Table Of ContentA lower bound for constant dimension codes from
multi-component lifted MRD codes
3
Anna-Lena Trautmann
1
0 Institute of Mathematics
2 University of Zurich, Switzerland
n
a
J
1 Introduction
9
] In this work we investigate unions of lifted MRD codes of a fixed dimension
T
and minimum distance and derive an explicit formula for the cardinality of
I
. such codes. This will then imply a lower bound on the cardinality of constant
s
dimension codes.
c
[ We will first repeat some known results needed in this section. In Section 2
we will explain the construction and derive the formula for the cardinality and
1
the lower bound. We conclude in Section 3.
v
8 Let Fq be the finite field with q elements. Gq(k,n) denotes the set of all
1 k-dimensional vector spaces over Fn, called the Grassmannian. A constant
q
9 dimension code is simply a subset of G (k,n). A metric on G (k,n) is the
q q
1
injection metric, given by
.
1
0 d (U,V)=k−dim(U ∩V)
I
3
1 for anyU,V ∈G (k,n). The minimum injection distance ofa code C ⊆G (k,n)
q q
:
v is defined as
i d (C)=min{d (U,V)|U,V ∈C,U 6=V}.
X I I
r ByAq(n,d,k)wedenotethemaximalsizeofacodeC ⊆Gq(k,n)withminimum
a
injection distance d.
There is a complete theory of matrix codes with the rank distance, which
can be used to construct constant dimension codes. We will now give a brief
overview on the most important definitions and results of this topic.
Let A,B ∈Fm×n be two matrices of the same size. It holds that
q
d :=rk(A−B)
R
is a metric on Fm×n, called the rank metric. A rank-metric code is simply a
q
subset of Fm×n. The minimum distance is defined in the usual way.
q
The following two theorems can be found in [1]:
1
Lemma 1. Let m,n ≥ d and C ∈ Fm×n be a rank-metric code with minimum
q
rank distance d. Then
|C|≤qmax(n,m)(min(n,m)−d+1).
Lemma2. Foranysetofparametersn,m≥d∈Nandarbitraryfieldsizethere
exist codes attaining the bound of Lemma 1. These codes are called maximum
rank distance (MRD) codes.
We willnowexplainhowto useMRDcodes forthe constructionofconstant
dimension codes.
Definition 3. For a given rank-metric code C ∈Fk×n the set
q
lift(C):={rowspace I A |A∈C}
k
(cid:2) (cid:3)
is called the lifting of C.
Lemma 4. [4] If C ∈ Fk×n−k is an MRD code of minimum rank distance d,
q
then lift(C) ∈ G (k,n) is a constant dimension code with minimum injection
q
distance d. Moreover,
|lift(C)|=q(n−k)(k−d+1).
Note,thatthisconstructioncorrespondstotheReed-Solomon-likeconstruc-
tionfromK¨otterandKschischang[3]. Wedonotwanttoexplainthisconstruc-
tion in detail here, but the interested reader is referred to [3].
2 Multi-component lifted MRD codes
Naturally, appending 0-columns in front of all code elements does not change
the minimum distance. Thus, if C ∈ Fk×n−k−l is an MRD code of minimum
q
rank distance d, then
rowspace 0 I M |M ∈C}⊆G (k,n)
k×l k×k q
(cid:2) (cid:3)
isaconstantdimensioncodewithminimuminjectiondistancedandcardinality
q(n−k−l)(k−d+1). This fact can be used to construct even larger codes, which
has also been observed by, among others, [2, 5]. We will now give our own
formulationof a multi-component construction and derive an exact formula for
the cardinality of these codes, which we call the multi-component lifted MRD
codes. Note, that our construction differs from the one in [5]. Moreover, the
constructionof [2] is more generaland does not give an explicit formula for the
size of such codes.
Theorem 5. Let C be some MRD code with minimum rank distance d in
j
Fk×n−k−jd for j =0,...,⌊n−k⌋. Then
q d
C ={rs 0 I M |M ∈C }
j k×j·d k×k j
(cid:2) (cid:3)
2
are called the component codes and the union
⌊n−dk⌋
C = C
j
j[=0
is an [n,N,d,k] -code, where
q
⌊n−d2k⌋ ⌊n−dk⌋
N = q(k−d+1)(n−k−di)+ ⌈qk(n−k+1−d(i+1))⌉.
Xi=0 i=⌊nX−d2k⌋+1
If k =d and n≡r mod k (such that 0≤r <k), it holds that
qn−qr+k qn−r −1
N = +1=qr( +q−r−1).
qk−1 qk−1
Proof. We will first prove the minimum distance. It is clear that the distance
between any elements of the same component C is greater than or equal to
i
d. Now let U ∈ C and V ∈ C . Since the identity blocks are shifted by d
i i+1
positions, the maximal intersection is (k−d)-dimensional. Thus
d (U,V)=k−dim(U ∩V)≥k−(k−d)=d.
I
Letus nowinvestigatethe size ofthe code. The subspacecomponentcode is as
large as the corresponding MRD code, thus
⌈q(n−k−di)(k−d+1)⌉ for n−di≥2k
|C |=
i (cid:26) ⌈qk(n−k+1−d(i+1))⌉ for n−di<2k
(followsfromLemma 2). As n−di≥2k ⇔i≤ n−2k andwelook atcodes with
d
n≥2k, we proved the general formula. For d=k it holds that
⌊nk⌋−2 ⌊nk⌋−1 ⌊nk⌋−2
qn−k(i+1)+ ⌈qk(n−k+1−k(i+1))⌉= qn−k(i+1)+⌈qk(n−k+1−k⌊nk⌋)⌉
Xi=0 i=⌊Xnk⌋−1 Xi=0
= qn−qk+n−k⌊nk⌋ +⌈qk(n+1−k⌊nk⌋)⌉.
qk−1
Note, that n−k⌊n⌋=r, if n≡r mod k. Thus, the expononent of the second
k
summand is non-positive and the formula for N follows.
We can now obtaina generallowerbound onthe size ofconstantdimension
codes:
Corollary 6.
⌊n−d2k⌋ ⌊n−dk⌋
A (n,d,k)≥ q(k−d+1)(n−k−di)+ ⌈qk(n−k+1−d(i+1))⌉.
q
Xi=0 i=⌊nX−d2k⌋+1
3
3 Conclusion
Lifte rank metric codes are a prominent family of constant dimension codes.
Taking unions of suchcodes of different length(with zerocolumns appended in
front) achieves codes of the same minimum distance but larger cardinality. In
this work we give an explicit formula for the size of such codes which gives rise
to a lower bound on the size of constant dimension codes ⊆ G (k,n) for any
q
q,k,n and minimum distance d.
References
[1] E`. M. Gabidulin, Theory of codes with maximum rank distance, Problemy
PeredachiInformatsii 21 (1985), no. 1, 3–16.MR MR791529 (87f:94036)
[2] E. M. Gabidulin and N. I. Pilipchuk, Multicomponent network coding, Pro-
ceedings of the Seventh International Workshop on Coding and Cryptogra-
phy (WCC) 2011,2011, pp. 443–452.
[3] R. K¨otter and F.R. Kschischang, Coding for errors and erasures in random
network coding, IEEETransactionsonInformationTheory54(2008),no.8,
3579–3591.
[4] D.Silva,F.R.Kschischang,andR.Koetter,Arank-metricapproach toerror
control in random network coding, Information Theory, IEEE Transactions
on 54 (2008), no. 9, 3951 –3967.
[5] V. Skachek, Recursive code construction for random networks,
arXiv:0806.3650,2008.
4
