constructions of constant dimension codes with ferrers … · 2018. 8. 13. · a ferrers diagram is...
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constructions of Constant Dimension Codeswith Ferrers Diagram Rank Metric Codes
Anna-Lena Horlemann-Trautmann
University of St. Gallen, Switzerland
July 13th, 2018
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Preliminaries
1 PreliminariesRank-metric codesSubspace codes
2 Constant Dimension Code ConstructionsFirst construction: lifted MRD codesSecond construction: lifted Ferrers diagram codesThird construction: pending dotsFourth construction: pending blocks
3 Summary and Outlook
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Preliminaries
Rank-metric codes
Rank-metric codes
Theorem
Let A,B P Fm�nq be two matrices. It holds that
dRpA,Bq :� rankpA�Bq
defines a metric on Fm�nq . It is called the rank distance.
Definition
A linear rank-metric code is simply a subspace of Fm�nq . The
minimum rank distance of a code C � Fm�nq is defined as
dRpCq :� mintdRpA,Bq | A,B P C,A � Bu.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Preliminaries
Rank-metric codes
Theorem
Let C � Fm�nq be a linear rank-metric code with minimum
rank distance δ. Then
|C| ¤ qmaxtm,nupmintm,nu�δ�1q.
For any set of parameters n,m ¥ δ P N and arbitrary fieldsize there exist codes attaining this bound. These codes arecalled maximum rank distance (MRD) codes.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Preliminaries
Subspace codes
Subspace codes
Theorem
Let U ,V � Fnq be two vector spaces. It holds that
dSpU ,Vq :� dimpU � Vq � dimpU X Vq
defines a metric on Pqpnq :� tU ¤ Fnq u. It is called the subspacedistance.
Definition
A subspace code is simply a subset of Pqpnq. A constantdimension code is simply a subset of Gqpk, nq :�tU ¤ Fnq | dimpUq � ku. The minimum subspace distance of acode C � Pqpnq is defined as
dSpCq :� mintdSpU ,Vq | U ,V P C,U � Vu.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Preliminaries
Subspace codes
Some boundsAqpn, δ, kq � max. cardinality of C � Gqpk, nq with dSpCq � 2δ.
1 Singleton:
Aqpn, δ, kq ¤ |Gqpn� δ � 1, k � δ � 1q| �
�n� δ � 1n� k
�q
2 Johnson-type I:
Aqpn, δ, kq ¤
Zpqn�k � qn�k�δqpqn � 1q
pqn�k � 1q2 � pqn � 1qpqn�k�δ � 1q
^
Johnson-type II:
Aqpn, δ, kq ¤
Zqn � 1
qk � 1
Z. . .
Zqn�k�δ � 1
qδ � 1
^. . .
^^
3 Etzion-Vardy: If k does not divide n, then
Aqpn, k, kq ¤
Zqn � 1
qk � 1
^� 1.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
First construction: lifted MRD codes
1 PreliminariesRank-metric codesSubspace codes
2 Constant Dimension Code ConstructionsFirst construction: lifted MRD codesSecond construction: lifted Ferrers diagram codesThird construction: pending dotsFourth construction: pending blocks
3 Summary and Outlook
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
First construction: lifted MRD codes
Lifted MRD codesFor a given rank-metric code C � Fk�nq the set
liftpCq :� rs�Ik A
�|A P C
(
is called the lifting of C.
Theorem
If C � Fk�pn�kqq is an MRD code of minimum rank distance δ,then liftpCq � Gqpk, nq is a constant dimension code withminimum subspace distance 2δ. Moreover,
|liftpCq| � qpn�kqpk�δ�1q.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
First construction: lifted MRD codes
Multi-component lifted MRD codes
Theorem
Let Cj be some MRD code with minimum rank distance δ in
Fk�pn�k�jδqq for j � 0, . . . , tn�kδ u. Then
Cj � rs�
0k�jδ Ik�k M�|M P Cj
(
are called the component codes and the union C ��tn�k
δu
j�0 Cj is asubspace code in Gqpk, nq with minimum subspace distance 2δand cardinality
N �
tn�2kδ
u¸i�0
qpk�δ�1qpn�k�δiq �
tn�kδ
u¸i�tn�2k
δu�1
rqkpn�k�1�δpi�1qqs.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
First construction: lifted MRD codes
Proof.The distance between any elements of the same component Cifollows from the MRD code.Now let U P Ci and V P Ci�1. Since the identity blocks areshifted by δ positions, the maximal intersection ispk � δq-dimensional. Thus
dSpU, V q � 2k � 2 dimpU X V q ¥ 2k � 2pk � δq � 2δ.
The size of the code follows from the formula for the size/dimension of the corresponding MRD code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
First construction: lifted MRD codes
Theorem
1 If δ � k and k|n, the MC-LMRD code construction isoptimal.
2 If δ � k, q � 2 and k|n� 1, the MC-LMRD codeconstruction is optimal.
Proof.
