New Lower Bounds for Constant Dimension Codes

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by  Natalia Silberstein

New Lower Bounds for Constant Dimension Codes
Natalia Silberstein Anna-Lena Trautmann
Dep. of Electrical & Computer Eng. Inst. of Mathematics
University of Texas at Austin University of Zurich
Austin, TX, USA Zurich, Switzerland
Email: [email protected] Email: [email protected]
Abstract—Thispaperprovidesnew constructivelower bounds II. PRELIMINARIES
for constant dimension codes, usingdifferent techniquessuch as
In this section we briefly provide the definitions and pre-
Ferrers diagram rank metric codes and pending blocks. Con-
vious results used in our constructions. More details can be
structions for two families of parameters of constant dimension
codes are presented. The examples of codes obtained by these found in [2], [3], [16].
constructions are the largest known constant dimension codes Let X be a k-dimensional subspace of Fn. We represent
q
3 for the given parameters. X by the matrix RE(X) in reduced row echelon form, such
1
that the rows of RE(X) form the basis of X. The identifying
0 I. INTRODUCTION
vector of X, denoted by v(X) is the binary vector of length
2
n Let Fq be the finite field of size q. Given two integers k,n, n and weight k, where the k ones of v(X) are exactly in
such that 0 ≤ k ≤ n, the set of all k-dimensional subspaces the positions where RE(X) has the leading coefficients (the
a
J of Fnq forms the Grassmannian over Fq, denoted by Gq(k,n). pivots).
5 It is well known that the cardinality of the Grassmannian is The Ferrers tableaux form of a subspace X, denoted by
2 given by the q-ary Gaussian coefficient F(X), is obtained from RE(X) first by removing from each
rowof RE(X)thezeroes to theleftof theleadingcoefficient;

T] n d=ef |G (k,n)|=k−1qn−i−1. andafterthatremovingthecolumnswhichcontaintheleading
(cid:20) k (cid:21) q qk−i−1 coefficients. All the remaining entries are shifted to the right.
.I q iY=0 The Ferrers diagram of X, denoted by F , is obtained from
s X
c TheGrassmannianspaceisametricspace,wherethesubspace F(X) by replacing the entries of F(X) with dots.
[ distance between any two subspaces X and Y in Gq(k,n), is Given F(X), the unique corresponding subspace X ∈
given by G (k,n) can be easily found. Also given v(X), the unique
1 q
v corresponding FX can be found. When we fill the dots of
1 dS(X,Y) d=ef dimX+dimY −2dim X∩Y . (1) a Ferrers diagram by elements of Fq we obtain a F(X) for
6 (cid:0) (cid:1) some X ∈G (k,n).
9 We say that C ⊆ Gq(k,n) is an (n,M,d,k)q code in the q
5 Grassmannian, or constant-dimension code, if M = |C| and Example 1. Let X be the subspace in G2(3,7) with the
1. dS(X,Y) ≥ d for all distinct elements X,Y ∈ C. Note, that following generator matrix in reduced row echelon form:
0 the minimumdistance d of C is always even.Aq(n,d,k) will 1 0 0 0 1 1 0
3 denote the maximum size of an (n,M,d,k)q code. RE(X)= 0 0 1 0 1 0 1 .
1 Constantdimensioncodeshavedrawnasignificantattention 0 0 0 1 0 1 1 !
:
v in the last five years due to the work by Koetter and Kschis- Its identifying vector is v(X) = 1011000, and its Ferrers
i chang [8], where they presented an application of such codes tableaux form and Ferrers diagram are given by
X
for error-correction in random network coding. Constructions
r 0 1 1 0 • • • •
a and bounds for constant dimension codes were given in [1], 1 0 1 , • • • ,
[2], [3], [4], [6], [7], [9], [11], [13], [16]. 0 1 1 • • •
In this paper we focus on constructions of large constant
respectively.
dimension codes. In particular, we generalize the idea of
construction of codes in the Grassmannian from [2], [3], [16] In the following we will consider Ferrers diagrams rank-
and obtain new lower bounds on A (n,d,k). In Section II metric codes which are closely related to constant dimension
q
we introducethenecessarydefinitionsandpresenttwo known codes. For two m×ℓ matrices A and B over Fq the rank
def
constructions which will be the starting point to our new distance,d (A,B), isdefinedbyd (A,B) = rank(A−B).
