Constructions of Constant Dimension Codes with Ferrers … · 2018. 8. 13. · A Ferrers diagram is a collection of dots such that the rows have a decreasing and the columns have

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


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


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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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C � tp111000q, p100110q, p010101qu


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

�


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C � tp111000q, p100110q, p010101qu


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

�


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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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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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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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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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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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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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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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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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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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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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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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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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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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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, ùu,

A3 �

"tp100||yq | y P Pi, q � 2 ¤ i ¤ ù 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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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�ì�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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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�ì�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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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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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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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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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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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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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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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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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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�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 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Summary and Outlook





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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http://subspacecodes.uni-bayreuth.de/


Summary and Outlook

Summary and Outlook







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Summary and Outlook

Summary and Outlook




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


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Summary and Outlook

Summary and Outlook







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Summary and Outlook

Summary and Outlook







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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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Summary and Outlook

Thank you for your attention!Questions? – Comments?

Documents

Constructions of Constant Dimension Codes with Ferrers … · 2018. 8. 13. · A Ferrers diagram is a collection of dots such that the rows have a decreasing and the columns have