2 , an−1 ) ∈ F[X ]
7→
F[X ]
hX n − 1i
[a0 + a1 X + . . . + an−2 X
n−2
+ an−1 X
n−1
].
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let
Rn =
F[X ]
;
hX n − 1i
We shall denote by [f ] the class of the polynomial f ∈ F[X ] in Rn .
The mapping:
ϕ : Fn →
(a0 , a1 , . . . , an−2 , an−1 ) ∈ F[X ]
7→
F[X ]
hX n − 1i
[a0 + a1 X + . . . + an−2 X
n−2
+ an−1 X
n−1
].
ϕ is an isomorphism of F-vector spaces. Hence A code C ⊂ Fn is
cyclic if and only if ϕ(C) is an ideal of Rn .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
In the case when Cn = ha | an = 1i = {1, a, a2 , . . . , an−1 } is a
cyclic group of order n, and F is a field, the elements of FCn are of
the form:
α = α0 + α1 a + α2 a2 + · · · + αn−1 an−1 .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
In the case when Cn = ha | an = 1i = {1, a, a2 , . . . , an−1 } is a
cyclic group of order n, and F is a field, the elements of FCn are of
the form:
α = α0 + α1 a + α2 a2 + · · · + αn−1 an−1 .
It is easy to show that
FCn ∼
= Rn =
F[X ]
;
hX n − 1i
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
In the case when Cn = ha | an = 1i = {1, a, a2 , . . . , an−1 } is a
cyclic group of order n, and F is a field, the elements of FCn are of
the form:
α = α0 + α1 a + α2 a2 + · · · + αn−1 an−1 .
It is easy to show that
FCn ∼
= Rn =
F[X ]
;
hX n − 1i
Hence, to study cyclic codes is equivalent to study
ideals of a group algebra of the form FCn .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Definition
A group code is an ideal of a finite group algebra.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Definition
A group code is an ideal of a finite group algebra.
If we assume that char (F) 6 | |G |, then to study
group codes is equivalent to study ideals in group
algebras, generated by idempotent elements
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Idempotents from subgroups
Let H be a subgroup of a finite group G and let F be a field such
that car (F) 6 | |G |. The element
eH =
1 X
h
|H|
h∈H
is an idempotent of the group algebra FG , called the idempotent
determined by H.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Idempotents from subgroups
Let H be a subgroup of a finite group G and let F be a field such
that car (F) 6 | |G |. The element
eH =
1 X
h
|H|
h∈H
is an idempotent of the group algebra FG , called the idempotent
determined by H.
eH is central if and only if H is normal in G .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
If H is a normal subgroup of a group G , we have that
b∼
FG · H
= F[G /H].
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
If H is a normal subgroup of a group G , we have that
b∼
FG · H
= F[G /H].
so
dimF
b
(FG ) · H
=
|G |
|H|
= [G : H].
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
If H is a normal subgroup of a group G , we have that
b∼
FG · H
= F[G /H].
so
dimF
b
(FG ) · H
=
|G |
|H|
= [G : H].
Set τ = {t1 , t2 , . . . , tk } a transversal of K in G (where k = [G : H]
and we choose t1 = 1),
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
If H is a normal subgroup of a group G , we have that
b∼
FG · H
= F[G /H].
so
dimF
b
(FG ) · H
=
|G |
|H|
= [G : H].
Set τ = {t1 , t2 , . . . , tk } a transversal of K in G (where k = [G : H]
and we choose t1 = 1), then
b | 1 ≤ i ≤ k}
{ti H
b
is a a basis of (FG ) · H.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Is it possible to determine the primitive central idempotents from
the idempotents determined by subgroups?
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem (Arora-Pruthi (1997), Ferraz-P.M. (2007))
Let F be a field with q elements and A a cyclic group of order p n
such that o(q) = ϕ(p n ) in U(Zpn ) (where ϕ denots Euler’s Totient
function). Let
A = A0 ⊃ A1 ⊃ · · · ⊃ An = {1}
be the descending chain of all subgroups of A. Then, the set of
primitive idempotents of FA is given by:
!
1 X
a
e0 = n
p
a∈A
ei = Abi − Ad
i−1 , 1 ≤ i ≤ n.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem (Arora and Pruthi (2002), Ferraz-PM (2007))
Let F be a field with q elements and A a cyclic group of order 2p n ,
p an odd prime, such that o(q) = ϕ(p n ) in U(Z2pn ).
Write G = C × A where A denotes the p-Sylow subgroup of G and
C = {1, t} is the 2-Sylow subgroup.
If ei , 0 ≤ i ≤ n denotes the set of primitive idempotents of FA,
then the primitive idempotents of FG are
(1 + t)
· ei
2
and
(1 − t)
· ei 0 ≤ i ≤ n.
2
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let A be an abelian p-group. For each subgroup H of A such that
A/H 6= {1} is cyclic, we shall construct an idempotent of FA.
Since A/H is a cyclic subgroup of order a power of p, there exists
a unique subgroup H ∗ of A, containing H, such that |H ∗ /H| = p.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let A be an abelian p-group. For each subgroup H of A such that
A/H 6= {1} is cyclic, we shall construct an idempotent of FA.