R R
constructions. In Section III we introduce the notation of Let F be a Ferrers diagram with m dots in the rightmost
pending blocks. In Sections IV and V we present our new column and ℓ dots in the top row. A code C is an [F,ρ,δ]
F
constructions. It appears that the codes obtained by these Ferrersdiagramrank-metric(FDRM)codeifallcodewordsof
constructionsare the largest known constant dimension codes C arem×ℓmatricesinwhichallentriesnotin F arezeroes,
F
for the given parameters. they form a linear subspace of dimension ρ of Fm×ℓ, and for
q
any two distinct codewords A and B, d (A,B)≥δ. If F is Lemma 5. [16] Let X and Y be two subspaces in G (k,n)
R q
a rectangular m×ℓ diagram with m·ℓ dots then the FDRM with d (v(X),v(Y)) = 2δ −2, such that the leftmost one
H
code is a classical rank-metric code [5], [12]. The following of v(X) is in the same position as the leftmost one of v(Y).
theorem provides an upper bound on the cardinality of C . Let P and P be the sets of pending dots of X and Y,
F X Y
respectively. If P ∩P 6= ∅ and the entries in P ∩P
Theorem 2. [2] Let F be a Ferrers diagram and C the X Y X Y
F (of their Ferrers tableaux forms) are assigned with different
correspondingFDRM code.Then|CF|≤qmini{wi}, wherewi values in at least one position, then d (X,Y)≥2δ.
isthenumberofdotsinF whicharenotcontainedinthefirst S
i rows and the rightmost δ−1−i columns (0≤i≤δ−1). Example 6. LetX andY besubspacesinGq(3,6)whichare
given by the following generator matrices:
A code which attains the bound of Theorem 2 is called a
Ferrers diagram maximum rank distance (FDMRD) code. 1 (cid:13)0 0 v1 v2 0 1 (cid:13)1 u1 0 u2 0
0 0 1 v3 v4 0 , 0 0 0 1 u3 0
0 0 0 0 0 1 ! 0 0 0 0 0 1 !
Remark 3. Maximumrankdistance(MRD)codesare aclass
of[F,ℓ(m−δ+1),δ] FDRMcodes,ℓ≥m, with a fullm×ℓ where v ,u ∈ F , and the pending dots are emphasized
i i q
diagram F, which attain the bound of Theorem 2 [5], [12]. by circles. Their identifying vectors are v(X) = 101001
and v(Y) = 100101. Clearly, d (v(X),v(Y)) = 2, while
It was proved in [2] that for general diagrams the bound H
d (X,Y)≥4.
of Theorem 2 is always attained for δ = 1,2. Some special S
cases, when this bound is attained for δ > 2, can be found The following lemma which follows from a one-
in [2]. factorization and near-one-factorization of a complete
For a codeword A ∈ CF ⊂ Fqk×(n−k) let AF denote the graph [10] will be used in our constructions.
part of A related to the entries of F in A. Given a FDMRD
code C , a lifted FDMRD code C is defined as follows: Lemma 7. Let D be the set of allbinary vectors of lengthm
F F
and weight 2.
CF ={X ∈Gq(k,n):F(X)=AF, A∈CF}. • Ifmiseven,Dcanbepartitionedintom−1classes,each
of m vectors with pairwise disjoint positions of ones;
2
This definition is the generalization of the definition of • If m is odd, D can be partitioned into m classes, each
a lifted MRD code [14]. Note, that all the codewords of a of m−1 vectors with pairwise disjoint positions of ones.
2
lifted MRD code have the same identifyingvectorof the type
(11...1000...00).Thefollowinglemma[2]isthegeneralization The following construction for k = 3 and d = 4 based on
pending dots [3] will be used as a base step of our recursive
k n−k
of the result given in [14]. construction proposed in the sequel.
|{z}| {z }
Construction 0. Let n ≥ 8 and q2 +q + 1 ≥ ℓ, where
Lemma 4. If C ⊂Fk×(n−k) is an [F,ρ,δ] Ferrers diagram
F q ℓ=n−4 for odd n and ℓ=n−3 for even n. In addition to
rank-metric code, then its lifted code C is an (n,qρ,2δ,k)
F q the lifted MRD code (which has the identifying vector v =
0
constant dimension code. (11100...0)), the final code C will contain the codewords
with identifying vectors of the form (x||y), where the prefix
A. The multilevel construction and pending dots construction x∈F3 is of weight 1 and the suffix y ∈Fn−3 is of weight2.
2 2
It was proved in [2] that for X,Y ∈ G (k,n) we have By Lemma 7, we partition the set of suffixes into ℓ classes
q
dS(X,Y) ≥ dH(v(X),v(Y)), where dH denotes the Ham- P1,P2,...,Pℓ and define the following three sets:
ming distance; and if v(X) = v(Y) then d (X,Y) =
S
A ={(001||y):y ∈P },
2d (RE(X),RE(Y)). The multilevelconstruction[2] of con- 1 1
R
stant dimension code is based on these properties of d .
