Journal of Sciences, Islamic Republic of Iran 26(1): 49 - 55 (2015) http://jsciences.ut.ac.ir
University of Tehran, ISSN 1016-1104
49
Cyclic Orbit Codes with the Normalizer of a Singer
Subgroup
F. Bardestani and A. Iranmanesh
*Department of pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Islamic Republic of Iran
Received: 16 November 2014 / Revised: 3 January 2015 / Accepted: 17 February 2015
Abstract
An algebraic construction for constant dimension subspace codes is called orbit code.
It arises as the orbits under the action of a subgroup of the general linear group on
subspaces in an ambient space. In particular orbit codes of a Singer subgroup of the
general linear group has investigated recently. In this paper, we consider the normalizer
of a Singer subgroup of the general linear group and its orbit codes. Several properties of
these codes are considered.
Key words: Linear network coding; Constant dimension codes; Group action; Orbit codes.
*
Cyclic Orbit Codes with the Normalizer of a Singer Subgroup
49
Introduction
Random linear network coding introduced in [1], is used to increase the information throughput by allowing the random linear combination of packets within a network i.e., the middle nodes combine packets linearly, where they choose the coefficients for linear combinations randomly within a finite field. Due the encoding method the receivers are able to reconstruct the original packets that have been injected into the network at its sources. Although this method is very effective but it is highly sensitive to the error propagation. In 2008, Kötter and Kschischang developed an algebraic approach to overcome this deficiency [9]. They suggested an error correcting network code as a family of subspaces of an ambient space
F
n. This pioneering article initiate intensive research effort on subspace codes, particularly for construction of large error-correcting codes with efficient encoding and decoding algorithms. In 2009, Etzion and Silberstein presented a construction of subspace codes with large distance and cardinality which is based on rank-metric codes [4]. In [5], Etzion and Vardy introduced the concept of a cyclic subspace code and present several optimal cyclic subspace codes.In [13], Trautmann et al., used group action and present an algebraic construction for subspace codes named orbit code. Especially by considering the group action of a certain group (Singer subgroup), they presented an algebraic construction for cyclic subspace codes.
In [6], Gluesing-Luerssen et al., introduced the notion of Stabilizer subfield and investigate the cardinality and distance of the orbit codes of the Singer subgroup.
While the orbit codes of a Singer subgroup are not large but taking the unions of them is a nice idea to obtain large cyclic subspace codes for a given minimum distance. The normalizer of a Singer subgroup is a candidate to have an algebraic structure for some certain unions of the orbits of the Singer subgroup.
In 2013, Braun et al., presented a nontrivial q -analog of Steiner systems, which also form an optimal cyclic code. This code has the normalizer of the Singer subgroup as its automorphism group [2].
In this paper, we will study the orbits of the normalizer of a Singer subgroup, which are a class of cyclic subspace codes. The structure of the orbits of this normalizer is closely related to the orbits of the Singer subgroup.
Notation and preliminary results
A subspace code of length is defined as a collection of subspaces in
F
qn , whereF
q is a finite field. If all the subspaces have the same dimension the code is called constant dimension code. The subspace distance between two subspaces is defined as follows [9]:Therefore, the minimum distance of a code
C
is defined as:As a consequence, if
C
is a constant dimension code, thend
min ) 2
C
k
.The dual of the subspace code
C
is denoted by
C
and is defined as . Since,
then
d
min(
C
)
d
min
C
[13].The set of all -dimensional subspaces of
F
qn is called Grassmannian, denoted by . It is well known that:Let be the general linear group , an orbit code, is defined as the orbit of the natural action of a subgroup of on the Grassmannian [13]. More precisely let
U
Mat
k n(
F
q)
be a full-rank matrix such that and be an element of . We can define, , then an orbit code byG
and starting pointU
is denoted by OrbitG( )U , so . Let Stab ( ) : {G U gG | .Ug U} be the stabilizer of the action, then simple facts in group theory leads that
and since ,
then:
n
k
( , ) : dim( ) dim( ) 2dim( ) s
d U V U V UV
min( ) : min{ s( , ) | , , }. d C d U V U VC UV
: {
|
}
C
U
U
C
(
,
)
( , )
s s
d
U V
d
U V
k
( , )
k n
G
1
1
(
1)
| ( , ) |
.