1 Meets Johnson II bound (spread codes).
2 Meets Etzion-Vardy bound (partial spread codes).
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
First construction: lifted MRD codes
Example:We want to construct a spread code in G2p2, 4q. Let C � F2�2
2
be an MRD code with dRpCq � 2, e.g.
C �
"�0 11 1
,
�1 11 0
,
�1 00 1
,
�0 00 0
*.
Then the component codes are
C1 � trsrI2�2 As | A P Cu and C2 � rsr 02�2 I2�2 s
and C � C1 Y C2 is the desired (spread) code with minimumsubspace distance 4 and cardinality 5.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
1 PreliminariesRank-metric codesSubspace codes
2 Constant Dimension Code ConstructionsFirst construction: lifted MRD codesSecond construction: lifted Ferrers diagram codesThird construction: pending dotsFourth construction: pending blocks
3 Summary and Outlook
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Idea: Break up the leading identiy blocks and allow anyreduced row echelon form (RREF).
Definition
The identifying vector of U P Gqpk, nq, denoted by vpUq isthe binary vector of length n and weight k, such that the kones of vpUq are exactly in the positions where RREFpUqhas the leading ones (also called the pivots).
A Ferrers diagram is a collection of dots such that the rowshave a decreasing and the columns have an increasingnumber of dots (from top to bottom and from left to right,respectively).
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Example:Let U be the subspace in G2p3, 8q with the following generatormatrix in reduced row echelon form:
RREFpUq �
�� 1 0 0 0 1 1 0 0
0 0 1 0 1 0 1 00 0 0 1 0 1 1 1
� .
Its identifying vector is vpUq � p10110000q, and its Ferrerstableaux form and Ferrers diagram are given by
FpUq �0 1 1 0 0
1 0 1 00 1 1 1
, FU �
.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Given vpUq, the unique corresponding FU can be found. Forthis, consider the zeros of vpUq – each zero after the first onerepresents a column in the Ferrers diagram, where the numberof dots in the column is equal to the number of ones before thezero in vpUq.
Example:Consider k � 3, n � 5 and vpUq � p11010q. Then thecorresponding Ferrers diagram is
.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Given vpUq, the unique corresponding FU can be found. Forthis, consider the zeros of vpUq – each zero after the first onerepresents a column in the Ferrers diagram, where the numberof dots in the column is equal to the number of ones before thezero in vpUq.
Example:Consider k � 3, n � 5 and vpUq � p11010q. Then thecorresponding Ferrers diagram is
.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Definition
Let F be a Ferrers diagram with m dots in the rightmostcolumn and ` dots in the top row. A code CF � Fm�`
q is anrF , ρ, δs-Ferrers diagram rank-metric (FD-RM) code if
for all codewords of CF , all entries not in F are zeros,
it forms a linear subspace of dimension ρ of Fm�`q , and
the rank distance between any two distinct codewords is atleast δ.
If F is a rectangular m� ` diagram with m` dots then theFD-RM code is a classical rank-metric code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Theorem
Let F be a Ferrers diagram and CF the corresponding FD-RMcode. Then |CF | ¤ qminitwiu, where wi is the number of dots inF which are not contained in the first i rows and the rightmostδ � 1� i columns (0 ¤ i ¤ δ � 1).
Definition
A code which attains this bound is called a Ferrers diagrammaximum rank distance (FD-MRD) code.
For many parameter sets constructions of FD-MRD codes areknown.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Example:We want to find a FD-MRD code over F2 with minimum rankdistance δ � 2 for the Ferrers diagram
F �
.
The code
CF �
$&%�� 0 0 0
0 0 00 0 0
� ,
�� 1 0 0
0 0 00 0 1
� ,
�� 0 1 0
0 0 10 0 0
� ,
�� 1 1 0
0 0 10 0 1
� ,.-
fulfills all the conditions, i.e. it fits F , forms a subspace ofdimension 2 and has minimum rank distance 2.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Definition
For a codeword A P CF � Fk�pn�kqq let AF denote the part of Arelated to the entries of F in A. Given a FD-RM code CF , alifted FD-RM code CF is defined as follows:
CF � liftpCF q :� tU P Gqpk, nq | FpUq � AF , A P CFu.
Theorem
If CF � Fk�pn�kqq is an rF , ρ, δs-Ferrers diagram rank-metriccode, then its lifted code CF is an pn, qρ, δ, kqq-constantdimension code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
As before, we can now construct multi-component liftedFD-RM codes. As a first step one needs to decide whichidentifying vectors one wants to use for the component codes.
Theorem
1 For U ,V P Gqpk, nq it holds that
dSpU ,Vq ¥ dHpvpUq, vpVqq :� |supppvpUq � vpVqq|.