S
A ={(010||y):y ∈P ,2≤i≤min{q+1,ℓ}},
Multilevel construction. First, a binary constant weight 2 i
code of length n, weight k, and Hamming distance 2δ is
{(100||y):y ∈P , q+2≤i≤ℓ} if ℓ>q+1
chosentobethesetoftheidentifyingvectorsforC.Then,for A = i .
3 (cid:26) ∅ if ℓ≤q+1
each identifying vector a corresponding lifted FDMRD code
with minimum rank distance δ is constructed. The union of The idea is that we use the same prefix for the suffixes of
these lifted FDMRD codes is an (n,M,2δ,k)q code. Hamming distance 4 (from the same class), and when we use
In the construction provided in [3], for k = 3 and δ = 2, the same prefix for two different classes P ,P , we use the
i j
in the stage of choosing identifying vectors for a code C, differentvaluesofFerrerstableauxformsin thependingdots.
the vectors of (Hamming) distance 2δ −2 = 2 are allowed, Then, the corresponding lifted FDMRD codes of distance 4
by using a method based on pending dots in a Ferrers are constructed, and their union with the lifted MRD code
diagram [16]. n−3
forms the final code C of size q2(n−3)+ .
The pending dots of a Ferrers diagram F are the leftmost (cid:20) 2 (cid:21)
q
dotsinthefirstrowof F whoseremovalhasnoimpactonthe In the following sections we will generalize this construc-
size of the corresponding Ferrers diagram rank-metric code. tion and obtain codes for any k ≥ 4 with d = 4 and with
The following lemma follows from [16]. d=2(k−1).
III. PENDINGBLOCKS Now we subtract the lower half from the upper one and get
1 ... 0 0 ...
To present the new constructions for constant dimension
codes, we first need to extend the definition of pending dots  ... ... ... ... BX ... ... ... 
of [16] to a two-dimensional setting.  0 ... 1 0 ... 0 0 ... 
 0 ... 0 0 ... 0 ... 0 1 ... 
 
Definition 8. Let F be a Ferrers diagram with m dots in  ... ... 
the rightmost column and ℓ dots in the top row. We say that =rank 0 ... 0 0 ... 
 
sthizeeℓℓ1 )<ifℓthleeftumpopsetrcbooluumndnsoonfthFefsoizremoaf FpeDnMdiRngDbcloodcekC(of  ... ... ... ... BX−BY ... ... ... 
1 F  0 ... 0 0 ... 0 0 ... 
from Theorem2 isequalto theupperboundonthesize ofCF  0 ... 0 0 ... 0 ... 0 0 ... 
without the ℓ1 leftmost columns.  ... ... 
 
Example 9. Consider the following Ferrers diagrams:  
The additional pivots of RE(X) and RE(Y) (to the right in
theaboverepresentation)thatwereindifferentcolumnsin the
• • • • • • • •
F1= • • • • , F2= • • • . beginning are still in different columns, hence it follows that
• • • • • •
RE(X) 1
rank ≥k+ d (v(X),v(Y))+rank(B −B ),
(cid:20) RE(Y) (cid:21) 2 H X Y
For δ = 3 by Theorem 2 both codes C and C have
F1 F2
|C | ≤ q3, i = 1,2. The diagram F has the pending block which implies the statement with the formula
Fi 1
• •
• and the diagram F2 has no pending block. d (X,Y)=2rank RE(X) −2k.
S (cid:20) RE(Y) (cid:21)
Definition10. LetF beaFerrersdiagramwithmdotsinthe
rightmost column and ℓ dots in the top row, and let ℓ < ℓ,
1 This theorem implies that for the construction of an
andm <m.Ifthe(m +1)strowofF haslessdotsthanthe
1 1 (n,M,2δ,k)-code, by filling the (quasi-)pending blocks with
m th row of F, then the ℓ leftmost columns of F are called
1 a suitable Ferrers diagram rank metric code, one can choose
a quasi-pending block (of size m ×ℓ ).
1 1 a set of identifying vectors with lower minimum Hamming
Note, that a pending block is also a quasi-pending block. distance than δ.
Theorem 11. Let X,Y ∈ Gq(k,n), such that RE(X) and IV. CONSTRUCTIONS FOR(n,M,4,k)q CODES
RE(Y) have a quasi-pending block of size m × ℓ in the
1 1 In this section we present a construction based on quasi-
same position and d (v(X),v(Y)) = d. Denote the subma-
H pending blocks for (n,M,4,k) codes with k ≥ 4 and
tricesofF(X)andF(Y)correspondingtothequasi-pending q
n ≥ 2k + 2. This construction will then give rise to new
blocks by B and B , respectively. Then d (X,Y) ≥ d+
X Y S lower bounds on the size of constant dimension codes with
2rank(B −B ).