(
1)
i n i i n k
i k i q
i
q
n
k n
k
q
G
GL( , )
n q
G
GL( , )
n q
: rowspace( )
U
U
g
G
U
. : rowspace(
g
Ug
)
( ) : { . | } G
Orbit U Ug gG
| | | ( ) |
| Stab ( ) | G
G
G Orbit U
U
1
(
,
)
( ,
)
s s
. Let
C
1 andC
2 be two subspace codes of length such that
C
1
(
C
2)
where
GL( , )
n q
, thenC
1 andC
2 are called linearly isometric or simply isometric. It is easy to see that isometric codes have the same minimum distance and cardinality [13]. Since orbit codes are a class of subspace codes, the isometry between them is defined in the same way. In [10], it is proved that two conjugate subgroups lead isometric orbit codes. For more information about isometry in subspace codes, the reader is refered to [12].If , we say
C
is a code with parameters , we may omit , where the cardinality is not the point. With these parameters it is easy to see that the dual code is an orbit code. The related orbit code to a cyclic subgroup of is called a cyclic orbit code.An element of order in is called a Singer cycle. Since the multiplicative group of the field
F
qn is cyclic of order and we can take itsgenerator as an element of , so for every and every there is a Singer cycle in . A cyclic group generated by a Singer cycle called a Singer subgroup. There are several ways to describe a Singer subgroup, for example, let be a primitive polynomial of degree over
F
q, and be its companion matrix, then is a Singer subgroup of[13].
We recall that if ,
then
.
Since the extension field n q
F is a vector space of dimension over
F
q, we may consider nq
F . as the
vector space. Fix as a primitive element of n q F , then
*
n q
C
F is a Singer subgroup of .Since all Singer subgroups are conjugate, in order to investigate orbits of a Singer subgroup, without loosing generality we investigate the orbits of
A collection of subspaces with this property that if is a codeword, then is also a codeword which is called a cyclic subspace code [5]. Whence one of the several interesting properties for , is that every is a cyclic subspace code.
According to the definition of isometric subspace codes, the auotomorphism group of is contained Recall that the Frobenius automorphism
:
qn qn
F
F
is defined by for all.
n
q
x
F
We mention, the following example which shows, in generalExample 1.1 Consider primitive polynomial
. Let
F
27 be a root ofp x
( )
and be a
subspace of
F
27, then , so, while .
In the next section, we will focus on the normalizer of a Singer subgroup and the related orbits.
Results and Discussion
Let be a subgroup of , the normalizer of in is the set of all elements in
such that commute with . That is: .
Let , then it is well known that is a subgroup of . It is straightforward to see that :
By this method, if we consider then we obtain a union of cyclic subspace codes as an orbit code. Through the paper, we denote the normalizer of the
min( G( )) : min{ ( ,s ) | \ Stab ( )}G
d Orbit U d U Ug gG U
n
( )
GOrbit
C
U
min[ , ,|
n k
C
|,
d
( )]
C
| |
C
C
min
[ ,
n n k
,| |,
C
d
( )]
C
GL( , )
n q
1
n
q
GL( , )
n q
1
n
q
GL( , )
n q
n
q
GL( , )
n q
( )
p x
n
M
pp
M
GL( , )n q
1
0 1 1
( )
...
n np x
p
p x
p
x
0 1 1
0 0 0 0
0 1 0 0
0 0 1
n p
p p p
M
n
GL( , )
n q
.
C
{0,
xi|
i
1,..2
k
1}
1{0,
xi|
i
1,..2
k
1}
( )
C
Orbit
U
(
)
C t
Orbit
U
( )
C
Orbit
U
.
C
( ) :
x
x
q
(
C( ))
.
Aut Orbit
U
C
7
( )
1
p x
x
x
8 24 56 120 121 123
{0,1,
,
,
,
,
,
}
U
8
U
U
(
C( ))
Aut Orbit
U
C
G
GL( , )
n q
G
GL( , )
n q
GL( , )
n q
G
GL
( ) : {
GL( , ) |
}
N
G
n
n q nG
Gn
GL
( )
n
N
G
,
.