2 If vpUq � vpVq then
dSpU ,Vq � 2dRpRREFpUq,RREFpVqq� 2dRpFpUq,FpVqq.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Multi-component lifted FD-RM codes
Theorem
The following construction produces a constant dimension codeC � Gqpk, nq with dSpCq � 2δ:
Choose a binary block code C � Fn2 of constant weight kand minimum Hamming distance 2δ.
Use each codeword vi P C as the identifying vector of acomponent code and construct the corresponding liftedFD-RM code Ci with minimum rank distance δ.
The union C ��|C|i�1Ci is the final code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Example:We want to construct a code in Gqp3, 6q of subspace distance 4,hence we start with a binary linear code of length 6, weight 3and Hamming distance 4:
C � tp111000q, p100110q, p010101qu
The corresponding reduced row echelon forms and Ferrersdiagrams are:��
1 0 0
0 1 0
0 0 1
� ,
��
1 0 0
0 0 0 1 0
0 0 0 0 1
� ,
��
0 1 0 00 0 0 1 00 0 0 0 0 1
�
We can fill the Ferrers diagrams with FD-MRD codes withminimum rank distance 2 of size q6, q2 and q. The union of thelifting of these codes is a code of subspace distance 4 withq6 � q2 � q elements.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Example:We want to construct a code in Gqp3, 6q of subspace distance 4,hence we start with a binary linear code of length 6, weight 3and Hamming distance 4:
C � tp111000q, p100110q, p010101qu
The corresponding reduced row echelon forms and Ferrersdiagrams are:��
1 0 0
0 1 0
0 0 1
� ,
��
1 0 0
0 0 0 1 0
0 0 0 0 1
� ,
��
0 1 0 00 0 0 1 00 0 0 0 0 1
�
We can fill the Ferrers diagrams with FD-MRD codes withminimum rank distance 2 of size q6, q2 and q. The union of thelifting of these codes is a code of subspace distance 4 withq6 � q2 � q elements.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Example:We want to construct a code in Gqp3, 6q of subspace distance 4,hence we start with a binary linear code of length 6, weight 3and Hamming distance 4:
C � tp111000q, p100110q, p010101qu
The corresponding reduced row echelon forms and Ferrersdiagrams are:��
1 0 0
0 1 0
0 0 1
� ,
��
1 0 0
0 0 0 1 0
0 0 0 0 1
� ,
��
0 1 0 00 0 0 1 00 0 0 0 0 1
�
We can fill the Ferrers diagrams with FD-MRD codes withminimum rank distance 2 of size q6, q2 and q. The union of thelifting of these codes is a code of subspace distance 4 withq6 � q2 � q elements.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Second construction: lifted Ferrers diagram codes
Remark:
The size of these codes depends mainly on the choice of theidentifying vectors.
It is conjectured that lexicographic binary constant weightcodes are the best choice, and for this choice one getsconstant dimension codes that are at least as large as therespective multi-component lifted MRD codes, whereequality is attained if dSpCq � 2k.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
1 PreliminariesRank-metric codesSubspace codes
2 Constant Dimension Code ConstructionsFirst construction: lifted MRD codesSecond construction: lifted Ferrers diagram codesThird construction: pending dotsFourth construction: pending blocks
3 Summary and Outlook
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Idea: Some identifying vectors lead to a Ferrers diagram whereone can remove dots and still achieve the same size of thecorresponding FD-RM code.
Example:All of the following Ferrers diagrams give rise to a FD-RM codewith minimum rank distance 2 of size q3, since the minimumnumber of dots not contained either in the first row or in thelast column is 3:
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Definition
Let F be a Ferrers diagram and fij be the dot in the i-th rowand j-th column of F . Fztfiju denotes the Ferrers diagram Fafter removing fij . We call a set of dots FP pending if the dotsare in the first row and the leftmost columns of F and
|CF | � |CFzFP |.
Example:
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Theorem
Let δ � 2 and v P Fn2 be an identifying vector of weight k withthe corresponding Ferrers diagram F . Moreover, let zi be thenumber of zeros between the i-th and the pi� 1q-th one of v forany 0 ¤ i ¤ k. Then the number of pending dots in the first rowof F is
n� k � z0 �maxti P t0, . . . , ku | zi ¡ 0u
if this value is positive. Otherwise, there are no pending dots inthe first row of F .