X Y this minimum distance. First we need the following results.
Proof: Since the quasi-pending blocks are in the same
Lemma 12. Let n≥2k+2. Let v be an identifyingvector of
position,ithastoholdthefirsthpivotsofRE(X)andRE(Y)
length n and weight k, such that there are k−2 many ones
RE(X)
arein thesamecolumns.Tocomputetherankof in the first k positions of v. Then the Ferrers diagram arising
(cid:20) RE(Y) (cid:21)
from v has more or equallymany dots in the first row than in
we permute the columns such that the h first pivot columns
the last column, and the upper bound for the dimension of a
are to the very left, then the columns of the pending block,
Ferrers diagramcode with minimumdistance 2 is the number
then the other pivot columns and then the rest (WLOG in the
of dots that are not in the first row.
following figure we assume that the h+1st pivots are also in
the same column): Proof: Because of the distribution of the ones, it holds
thatthe numberofdotsin the first rowof the Ferrersdiagram
is
1 ... 0 0 ...
n−k−2+i, i∈{0,1,2}
 ... ... ... ... BX ... ... ... 
0 ... 1 0 ... 0 0 ... and the number of dots in the last column of the Ferrers
 
 0 ... 0 0 ... 0 ... 0 1 ...  diagram is
 
 ... ...  k−2+j, j ∈{0,1,2}.
rank 1 ... 0 0 ... 
 
 ... ... ... ... BY ... ... ...  Sfiirnstcerowweisaasslwumayestghraetatner≥or2ekq+ual2,toththeennuummbbeerrooffddoottssiinntthhee
 0 ... 1 0 ... 0 0 ... 
 0 ... 0 0 ... 0 ... 0 1 ...  last column.Then,the upperboundon the dimensiondirectly
 ... ...  follows from Theorem 2.
 
 
Lemma 13. The number of all matrices filling the Ferrers Proof:Itholdsthat|C |=q(k−1)(n−k) and|C¯|=A (n−
0 q
diagrams arising from all elements of Fk of weight k−2 as k,4,k).Becauseoftheassumptiononk andq itfollowsfrom
q
identifying vectors is ν := jk=−02 ik=−j2qi+j −1. Lemma 13 that all the yi ∈F2n−k are used for the identifying
vectors, hence a cardinality of |G (2,n − k)| for the lower
P P q
Proof:Assumethefirstzeroisinthej-thandthesecond
two rows. Moreover,we can fill the second to (k−2)-ndrow
zero is in the i-th position of the identifying vector. Then
of the Ferrers diagrams with anything in the construction of
the corresponding Ferrers diagram has j−1 dots in the first the FDMRD code,henceq(n−k−2)(k−3) possibilitiesforthese
column and i−2 dots in the second column. I.e. there are
dots. Togetherwith 4) from Construction I we have the lower
k−1 k k−2k−2 bound on the code size.
(j−1)+(i−2)= i+j Let X,Y ∈ C be two codewords. If both are from C¯,
Xj=1i=Xj+1 Xj=0Xi=j the distance is clear. If X is from C¯ and Y is not, then
dots overall and we can fill each diagram with i dots with qi dH(v(X),v(Y)) ≥ 2(k − 2). Since k ≥ 4, it follows that
many differentmatrices. Hence the formulafollows, since we dS(X,Y)≥4.Fortherestwedistinguishfourdifferentcases:
have to subtract 1 for the summand where i=j =0. 1) If v(X) = v(Y), then the FDMRD code implies the
We can now describe the construction(k≥4,n≥2k+2): distance.
Construction Ia. First, by Lemma 7, we partition the 2) If v(X) 6= v(Y) and v(X),v(Y) are in the same set
weight-2 vectors of F2n−k into classes P1,...,Pℓ of size 2ℓ¯ Ai for some i, then dH(v(X),v(Y)) ≥ 2 (because of
(whereℓ=ℓ¯−1=n−k−1ifn−kevenandℓ=ℓ¯+1=n−k thestructureofthePi’s).Thequasi-pendingblocksthen
if n−k odd) with pairwise disjoint positions of the ones. imply by Theorem 11 that dS(X,Y)≥2+2=4.
1) We define the following sets of identifying vectors (of 3) If v(X) ∈ A0,v(Y) ∈ Aj, where j > 0, then
weight k): dH(v(X),v(Y)≥4. Hence, dS(X,Y)≥4.