G n
G n
GL( , )
n q
| | . 0
( )
(
).
i n iG n G
i
Orbit
Orbit
n
U
U
,
Cyclic Orbit Codes with the Normalizer of a Singer Subgroup
51 Singer subgroup in by . Galois group
Gal
(
F
qn/
F
q)
is a cyclic group which is generated by the Frobenius automorphism of order . We mention a well known construction for as a semidirect product [7]. We recall the definition of a semidirect product as the following:Definition 2.1. Let and be two arbitrary
groups and be a homomorphism (denote the image of every
h
under
by
h). On the Cartesian product define, the following operation:.
The set with this operation is a group, which is called the semidirect product of and , denoted by .
Theorem 2.2. [7] has order and is
isomorphic to the semidirect product of the Galois group
Gal
(
F
qn/
F
q)
andC
.As a consequence of Theorem 2.2,
and where is the
smallest integer such that . It turns that the orbit of is the union of cyclic subspace codes and so is cyclic too.
We mention two other important results which can be found in [2] and [3].
Lemma 2.3. [2; Lemma 4] The normalizer of a
Singer subgroup is self-normalizing in , that is
The following theorem mentions when the normalizer of a Singer subgroup is a maximal subgroup
Theorem 2.4. [3] Let be an odd prime. Then the
normalizer of a Singer subgroup is a maximal subgroup
of .
As a consequence of Theorem 2.4, if is a prime, then and we have the
following result:
Theorem 2.5. Suppose that is a prime. Let
be the normalizer of a Singer subgroup in . Then distinct orbit codes by are nonisomorphic.
Proof. Let and be two distinct orbit codes, and there is such that Then for every , we have:
that is and hence
It follows that , which is a contradiction.
Remark 2.6. Let be an arbitrary -dimensional
subspace of
F
qn according to [6], there exist a-dimensional subspace such that and Then
( i) ( ( )) i ( i) ( i)
C C C C
Orbit U Orbit U Orbit U Orbit U
So . Therefore we
can restrict ourselves to the subspaces that contain the identity
1
nq
F
.According to the above remark, if , then since ,
A spread code in is a set of subspaces of dimension k such that they pairwise intersect only trivially and they cover the whole vector space
F
qn. Spread code is optimal since its minimum distance is2
k
and its cardinality is 1 1n k
q q
[12]. By Corollary 3.8
in [6], an is a spread code if and only if
k q
F
U=
. Then since
(
F
qk)
F
qk,C
(
k)
(
k)
N q q
O
r t
bi
F
Orbi
t
F
. Therefore an interesting case is to focus on orbit codes byN
with parameters,In the following examples, we mention several
C
GL( , )
n q
N
n
N
H
K
:
H
K
H
K
2
1 1 2 2 1 2 1 2
( , k )( ,
h
h k
)
(
h h
, (
k
h) )
k
H
K
H
K
H
ã
K
N
n q
(
n
1)
.
N
C
0
( )
(
),
s
i
N C
i
Orbit
Orbit
U
U
s
( ) ( s)
C C
Orbit U Orbit U
N
N
GL( , )
n q
GL
(
)
.
N
N
N
n
GL( , )
n q
2
n
(
N( ))
Aut Orbit
U
N
3
n
N
GL( , )
n q
N
( )
N
Orbit
U
Orbit
N(
U
)
GL( , )
g
n q
( )
(
(
)) .
N N
Orbit
U
Orbit
U
g
n
N
1 1 1
(OrbitN(U))gng (OrbitN( ))U ng (OrbitN( )U)g OrbitN(U),
1
gng
N
g
N
GL(
N
)
N
.
( )
(
)
N N
Orbit
U
Orbit
U
U
k
k
U
1
U
( )
(
).
C C
Orbit
U
Orbit
U
NOrbit
U
Orbit
N(
U
)
U
U U
(1) 1
d
min(
Orbit
N( ))
U
2(
k
1).
( , )
k n
G
( )
C
Orbit
U
, , 2 1 . n k k
orbits with parameters , where
Example 2.7. Let be a subspace of
2n
F
where Then is acode with parameters .
Example 2.8.
1) Let then is a
code.