Proof. The number of dots in the first row of F is n� k � z0and the number of dots in the last column ismaxti P t0, . . . , ku | zi ¡ 0u, which implies the statement.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Main result:
Theorem
Let U and V be two subspaces in Gqpk, nq withdHpvpUq, vpVqq � 2δ � 2, such that the leftmost one of vpUq isin the same position as the leftmost one of vpVq. If U and Vhave a common pending dot and this dot is assigned withdifferent values, respectively, then dSpU ,Vq ¥ 2δ.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Example:Let n � 7, k � 3, δ � 2 and consider the identifying vectorp1001100q, that gives rise to the matrix
�� 1 0 0
0 0 0 1 0 0 0 0 0 1
�
where the dot in the box marks the position of the pending dot.We fix it once as 0 and once as 1 and assign
U � rs
��
1 0 0 0
0 0 0 1 0
0 0 0 0 1
� ,V � rs
��
1 1 0 0
0 0 0 1 0
0 0 0 0 1
� .
Then dHpvpUq, vpVqq � 0 but dSpU ,Vq ¥ 2 for any values fillingthe remaining dots.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
The pending dots construction
Theorem
The following construction produces a code C � Gqpk, nq withdSpCq � 2δ:
Choose a binary block code C � Fn2 of constant weight kand minimum Hamming distance 2δ � 2 , such that anyelements v, w P C with dHpv, wq � 2δ � 2 have the first onein the same position and a common pending dot in thecorresponding Ferrers diagram, which can be fixed withdistinct values from Fq for each distinct element,respectively.
Use each codeword vi P C as the identifying vector of acomponent code and construct the corresponding liftedFD-RM code Ci with minimum rank distance δ.
The union C ��|C|i�1 Ci is the final code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Proof.Let U ,V P C be two codewords.
If dHpvpUq, vpVqq ¥ 2δ, then dSpU ,Vq ¥ 2δ.
If dHpvpUq, vpVqq � 2δ � 2, then the pending dots implythe minimum distance.
If dHpvpUq, vpVqq � 0, then the FD-RM code implies thedistance.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Example:We want to construct a code in Gqp3, 7q with dSpCq � 4.
1 We choose the first identifying vector v1 � p1110000q,whose Ferrers diagram has no pending dot and it can befilled with an FD-MRD code of size q8.
2 The second identifying vector v2 � p1001100q (withdHpv1, v2q � 4) leads to a Ferrers diagram with one pendingdot. Fix the pending dot as 0 and fill the remaining Ferrersdiagram with an FD-MRD code of size q4:
�� 1 0 0 0
0 0 0 1 0 0 0 0 0 1
�
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
3 The next identifying vector v3 � p1001010q (withdHpv1, v3q � 4, dHpv2, v3q � 2) leads to a Ferrers diagramwith a pending dot in the same position as before. Fix thepending dot as 1 and fill the remaining Ferrers diagramwith an FD-MRD code of size q3:�
� 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1
�
4 The next identifying vector v4 � p1000101q (withdHpv1, v4q � 4, dHpv2, v4q � 2, dHpv3, v4q � 4) leads to aFerrers diagram with a pending dot in the same position asbefore. Fix the pending dot as 1. The echelon-Ferrers formcan be filled with a Ferrers diagram code of size q.�
� 1 1 0 00 0 0 0 1 00 0 0 0 0 0 1
�
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
5 The next identifying vectorsp0101001q, p0100110q, p0010011q have Hamming distance 4to any other identifying vector and lead to FD-MRD codesof size q2, q2 and 1, respectively.
Hence we constructed a code C � Gqp3, 7q with dSpCq � 4. It hascardinality q8 � q4 � q3 � 2q2 � q � 1 which is larger than thecode constructed by the classical multi-level construction.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Table of improved code sizes:
n lifted FD-RM construction new pending dots construction
7 q8 � q4 � q3 � 2q2 � 1 q8 � q4 � q3 � 2q2 � q � 18 q10 � q6 � q5 � 2q4 � q3 � q2 q10 � q6 � q5 � 2q4 � 2q3 � 2q2 � q � 19 q12 � q8 � q7 � 2q6 � q5 � q4 � 1 q12 � q8 � q7 � 2q6 � 2q5 � 3q4 � 2q3�
2q2 � q � 1
Tabelle: Sizes of codes � Gqp3, nq with minimum subspace distance 4.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
A special pending dots construction for k � 3, δ � 2
Lemma
Let D be the set of all binary vectors of length m and weight 2.
If m is even, D can be partitioned into m� 1 classes, eachof m
2 vectors with pairwise disjoint positions of ones;
If m is odd, D can be partitioned into m classes, each ofm�12 vectors with pairwise disjoint positions of ones.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
Construction:
We partition the set of weight-2 vectors of Fn�32 into
` � n� 3 (or ` � n� 4) classes P1, P2, . . . , P` and definethe following four sets of identifying vectors:
A0 � tp111||0 . . . 0qu,
A1 � tp001||yq | y P P1u,
A2 � tp010||yq | y P Pi, 2 ¤ i ¤ mintq � 1, `uu,
A3 �
"tp100||yq | y P Pi, q � 2 ¤ i ¤ `u if ` ¡ q � 1
∅ if ` ¤ q � 1.