4) If v(X) ∈ A ,v(Y) ∈ A , where i 6= j and i,j > 0,
i j
A0 ={(1...1||0...0)} thendH(v(X),v(Y)≥4becausethefirstk coordinates
A ={(0011...1||y):y ∈P }, have minimum distance ≥2 and the last n−k coordi-
1 1
A ={(0101...1||y):y ∈P ,...,P }, nates have minimum distance ≥ 2, since they are in
2 2 q+1
different P ’s. Hence, d (X,Y)≥4.
. i S
.
.
A ={(1...1100||y):y ∈P ,...,P }. We can now retrieve a lower bound on the size of constant
(k) µ ν
2 dimension codes for minimum subspace distance 4.
such that the prefixesin A ,...,A are all vectorsof
1 (k2) Corollary16. Letk ≥4,n≥2k+2and k−2 k−2qi+j−
Fk ofweightk−2.ThenumberofP ’susedineachset j=0 i=j
2 i 1≥n−k if n−k is odd (otherwise ≥nP−k−P1). Then
dependsonthesizeofthequasi-pendingblockarisingin
tνheiskthleeftvmalousetcfroolmumLnesmomftahe13reaspnedctµiv:e=mνat−ricqe2s(.k−T2h)u.s, Aq(n,4,k)≥q(k−1)(n−k)+q(n−k−2)(k−3)(cid:20) n−2 k (cid:21)q+Aq(n−k,4,k).
2) For each vector vj in a given Ai for i ∈ {2,..., k2 } This bound is always tighter than the ones given by the
assign a different matrix filling for the quasi-pen(cid:0)din(cid:1)g Reed-Solomon like construction [8] and the multicomponent
block in the k leftmost columnsof the respective matri- extension of this [6], [15].
ces.FilltheremainingpartoftheFerrersdiagramwitha Note, that in the construction we did not use the dots in
suitable FDMRD code of the minimum rank distance 2 the quasi-pending blocks for the calculation of the size of a
andliftthecodetoobtainCi,j. DefineCi = |jA=i1|Ci,j. FDMRD code. Thus, the bound of Corollary 16 is not tight.
3) Take the largest known code C¯ ⊆ Gq(k,nS− k) with To make ittighter,onecan use less pendingblocksandlarger
minimumdistance4andappendkzerocolumnsinfront FDMRDcodes,asillustratedinthefollowingconstruction.We
of every matrix representation of the codewords. denote by P the class of suffixes which contains the suffix
y
4) The following union of codes form the final code C: vector y (in the partition of Lemma 7).
(k) ConstructionIb.First,inadditiontoA0 ofConstructionIa,
2
we define the following sets of identifying vectors:
C= C ∪C¯
i
i[=0 A¯ ={(11...1100||y):y ∈P },
1 1100...00
where C is the lifted MRD code correspondingto A .
0 0 A¯ ={(11...1010||y):y ∈P },
2 1010...00
Remark 14. If ℓ < ν, then we use only the sets A ,...,A
0 i
(i≤ k2 ) such that all of P1,...,Pℓ are used once. A¯3 ={(11...0110||y):y ∈P1001...00},
Theo(cid:0)re(cid:1)m 15. If ℓ ≤ ν, a code C ⊆ Gq(k,n) constructed A¯4 ={(11...1001||y):y ∈P0110...00}.
according to Construction I has minimum distance 4 and
cardinality AlltheotheridentifyingvectorsaredistributedasinConstruc-
|C|=q(k−1)(n−k)+q(n−k−2)(k−3) n−2 k +Aq(n−k,4,k). TtiohnenIat.heThloewseterpbso2u)n−d4o)nothfeCcoanrsdtrinuacltiitoynbIeacroemmeasinthe same.
(cid:20) (cid:21)q
Corollary17. If k−2 k−2qi+j− 5 q2k−i−2q2k−6 ≥ The codes obtained by Construction 0 attain this bound for
j=0 i=j i=4
n−k, then P P P k =3.Ourrecursiveconstructionprovidesanewlowerbound
on the cardinality of such codes for general k.
Aq(n,4,k)≥q(k−1)(n−k)+q(n−k−2)(k−3) n−2 k First, we need the following lemma which is a simple
(cid:20) (cid:21)q
generalization of Lemma 12.