2) Let then is a
-code.
3) Let , then is a
-code.
4) Let , then is a
-code.
Remark 2.9. Let divides and r
q
F
be the largest subfield inF
qn such that is a vector spaceover
F
qr. In [6], the authors calledF
qr, the best friendfor . Consider 1
0 r t il q i
F
U= , then
[6]. In fact this result is based on this property that for every , is a vector sapcae over
F
qr too. It is obvious that this is not truefor every element of . However, since
1 1
0 0
(
r)
rt t
il ilq
q q
i
i
F
F
, for every ,is a vector space over r q
F
therefore we have proved the following theorem:Theorem 2.10. Let divides and 1
0 r t il q i
F
U=
be a nontrivial subspace of
F
qn, where . Ifrt q
F
U=
, then .Example 2.11. Let 2 2
3 1
F
2
F
2U=
and2 2
5 2F2
F2U= , then for we have,
. Also and
In [6], the authors proved that when is a prime, *
Stab ( )C U Fq and so . Hence
we have the following result:
Lemma 2.12. If is a prime, then
N
( )
C( )
O
rbit
U
Orb
it
U
. or .Let
F
q
F
2, if and then . However as a consequence of the above lemma, in the following Corollary, we will prove that, if is a prime, then there is not with parametersCorollary 2.13. Let be a prime number,
2
q
F
F
and , thenor
Proof. Supppose and
Then Singelton bound for subspace codes [9; Theorem 9], leads that:
therefore
On the other hand if
k
3
, then thus for ,which is a contradiction.
Consider , and then
NOrbit
U
[ , , 2(
n k
k
1)]
|
Orbit
N( ) | |
U
Orbit
C( ) |:
U
2 19
1,
,
U
{12
20}.
n
Orbit
N
U
[ ,3, (2
n
n
n
1), 4]
5 93 1, , ,
U
Orbit
N
U
8[8,3, 2(2
1), 4]
1 47 1, , ,
U
Orbit
N
U
9[9,3,9(2
1), 4]
7 87 1, , ,
U
Orbit
N
U
13
[13, 4,13(2
1), 6]
5 7 87 1,
, ,
U
Orbit
N
U
17[17, 4,17(2
1), 6]
r
n
U
U
min
(
C( ))
2
d
Orbit
U
r
J
U
JGL( , )
n q
g
N
U
g
r
n
1 1 n r q l q min
(
N( ))
2
d
Orbit
U
r
10,
n
1, 2
i
min
(
N(
i))
4
d
Orbit
U
10
1 2
2 1
| ( ) | 2
2 1 N Orbit U 10 2 2
2
1
|
(
) | 10
.
2
1
N
Orbit
U
n
1
|
( ) |
1
n Cq
Orbit
q
U
n
1 | ( ) | ( ) 1 n N q Orbit n q U5
n
k
2,
min
(
N( ))
2
2(
1)
d
Orbit
U
k
7
n
( )
N
Orbit
U
1 1
[ , , (2 1), 2( 1)] :
2 2
n
n n
n n
7 n
1
2
n
k
( )
( )
N C
Orbit
U
Orbit
U
min
(
N( ))
2(
2).
d
Orbit
U
k
( )
( )
N C
Orbit
U
Orbit
U
min
(
N( ))
2(
1).
d
Orbit
U
k
2 (2 4) / 2
(2 1) ,
max{ , }
n n k
n
n n k
3 2
2 1
(2
1)(2
1)
(2
1)(2
1)
.
3
k k kk
3 2 4 2 1
(2k 1)(2k 1) 2 (2k 1),
n
7
3 2 2 1
(2
k
1)(2
k
1)
3(2
k
1)(2
k
1),
8
n
q
2
17 341, , ,
Cyclic Orbit Codes with the Normalizer of a Singer Subgroup
53 . In general it is an interesting question that when the Frobenius automorphism is a shift on the elements of ? In the other words, when is not equal to In the following theorem, we give an answer to this question:
Theorem 2.14. Let be a prime, and
, such that , and ,
then .
Proof. Since is a prime, then is a prime,
therefore by Lemma 2.12, if we suppose
, then .
Since , we have, for some .