When we use the same prefix for two different classesPi, Pj , we assign different values in the pending dots of theFerrers tableaux forms.(Elements with the same prefix and distinct suffices fromthe same class Pi have Hamming distance 4.)
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
For each identifying vector from A0, . . . ,A3 construct thecorresponding lifted FD-MRD code Ci with rank distance2. Note that the Ferrers diagrams used here are withoutthe pending dots used in the step before.
The union�`i�0 Ci forms the final code C.
Theorem
1 The code C � Gqp3, nq has minimum distance dSpCq � 4.
2 The cardinality of C is
q2pn�3q �
�n� 3
2
�q
,
which attains the upper bound on codes containing aLMRD code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Third construction: pending dots
For each identifying vector from A0, . . . ,A3 construct thecorresponding lifted FD-MRD code Ci with rank distance2. Note that the Ferrers diagrams used here are withoutthe pending dots used in the step before.
The union�`i�0 Ci forms the final code C.
Theorem
1 The code C � Gqp3, nq has minimum distance dSpCq � 4.
2 The cardinality of C is
q2pn�3q �
�n� 3
2
�q
,
which attains the upper bound on codes containing aLMRD code.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
1 PreliminariesRank-metric codesSubspace codes
2 Constant Dimension Code ConstructionsFirst construction: lifted MRD codesSecond construction: lifted Ferrers diagram codesThird construction: pending dotsFourth construction: pending blocks
3 Summary and Outlook
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Idea: extend the definition of pending dots to a 2-dim setting
Definition
Let F be a Ferrers diagram with m dots in the rightmostcolumn and ` dots in the top row. We say that the `1 `leftmost columns of F form a pending block if the upper boundon the size of FD-MRD code CF is equal to the upper bound onthe size of CF without the `1 leftmost columns.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Example:Consider the following Ferrers diagrams:
F1 �
, F2 �
.
For δ � 3 both codes CF1 and CF2 have |CFi | ¤ q3, i � 1, 2.
The diagram F1 has the pending block
and the diagram F2 has no pending block.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Example:Consider the following Ferrers diagrams:
F1 �
, F2 �
.
For δ � 3 both codes CF1 and CF2 have |CFi | ¤ q3, i � 1, 2.The diagram F1 has the pending block
and the diagram F2 has no pending block.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Definition
Let F be a Ferrers diagram with m dots in the rightmostcolumn and ` dots in the top row, and let `1 ¤ `,m1 m suchthat the m1-th row has `� `1 � 1 many dots. If the pm1 � 1q-throw of F has less dots than the m1-th row of F , then the `1leftmost columns of F are called a quasi-pending block (of sizem1 � `1).
A pending block is also a quasi-pending block.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Main result:
Theorem
Let U ,V P Gqpk, nq, such that the first m1 ones of vpUq and vpVqare in the same positions and RREFpUq, RREFpVq have aquasi-pending block of size m1 � `1 in the same position. Denotethe submatrices of FpUq and FpVq corresponding to thequasi-pending blocks by BU and BV , respectively. Then
dSpU ,Vq ¥ dHpvpUq, vpVqq � 2dRpBU , BVq.
ùñ by filling the (quasi-)pending blocks with a suitableFD-RM code, one can choose a set of identifying vectors withlower minimum Hamming distance than 2δ
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Main result:
Theorem
Let U ,V P Gqpk, nq, such that the first m1 ones of vpUq and vpVqare in the same positions and RREFpUq, RREFpVq have aquasi-pending block of size m1 � `1 in the same position. Denotethe submatrices of FpUq and FpVq corresponding to thequasi-pending blocks by BU and BV , respectively. Then
dSpU ,Vq ¥ dHpvpUq, vpVqq � 2dRpBU , BVq.
ùñ by filling the (quasi-)pending blocks with a suitableFD-RM code, one can choose a set of identifying vectors withlower minimum Hamming distance than 2δ
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Proof.Since the first ones of the identifying vectors are in the sameposition, it has to hold that the first m1 pivots of RREFpUq andRREFpVq are in the same columns. To compute the rank of
�RREFpUqRREFpVq
�
we permute the columns such that the m1 first pivot columnsare to the very left, then the columns of the pending block, thenthe other pivot columns and then the rest (WLOG in thefollowing figure we assume that the pm1 � 1q-th pivots are alsoin the same column):
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
rank
��������������������
1 . . . 0 0 . . ....
. . ....
. . . BU...
......
0 . . . 1 0 . . . 0 0 . . .0 . . . 0 0 . . . 0 . . . 0 1 . . ....
...
1 . . . 0 0 . . ....
. . ....
. . . BV...
......