+(q2(k−3)−1)q(k−1)(n−k−2)+(q2(k−3)−1−1)q(k−1)(n−k−2)−1
Lemma21. Letn−k−2≥n ≥k−2andvbeanidentifying
+2(q2(k−4)−1)q(k−1)(n−k−2)−2+Aq(n−k,4,k). vectoroflengthnandweight1k,suchthattherearek−2many
Note, that one can use this idea on more A ’s, as long as ones in the first n positions of v. Then the Ferrers diagram
i 1
there are enough pending blocks such that all P ’s are used. arising from v has more or equally many dots in any of the
i
Moreover, instead of using all the classes P we can use first k−2 rows than in the last column, and the upper bound
i
the classes which contribute more codewords more then once for the dimension of a Ferrers diagram code with minimum
with the disjoint prefixes. We illustrate this idea for a code distance k−1 is the number of dots that are not in the first
having k = 4 and n = 10. It appears, that the code obtained k−2 rows.
by this construction is the largest known code.
Proof: Naturally, the last column of the Ferrers diagram
Example 18. Let q = 2, k = 4, n = 10. We partition has at most k many dots. Since there are k−2 many ones in
the binary vectors of length 6 and weight 2 into the fol- the first n positionsof v, it follows that there are n−n −2
1 1
lowing 5 classes: P = {110000,001010,000101},P = zerosin the last n−n positionsof v. Thus,there are at least
1 2 1
{101000,010001,000110},P ={011000,100100,000011}, n−n −2 many dots in any but the lower two rows of the
3 1
P ={010100,100010,001001},P ={100001,010010, Ferrers diagram arising from v. Therefore, if n−n −2 ≥
4 5 1
001100}. We define A as previously and k ⇐⇒ n−k−2 ≥ n the Ferrers diagram arising from v
0 1
has more or equally many dots in any of the first k−2 rows
A ={(1100||y):y ∈P },A ={(0011||y):y ∈P },
1 1 2 1 than in the last column. It holds that any column has at most
A ={(0110||y):y ∈P },A ={(1001||y):y ∈P }, as many dots as the last one.
3 4 4 4
FromTheorem2weknowthattheboundonthedimension
A ={(1010||y):y∈P ∪P },A ={(0101||y):y∈P ∪P },
5 2 3 6 2 3 of the FDRM code is given by the minimum of dots not
whereweusethependingdotinA andA .Note,thatwedo contained in the first i rows and last k−2−i columns for
5 6
not use P . Also, the FDMRD codes are now constructed for i=0,...,k−2.Since,forthegiveni’s,thepreviousstatement
5
the whole Ferrers diagrams (without the pending dot), and holds, the minimum is attained for i=k−2.
not only for the last 6 columns. We can add A (6,4,4) =
2 Remark 22. If a m × ℓ-Ferrers diagram has δ rows with
A (6,4,2)=(26−1)/(22−1)=21codewordscorresponding
2 ℓ dots each, then the construction of [2] provides respective
toset4)inConstructionIa.Thesize ofthefinalcodeis 218+
FDMRDcodesofminimumdistanceδ+1attainingthebound
37477. The largest previously known code was obtained by
of Theorem 2.
the multilevel construction and has size 218+34768 [2].
Lemma 23. For a m×ℓ-Ferrers diagram where the jth row
Inthefollowingwediscussaconstructionofanewconstant
has at least x more dots than the (j+1)th row for 1 ≤j ≤
dimension code with minimum distance 4 from a given one.
m−1andthelowestrowhasxmanydots,onecanconstructa
Theorem 19. Let C be an (n,M,4,k)q constant dimension FDMRD code with minimum rankdistance m andcardinality
code.Let ∆ be anintegersuch that∆≥k. Then,there exists qx asfollows.Foreachcodewordtakeadifferentw ∈Fx and
q
an (n′ =n+∆,M′,4,k) code C′ with M′ =Mq∆(k−1). fill the first x dots of every row with this vector, whereas all
other dots are filled with zeros.
Proof: To the generator matrix of each codeword of
C we append a [k × ∆,∆(k − 1),2]-MRD code in the Proof:Theminimumdistancefollowseasilyfromthefact
additional columns. This MRD code has cardinality q∆(k−1) thatthepositionsofthew’sineachrowhavenocolumn-wise
by Theorem 2. intersection.Sincetheyarealldifferent,anydifference oftwo
Example 20. We take the (8,212 +701,4,4) code C con- codewords has a non-zero entry in each row and it is already
2
row-reduced.
structed in [3] and apply on it Theorem 19 with ∆=4. Then
the code |C′|=224+701·212 =224+2871296.The largest The cardinality is clear, hence it remains to show that this
previously known code of size 224 +2290845 was obtained attains the bound of Theorem 2. Plugging in i = k −1 in
Theorem2wegetthatthedimensionofthecodeislessthanor
in [2].