Since A/H is a cyclic subgroup of order a power of p, there exists
a unique subgroup H ∗ of A, containing H, such that |H ∗ /H| = p.
We set
b −H
c∗ .
eH = H
and also
eG =
1 X
g.
|G |
g ∈G
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem (Ferraz-PM (2007))
Let p be an odd prime and let A be an Abelian p-group of
exponent p r . Then, the set of idemponts above is the set of
primitive idempotents of FA if and only if one of the following
holds:
(i) p r = 2, and q is odd.
(ii) p r = 4 and q ≡ 3 (mod 4).
(iii) o(q) = ϕ(p n ) in U(Zpn ).
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem (Ferraz-PM (2007))
Let p be an odd prime and let A be an Abelian p-group of
exponent p r . Then, the set of idemponts above is the set of
primitive idempotents of FA if and only if one of the following
holds:
(i) p r = 2, and q is odd.
(ii) p r = 4 and q ≡ 3 (mod 4).
(iii) o(q) = ϕ(p n ) in U(Zpn ).
Theorem (Ferraz-PM (2007))
Let p be an odd prime and let A be an abelian p-group of
exponent 2p r . Write A = E × B, where E is an elementary abelian
2-group and B a p-group. Then the primitive idempotents of FA
are products of the form e.f , where e is a primitive idempotent of
FE and f a primitive idempotent of FB.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem (Ferraz-PM (2007))
Let F be a finite field with |F| = q, and let A be a finite abelian
group, of exponent e. Then the primitive central idempotents can
be constructed as above if and only if one of the following holds:
(i) e = 2 and q is odd.
(ii) e = 4 and q ≡ 3 (mod 4).
(iii) e = p n and o(q) = ϕ(p n ) in U(Zpn ).
(iv) e = 2p n and o(q) = ϕ(p n ) in U(Z2pn ).
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let F be a finite field with q elements and let Dn be the dihedral
group of order 2n, i.e.
Dn = ha, b | an = b 2 = 1, bab = a−1 i.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let F be a finite field with q elements and let Dn be the dihedral
group of order 2n, i.e.
Dn = ha, b | an = b 2 = 1, bab = a−1 i.
We shall assume throughout that char (F) does not divide 2n.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Lemma
Let F be a finite field and let ζ be an e th -root of unity, where e is
the exponent of Dn . Then, the number of simple components of
FDn equals the number of simple components of QDn if and only
if denoting by q the residue class of q in Zn , we have that either
hqi = U(Zn ) or hqi is a subgroup of index 2 in U(Zn ) and
−1 6∈ hqi.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem (Ferraz, Dutra, PM)
The idempotents of FDn can be written from idempotents
determined by subgroups (in a way similar to that of the abelian
case) if and only if one of the following holds:
(i) n = 2 or 4 and q is odd.
(ii) n = 2m , with m ≥ 3 and q is congruent to either 3 or 5,
modulo 8.
(iii) n = p m with p an odd prime and the class q is a generator of
the group U(Zpm ).
(iv) n = p m with p an odd prime, the class q is a generator of the
group U 2 (Zpm ) = {x 2 | x ∈ U(Zpm }) and −1 is not a square
modulo p m .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
(v) n = 2p m with p an odd prime and the class q is a generator of
the group U(Z2pm ).
(vi) n = 2p m with p an odd prime, the class q is a generator of
the group U 2 (Zpm ) = {x 2 | x ∈ U(Z2pm }) and −1 is not a
square modulo 2p m .
(vii) n = 4p m with p an odd prime and both q and −q have order
ϕ(p m ) modulo 4p m .
(viii) n = p1m1 p2m2 with p1 , p2 odd primes, (ϕ(p1m1 ), ϕ(p2m2 )) = 2 and
both q and −q have order ϕ(p1m1 )ϕ(p2m2 )/2 modulo p1m1 p2m2 .
(ix) n = 2p1m1 p2m2 with p1 , p2 odd primes, (ϕ(p1m1 ), ϕ(p2m2 )) = 2
and both q and −q have order ϕ(p1m1 )ϕ(p2m2 )/2 modulo
p1m1 p2m2 .
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let Qn with n even be a generalized quaternion group; i.e, a group
given by the presentation:
Qn = hα, β | αn = 1 β 2 = αn/2 , β −1 αβ = α−1 i
with m ≥ 2.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Let Qn with n even be a generalized quaternion group; i.e, a group
given by the presentation:
Qn = hα, β | αn = 1 β 2 = αn/2 , β −1 αβ = α−1 i
with m ≥ 2.
Theorem
The group algebras FDn and FQn are isomorphic if and only if
4 | n or q ≡ 1 (mod 4).