0 . . . 1 0 . . . 0 0 . . .0 . . . 0 0 . . . 0 . . . 0 1 . . ....
...
��������������������
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
�rank
��������������������
1 . . . 0 0 . . ....
. . ....
. . . BU...
......
0 . . . 1 0 . . . 0 0 . . .0 . . . 0 0 . . . 0 . . . 0 1 . . ....
...
0 . . . 0 0 . . ....
. . ....
. . . BU �BV...
......
0 . . . 0 0 . . . 0 0 . . .0 . . . 0 0 . . . 0 . . . 0 0 . . ....
...
��������������������
¥k �1
2dHpvpUq, vpVqq � rankpBU �BVq.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Constructions for dSpCq � 4
Lemma
Let n ¥ 2k � 2 and v P Fn2 be an identifying vector of weight k,such that there are k � 2 many ones in the first k positions of v.Then the Ferrers diagram arising from v has more or equallymany dots in the first row than in the last column.
Theorem
The upper bound for the dimension of a FD-RM code withminimum rank distance 2 in the setting above is the number ofdots that are not in the first row.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Construction:Partition the weight-2 vectors of Fn�k2 into classes P1, . . . , P`(where ` � n� k � 1 or ` � n� k) with pairwise disjointpositions of the ones.
We define the following sets of identifying vectors (ofweight k):
A0 � tp1 . . . 1||0 . . . 0qu
A1 � tp0011 . . . 1||yq | y P P1u,
A2 � tp0101 . . . 1||yq | y P P2, . . . , Pq�1u,
...
Apk2q � tp1 . . . 1100||yq | y P Pµ, . . . , Pνu.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
For each vector vj in a given Ai for i P t2, . . . ,�k2
�u assign a
different matrix filling for the quasi-pending block in the kleftmost columns of the respective matrices. Fill theremaining part of the Ferrers diagram with a suitableFD-MRD code of the minimum rank distance 2 and lift thecode to obtain Ci,j . Define Ci �
�|Ai|j�1 Ci,j .
Take the largest known code C � Gqpk, n� kq withdSpCq � 4 and append k zero columns in front of everymatrix representation of the codewords. Call this code C.
The union
C �pk2q¤i�0
Ci Y C
forms the final code C, where C0 is the lifted MRD codecorresponding to A0.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Theorem
A code C � Gqpk, nq constructed as before has minimumsubspace distance 4 and cardinality
|C| � qpk�1qpn�kq � qpn�k�2qpk�3q
�n� k
2
�q
�Aqpn� k, 2, kq.
ùñ closed cardinality formula
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Theorem
A code C � Gqpk, nq constructed as before has minimumsubspace distance 4 and cardinality
|C| � qpk�1qpn�kq � qpn�k�2qpk�3q
�n� k
2
�q
�Aqpn� k, 2, kq.
ùñ closed cardinality formula
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Example:We want to construct a code C � G2p4, 10q with dSpCq � 4. Wepartition the binary vectors of length 6 and weight 2:
P1 � t110000, 001010, 000101u, P2 � t101000, 010001, 000110u,
P3 � t011000, 100100, 000011u, P4 � t010100, 100010, 001001u,
P5 � t100001, 010010, 001100u.
Then we define the identifying vectors as
A0 � tp1111||000000qu,
A1 � tp0011||110000q, p0011||001010q, p0011||000101qu,
A2 � tp0101||yq | y P P2 Y P3u, using one pending dot,
A3 � tp1001||yq | y P P4 Y P5u, using one of the two pending dots.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
The lifted FD-MRD code for A0 has 218, the one for A1 has212 � 27 � 25 elements, etc. The union of all these liftedFD-MRD codes has cardinality
218 � 24�
62
�2
� 218 � 10416.
We can then add a p6, 21, 2, 4q2-code (the dual of ap6, 21, 2, 2q-spread code) with four zero columns appended infront of each codeword and get a final code size of 218 � 10437.In comparison, the multi-component lifted MRD code hascardinality 218 � 4113.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Recursively applying the construction gives:
Theorem
Let k ¥ 4, n ¥ 2k � 2 and°k�2j�0
°k�2i�j q
i�j � 1 ¥ n� k if n� kis odd (otherwise ¥ n� k � 1). Then
Aqpn, 4, kq ¥ qpk�1qpn�kq�qpn�k�2qpk�3q
�n� k
2
�q
�Aqpn�k, 2, kq.
This bound is always tighter than the one given by themulti-component lifted MRD code construction!
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Note:
We did not use the dots in the quasi-pending blocks!
Thus, the lower cardinality bound is not tight.
To make it tighter, one can use less pending blocks andlarger FD-MRD codes.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Py � the class of suffixes which contains the suffix vector y
Construction:First, in addition to A0, we define the following sets ofidentifying vectors:
A1 � tp11...1100||yq | y P P1100...00u
A2 � tp11...1010||yq | y P P1010...00u,
A3 � tp11...0110||yq | y P P1001...00u
A4 � tp11...1001||yq | y P P0110...00u.