equalto the numberof dots in the last row,which is achieved
V. CONSTRUCTION FOR(n,M,2(k−1),k)q CODES by this construction.
In this section we provide a recursive construction for Construction II. Let s = k i, n ≥ s + 2 + k and
i=3
(n,M,2(k−1),k)q codes,whichusesthependingdotsbased q2+q+1≥ℓ,whereℓ=n−sPforoddn−s(orℓ=n−s−1
construction described in Section II as an initial step. Codes for even n−s).
obtainedbythisconstructioncontaintheliftedMRDcode.An Identifying vectors: In addition to the identifying vector
upper bound on the cardinality of such codes is given in [3]. vk = (11...1100...0) of the lifted MRD code Ck (of size
00 ∗
q2(n−k) anddistance2(k−1)),theotheridentifyingvectorsof Ferrers tableaux forms: On the dots corresponding to the
the codewords are defined as follows. First, by Lemma 7, we lastn−s−2 columnsof theFerrersdiagramsforeachvector
partition the weight-2vectorsof Fn−s into classes P ,...,P v in a given Ak, 0 ≤ i ≤ 3, we construct a FDMRD code
2 1 ℓ j i
of size ℓ¯ (where ℓ = ℓ¯− 1 = n − s − 1 if n − s even with minimum distance k−1 (according to Remark 22) and
and ℓ =2ℓ¯+1 = n−s if n−s odd) with pairwise disjoint lift it to obtain Ck . We define Ck = |Aki|Ck .
i,j i j=1 i,j
positionsoftheones.We definethesetsofidentifyingvectors Code: The final code is defined as S
by a recursion. Let v and A ,A ,A ⊆Fn−s+3, as defined
0 1 2 3 q 3
in Construction 0. Then v030 =v0, Ck = Ck∪Ck.
i ∗
A3 =∅, A3 =A , 1≤i≤3. i[=0
0 i i
For k ≥4 we define: Theorem 24. The code Ck obtained by Construction II has
minimum distance 2(k−1) and cardinality |Ck|=q2(n−k)+
Ak0 ={v0k1,...,v0kk−3}, q2(n−(k+(k−1)))+...+q2(n−(Pki=3i))+ n−( ki=3i) .
where vk =(000wk ||vk−1 ) (1≤j ≤k−3), such that the (cid:20) P2 (cid:21)q
0j j 0j−1
wk arealldifferentweight-1vectorsofFk−3.Furthermorewe Proof:Firstobservethat,forallidentifyingvectorsexcept
j 2
define: vk , the additional line of dots of the corresponding Ferrers
00
diagrams does not increase the cardinality compared to the
Ak ={(0010...00||z):z ∈Ak−1},
1 1 previous recursion step, due to Lemma 21. The only identi-
Ak ={(0100...00||z):z ∈Ak−1}, fying vector that contributes additional words to Ck is vk ,
2 2 00
Ak ={(1000...00||z):z ∈Ak−1}, and thus |Ck|=|Ck−1|+q2(n−k) for any k ≥4. Inductively,
3 3
the cardinality formula follows, together with the cardinality
such that the prefixes of the vectors in ∪3i=0Aki are vectors formula for k =3 from Construction 0.
of Fk2 of weight 1. Note, that the suffix y ∈ Fqn−s (from NextweprovethattheminimumdistanceofCk is2(k−1).
Construction 0) in all the vectors from Ak1 belongs to P1, the Let X,Y ∈ Ck, X 6= Y. If v(X) = v(Y) then by Lemma 4
suffix y in all the vectors from Ak belongs to ∪min{q+1,ℓ}P , d (X,Y)≥2(k−1).
2 i=2 i S
andthesuffixyinallthevectorsfromAkbelongsto∪ℓ P Now we assume that v(X) 6= v(Y). Note, that according
3 i=q+2 i
(the set Ak is empty if ℓ≤q+1). to the definition of the identifying vectors, d (X,Y) ≥
3 S
Pending blocks: d (v(X,v(Y))=2(k−1)for(X,Y)∈Ck×Ck,0≤i≤3,
H ∗ i
• All Ferrers diagrams that correspond to the vectors in for (X,Y)∈Ck0 ×Ck0, and for (X,Y)∈Cki ×Ckj, i6=j.
Ak1 have a common pending block with k−3 rows and Now let X,Y ∈Cki, for some 1≤i≤3.
ik=−3ji dots in the jth row, for 1 ≤ j ≤ k − 3. We • If the suffixes of v(X) and v(Y) of length n − s
Pfill eachofthese pendingblockswith a differentelement belong to the same class Pt, then dH(v(X),v(Y)) = 4
of a suitable FDMRD code with minimum rank distance and d (B ,B ) = k − 3, for the common pending
R X Y
k−3andsizeq3,accordingto Lemma23. Note,thatthe blocks submatrices B ,B of F(X),F(Y). Then by
X Y
initial conditions imply that q3 ≥ℓ¯, i.e. we always have Theorem 11, d (X,Y)≥4+2(k−3)=2(k−1).