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Code Parameters
Theorem
Let G be a finite group and let F be a field such that
char (F) 6 | |G |. Let H and H ∗ be normal subgroups of G such that
b −H
c∗ . Then,
H ⊂ H ∗ and set e = H
|G |
|H|
∗
dimF (FG )e = |G /H| − |G /H | =
1− ∗
|H|
|H |
and
w ((FG )e) = 2|H|
where w ((FG )e) denotes the minimal distance of (FG )e.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem
Let G be a finite group and let F be a field such that
char (F) 6 | |G |. Let H and H ∗ be normal subgroups of G such that
b −H
c∗ . Let A be a transversal of H ∗ in G
H ⊂ H ∗ and set e = H
and τ a transversal of H in H ∗ containing 1. Then
b | a ∈ A, t ∈ τ \ {1}}
B = {a(1 − t)H
is a basis of (FG )e over F.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem
Let Hi ⊂ Hi∗ , be normal subgroups of a group G , 1 ≤ i ≤ k, such
that Hi∗ ∩ Ni∗ = {1}, where Ni denotes the subgroup generated by
c1 − H
c∗ )(H
c2 − H
c∗ ) · · · (H
ck − H
c∗ ).
all Hj∗ with j 6= i. Set e = (H
1
2
k
Then,
|H1 |
|H2 |
|Hk |
|G |
1− ∗
1 − ∗ ··· 1 − ∗
dimF (FG )e =
|H1 H2 · · · Hk |
|H1 |
|H2 |
|Hk |
and
w ((FG )e) = 2k |H1 H2 · · · Hk |.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Theorem
Let Hi ⊂ Hi∗ , be normal subgroups of a group G ,
1 ≤ i ≤ k, such that Hi∗ ∩ Ni∗ = {1}, where Ni denotes the
subgroup generated by all Hj∗ with j 6= i. Set
c1 − H
c∗ )(H
c2 − H
c∗ ) · · · (H
ck − H
c∗ ). Let A be a transversal of
e = (H
1
2
k
H ∗ in G and τi a transversal of Hi in Hi∗ containing 1, 1 ≤ i ≤ k.
Then
b | a ∈ A, ti ∈ τi , ti 6= 1, 1 ≤ i ≤ k}
B = {a(1−t1 )(1−t2 ) · · · (1−tk )H
is a basis of (FG )e over F.
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Dimensions and weights when n involves only one prime.
n
4
2m
pm
e
1 − a2
[
e2i = Cc
2i − C2i+1
[
epi = Cc
p i − Cp i+1
dim[(FDn )e]
4
m−i
2
2ϕ(p i−i )
w [(FDn )e]
2
i+1
2
2p i
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Dimensions and weights when n involves two primes.
n
2p m
4p m
ep i = Cc
pi
ep =
e i = Cci
c2
e2 = C
c2
e2 = 1 − C
c2
e2 = 1 − C
c4
e =C
e2i = Cc
−
C[
,
2i
2i+1
e i = Cci − C[
i+1 ,
2
m m
p1 1 p2 2
2
2
[
ep1 = C
m1
p
d
e i =C
− C[
p
pi
p i+1
1
1
1
d
[
e i =C
i − C i+1
p
1
p
1
p
1
p
− C[
p
p
p i+1
epi = Cc
− C[
pi
p i+1
ep = Cd
m
p
ep i = Cc
− C[
pi
p i+1
d
\
e j =C
j − C j+1
p
2
1
− C[
p i+1
Cd
m
p
p
2
2
[
e p2 = C
m2
p
2
d
\
e j =C
j − C j+1
p
p
p
2
2
2
dim[I]
w [I]
2ϕ(p m−i )
4p i
2
2ϕ(p m−i )
2p i
4p i
2ϕ(p m−i )
8p i
ϕ(2i )
2 . 2i p m
2ϕ(2i )ϕ(p j )
4 . 2i p j
m −j
2ϕ(p2 2 )
m j
2p1 1 p2
m −i
2ϕ(p1 1 )
m1 −i
m −j
2ϕ(p1
)ϕ(p2 2 )
m
2p1i p2 2
j
4p1i p2
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
Dimensions and weights when n = 2p1m1 p2m2 .
e
e1
e2
dim[FDn )ee1 e2 ]
w [(FDn )ee1 e2 ]
c2
C
m
d
Cp11
d
\
C
j − C j+1
m −j
2ϕ(p2 2 )
m j
4p1 1 p2
m
d
Cp22
m −i
2ϕ(p1 1 )
m
4p1i p2 2
d
\
C
j − C j+1
m −j
2ϕ(p2 2 )
m j
4p1 1 p2
[
C
m2
m −i
2ϕ(p1 1 )
m
4p1i p2 2
d
\
C
j − C j+1
m −i
m −j
2ϕ(p1 1 ϕ(p2 2 )
8p1i p2
m −i
m −j
2ϕ(p1 1 ϕ(p2 2 )
j
8p1i p2
c2
C
p
p
1
1
[
C
m1
p
p
1
1
p
d
[
C
i − C j+i
c2
1−C
d
[
C
i − C j+i
p
p
1
1
p
p
p
1
c2
C
1
1
p
2
d
[
C
i − C j+i
p
p
2
d
[
C
i − C i+1
c2
1−C
c2
1−C
p
p
2
2
2
2
p
2
d
\
C
j − C j+1
p
2
p
2
j
Basic Facts
Linear and Cyclic codes
Group Codes
Dihedral Codes
THANK YOU!!