All the other identifying vectors are distributed as before, usingpossible pending blocks. Then we construct the respective liftedFD-MRD codes, where we now consider the whole Ferrersdiagram (and not only the one of the last n� k columns) forA1, . . . , A4. The other identifying vectors are treated as before.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
With this construction, we get larger FD-MRD codes forA1, . . . , A4 but we also have a stricter condition on q and k suchthat all Pi’s are used.
Theorem
If°k�2j�0
°k�2i�j q
i�j �°5i�4 q
2k�i � 2q2k�6 ¥ n� k, then
Aqpn, 4, kq ¥ qpk�1qpn�kq � qpn�k�2qpk�3q
�n� k
2
�q
�pq2pk�3q � 1qqpk�1qpn�k�2q � pq2pk�3q�1 � 1qqpk�1qpn�k�2q�1
�2pq2pk�4q � 1qqpk�1qpn�k�2q�2 �Aqpn� k, 4, kq.
One can use the idea of this construction on more Ai’s, as longas there are enough pending blocks such that all Pi’s are used.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Another tweak: Instead of using all the Pi-classes, use theclasses which contribute the most codewords more than oncewith disjoint prefixes.Example:We want to again construct C � G2p4, 10q. We define A0 aspreviously and
A1 � tp1100||yq | y P P1u,A2 � tp0011||yq | y P P1u,
A3 � tp0110||yq | y P P4u,A4 � tp1001||yq | y P P4u,
A5 � tp1010||yq | y P P2 Y P3u,A6 � tp0101||yq | y P P2 Y P3u,
where we use the pending dot in A5 and A6. Note that we donot use P5. The FD-MRD codes are now constructed for thewhole Ferrers diagram. We add A2p6, 2, 4q � 21 codewords. Thesize of the final code is 218 � 37477. The largest previouslyknown code was a LFD-MRD code and had size 218 � 34768.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
Similar construction for dSpCq � 2k � 2
Theorem
Let n ¥ s� 2� k and q2 � q � 1 ¥ `, where s �°ki�3 i and
` � n� s for odd n� s (or ` � n� s� 1 for even n� s). Then
Aqpn, k � 1, kq ¥k
j�3
q2pn�°ki�j iq �
�n� p
°ki�3 iq
2
�q
.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
ExampleLet k � 5, 2δ � 8, n � 19, and q � 2.First, we partition the set of suffixes y P F7
2 of weight 2 into 7classes, P1, . . . , P7 of size 3 each. The identifying vectors of thecode are partitioned as follows:
v500 �p11111||0000||000||0000000q,
A50 �tp00001||1111||000||0000000q, p00010||0001||111||0000000qu
A51 �tp00100||0010||001||yq | y P P1u
A52 �tp01000||0100||010||yq | y P tP2, P3uu
A53 �tp10000||1000||100||yq | y P tP4, P5, P6, P7uu
To demonstrate the idea of the construction we will considerthe set A5
2. Common pending block :
B �
.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
To distinguish between the two classes P2 and P3, we assign thefollowing values to B, respectively:
1 1
1
0 0
0
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
For the remaining dots of B we construct a FD-MRD code ofdistance 2 (here a � 0 for P2 or a � 1 for P3):
a 0 0 0 0 0 0 0a 0 0 0
a,a 1 0 0 0 0 0 0
a 1 0 0a,
a 0 1 0 0 0 0 0a 0 1 0
a,a 0 0 1 0 0 0 0
a 0 0 1a,
a 1 1 0 0 0 0 0a 1 1 0
a,a 1 0 1 0 0 0 0
a 1 0 1a,
a 0 1 1 0 0 0 0a 0 1 1
a,a 1 1 1 0 0 0 0
a 1 1 1a.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Constant Dimension Code Constructions
Fourth construction: pending blocks
1 Since both P2 and P3 contain only three elements, we onlyneed to use three of the above tableaux to assign a differentfilling of the common pending block for the differentelements corresponding to A5
2.
2 We proceed analogously for the pending blocks of A51,A5
3.
3 Then we fill the Ferrers diagrams corresponding to the last7 columns of the identifying vectors with an FD-MRD codeof minimum rank distance 4 and lift these elements.
4 Moreover, we add the lifted MRD code corresponding tov500, which has cardinality 228. The number of codewordswhich corresponds to the set A5
0 is 220 � 214.