S
enough fillings for the pending block to use all elements • If the suffixes of v(X) and v(Y) of length n − s
of the given Pi. belong to different classes, say Pt1,Pt2 respectively,
• All Ferrers diagrams that correspond to the vectors in then dH(v(X),v(Y)) ≥ 2 and dR(BX,BY) = k −2,
Ak have a common pending block with k−2 rows and for the common pending blocks submatrices B ,B
2 X Y
k−ji+1 dots in the jth row, 1 ≤ j ≤ k−2. Every of F(X),F(Y). Then by Theorem 11, d (X,Y) ≥
i=3 S
vPectorwhichhasasuffixyfromthesamePiwillhavethe 2+2(k−2)=2(k−1).
same value ai ∈Fq in the first entry in each row of the Hence,foranyX,Y ∈Ck itholdsthatdS(X,Y)≥2(k−1).
commonpendingblock,s.t.thevectorswithsuffixesfrom
the different classes will have different values in these
entries. (This correspondsto a FDMRD code of distance Corollary 25. Let n≥s+2+k and q2+q+1≥ℓ, where
k−2 and size q.) Given the filling of the first entries of s = ki=3i and ℓ = n−s for odd n−s (or ℓ = n−s−1
every row, all the otherentries of the pendingblocks are for ePven n−s). Then
filled by a FDMRD code with minimum distance k−3,
k k
according to Lemma 23. Aq(n,2(k−1),k)≥ q2(n−Pki=ji)+ n−( i=3i) .
• All Ferrers diagrams that correspond to the vectors in Xj=3 (cid:20) P2 (cid:21)q
Ak have a common pending block with k−2 rows and
3
k−ji+2dotsinthejthrow,1≤j ≤k−2.Thefilling Example 26. Let k =4, d=6, n=13, andq =2. The code
Pofit=he3sependingblocksisanalogoustothepreviouscase, C4 obtained by Construction II has cardinality 218 +212 +
6
butforthesuffixesfromthedifferentPi-classeswefixthe = 218 +4747 (the largest previously known code is
first two entries in each row of a pending block. Hence, (cid:20) 2 (cid:21)q
there are q2 different possibilities. of cardinality 218+4357 [2]).
Example 27. Let k =5, d=8, n=19, and q =2. The code VI. CONCLUSION
C5 obtained by Construction II has cardinality 228 +220 + In this work we presented new constructions (based on
7
214+ =228+1067627(the largest previously known the ideas of [2], [3], [16]) of constant dimension codes
(cid:20) 2 (cid:21)q in Gq(k,n) with minimum subspace distance 4 or 2k − 2,
code is of cardinality 228+1052778 [2]). We illustrate now respectively. These constructions give rise to lower bounds
the construction. on the cardinality of such codes, which are tighter than other
First,wepartitionthesetofsuffixesy ∈F27 ofweight2into knownboundsforgeneralparameters.Ontheotherhandthere
7 classes, P1,...,P7 of size 3 each. The identifying vectors existsomeparametersetswhereweknowbetterconstructions,
of the code are partitioned as follows: hence these bounds are not tight in general. Then again, we
v5 =(11111||0000||000||0000000), show some examples where our constructions come up with
00
the largest codes known so far for the given parameters.
A5 ={(00001||1111||000||0000000),
0 Forfutureworkone cantry to applythe ideasofthis paper
(00010||0001||111||0000000)} to constant dimension codes with other minimum subspace
A5 ={(00100||0010||001||y):y ∈P } distance than 4 or 2k−2 to come up with better codes than
1 1
known so far.
A5 ={(01000||0100||010||y):y ∈{P ,P }}
2 2 3
A5 ={(10000||1000||100||y):y ∈{P ,P ,P ,P }} ACKNOWLEDGMENT
3 4 5 6 7
Todemonstratetheideaoftheconstructionwewillconsider The second author war partially supported by Swiss Na-
only the set A5. All the codewords correspondingto A5 have tional Science Foundation grant no. 138080 and Forschungs-
2 2
the following common pending block B: kredit of the University of Zurich, grant no. 57104103.
• • • • • • • •
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00
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0
The number of codewords that correspond to A5∪A5∪A5
1 2 3
7
is .
(cid:20) 2 (cid:21)
q