Suppose and denote the
powers of the elements of U
0 byX
U: .It is obvious that and all the operations over the elements are done module . We have
Let , if , then
3
J
0
. Since and is a prime number, then , a contradiction. Suppose , so . If , then , which is a contradiction. Therefore so . In general suppose , thenSince , then and we have the following two cases which we obtain a contradiction in both of them:
Case 1. If , then there is not an element
such that and it is a contradiction.
Case 2. If , then for all ,
. Since
n
is a prime, thenn
2
k
2
, hence there is at least two elemets in the following equal sets:{
|
,..., 2
k2} {2
|
,...2
k2}.
i i
x
J i
n
x i
n
If
x
n
J
2
x
n , then and for all, . Let ,
then we have
Therefore it is easy to see that which is a contradiction. If we suppose that there is not any i such that , then we have and in the same way we have, which is a contradiction and the proof is completed.
By Example 2.7 ( ), we can see that the converse of the above theorem is not true.
Remark 2.15. In the proof of Theorem 2.14, if
, then where
for we have . This gives the structure of the subspace , which
.
We know that if is a prime number, then
is a prime, so , also if , then for every we have , therefore we can prove the following theorem:
Theorem 2.16. Let be a prime number,
and . Then
Proof. Since the proof is similar to Theorem 2.14, we
omit it.
Conclusion
We considered orbit codes of the normalizer of a Singer subgroup. The structure of this normalizer leads a close relation between its orbits and the orbits of the Singer subgroup. Since these orbits are union of the cyclic subspace codes, they are cyclic subspace codes as well. Despite the orbits of the normalizer of a Singer subgroup are not large, taking the union of these orbits is suitable to form large cyclic subspace codes. For further work the main aid is to use the normalizer of the
( )
( )
N C
Orbit
U
Orbit
U
J
U
( )
N
Orbit
U
( ).
C
Orbit
U
2
n
1
n
5
dim( )
U
k
k
log (
2n
1)
U
U
|
Orbit
N( ) |
U
n
(2
n
1)
2
n
1
n
| ( ) | (2n 1)
N
Orbit U n OrbitN( )U OrbitC( )U
U U
U
U
JJ
0
{0,1,
xi|
i
1,..., 2
k2}
U
{0, |
i1, 2,..., 2
k2}
X
U
x i
2n
X
U
Z
2
n
1
{ ,
|
1, 2,..., 2
2}
J
k i
X
U
J x
J i
{0, 2 |
i1, 2,..., 2
k2}.
X
x i
U
1
0
x
J
2
x
1
J
2
n
1
3
J
0
1 2
2
x
x
J
x
2
3
x
12
2
x
J
J
0
2 3
2
x
x
J
x
3
7
x
11
(2
i1)
i
x
x
1 12
x
iJ
x
(2
i1)
0
n i
1.
∣
(
1)
q
k
log n
n
1
q
k1 2
kn
x
i2
x
i
J
2
k
n
1
1
i
n
1
1
(2
i1)
i
x
x
n
x
J
2
i2
n i
x
i
J
2
x
ix
n1
J
2
x
n21
2
2,
22
3,...,
2
1.
n n n n n i n
x
J
x
x
J
x
x
J
x
1 n
x
J
2
i i
x
J
x
x
n
J
2
x
n1n
x
J
2
1
2
kn
X
U
{0,
x i
i|
1, 2,..., 2
k
2}
1
i
2
k
2
x
i
(2
i
1)
x
1U
( )
( )
N C
Orbit
U
Orbit
U
1
1
nq
q
n
1
|
( ) |
1
n Cq
Orbit
q
U
q
2
k
k
log (
qn
1)
1
1
nq
q
5
n
q
2
|
( ) |
1
.
Singer subgroup to construct optimal subspace codes. Also we will consider the normalizer of an irreducible cyclic subgroup of the general linear group and investigate its orbits.
Acknowledgement
The authors are grateful to the referee for the useful comments which improved the presentation of manuscript. This work was partially supported by the Center of Excellence of Algebraic Hyperstructures and its Applications of Tarbiat Modares University (CEAHA). Partial support by the Iran Telecommunication Research Center is gratefully acknowledged by the first author (FB).
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