The code obtained has cardinality 228 � 1067627. The largestpreviously known code was of cardinality 228 � 1052778.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
1 PreliminariesRank-metric codesSubspace codes
2 Constant Dimension Code ConstructionsFirst construction: lifted MRD codesSecond construction: lifted Ferrers diagram codesThird construction: pending dotsFourth construction: pending blocks
3 Summary and Outlook
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
Summary and Outlook
1 We described several construction for constant dimensioncodes, each a generalization of the previous:
1 (multi-component) lifted MRD codes2 lifted Ferrers diagram codes3 pending dots4 pending blocks
2 One problem is the choice of identifying vectors; theoptimal choice is still unknown.
3 ùñ no closed formula for the cardinality4 ùñ some constructions for dSpCq � 4 and dSpCq � 2k � 2,
where we prescribe the identifying vectors, based on resultsfrom graph theory – with closed cardinality formula
5 In general, the pending dots/blocks construction, and inparticular these latter constructions, are still the bestknown for many parameter sets, seehttp://subspacecodes.uni-bayreuth.de/.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
Summary and Outlook
1 We described several construction for constant dimensioncodes, each a generalization of the previous:
1 (multi-component) lifted MRD codes2 lifted Ferrers diagram codes3 pending dots4 pending blocks
2 One problem is the choice of identifying vectors; theoptimal choice is still unknown.
3 ùñ no closed formula for the cardinality4 ùñ some constructions for dSpCq � 4 and dSpCq � 2k � 2,
where we prescribe the identifying vectors, based on resultsfrom graph theory – with closed cardinality formula
5 In general, the pending dots/blocks construction, and inparticular these latter constructions, are still the bestknown for many parameter sets, seehttp://subspacecodes.uni-bayreuth.de/.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
Summary and Outlook
1 We described several construction for constant dimensioncodes, each a generalization of the previous:
1 (multi-component) lifted MRD codes2 lifted Ferrers diagram codes3 pending dots4 pending blocks
2 One problem is the choice of identifying vectors; theoptimal choice is still unknown.
3 ùñ no closed formula for the cardinality
4 ùñ some constructions for dSpCq � 4 and dSpCq � 2k � 2,where we prescribe the identifying vectors, based on resultsfrom graph theory – with closed cardinality formula
5 In general, the pending dots/blocks construction, and inparticular these latter constructions, are still the bestknown for many parameter sets, seehttp://subspacecodes.uni-bayreuth.de/.
58 / 59
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
Summary and Outlook
1 We described several construction for constant dimensioncodes, each a generalization of the previous:
1 (multi-component) lifted MRD codes2 lifted Ferrers diagram codes3 pending dots4 pending blocks
2 One problem is the choice of identifying vectors; theoptimal choice is still unknown.
3 ùñ no closed formula for the cardinality4 ùñ some constructions for dSpCq � 4 and dSpCq � 2k � 2,
where we prescribe the identifying vectors, based on resultsfrom graph theory – with closed cardinality formula
5 In general, the pending dots/blocks construction, and inparticular these latter constructions, are still the bestknown for many parameter sets, seehttp://subspacecodes.uni-bayreuth.de/.
58 / 59
Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
Summary and Outlook
1 We described several construction for constant dimensioncodes, each a generalization of the previous:
1 (multi-component) lifted MRD codes2 lifted Ferrers diagram codes3 pending dots4 pending blocks
2 One problem is the choice of identifying vectors; theoptimal choice is still unknown.
3 ùñ no closed formula for the cardinality4 ùñ some constructions for dSpCq � 4 and dSpCq � 2k � 2,
where we prescribe the identifying vectors, based on resultsfrom graph theory – with closed cardinality formula
5 In general, the pending dots/blocks construction, and inparticular these latter constructions, are still the bestknown for many parameter sets, seehttp://subspacecodes.uni-bayreuth.de/.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
References
D. Silva, F. R. Kschischang, and R. Koetter, A rank-metricapproach to error control in random network coding, IEEETransactions on Information Theory 54 (2008), no. 9, 3951 3967.
T. Etzion and N. Silberstein, Error-correcting codes in projectivespaces via rank-metric codes and Ferrers diagrams, IEEETransactions on Information Theory 55 (2009), no. 7, 29092919.
A.-L. Trautmann and J. Rosenthal, New improvements on theechelon-Ferrers construction, Proceedings of the 19thInternational Symposium on Mathematical Theory of Networksand Systems MTNS (Budapest, Hungary), 2010, pp. 405 408.
T. Etzion and N. Silberstein, Codes and designs related to liftedMRD codes, IEEE Transactions on Information Theory 59(2013), no. 2, 1004 1017.
N. Silberstein and A.-L. Trautmann, Subspace Codes Based onGraph Matchings, Ferrers Diagrams, and Pending Blocks, IEEETransactions on Information Theory 61 (2015), no. 7, 3937-3953.
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Constructions of Constant Dimension Codes with Ferrers Diagram Rank Metric Codes
Summary and Outlook
Thank you for your attention!Questions? – Comments?