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ISSN 2148-838X
https://doi.org/10.13069/jacodesmath.858732
J. Algebra Comb. Discrete Appl.
8(1) •31–39
Received: 16 February 2020
Accepted: 13 September 2020
Journal of Algebra Combinatorics Discrete Structures and Applications
A new construction of anticode-optimal Grassmannian
codes
Research Article
Ben Paul Dela Cruz,John Mark Lampos,Herbert Palines,Virgilio Sison
Abstract: In this paper, we consider the well-known unital embedding from Fqkinto Mk(Fq)seen as a map of
vector spaces over Fqand apply this map in a linear block code of rate ρ/` over Fqk. This natural
extension gives rise to a rank-metric code with krows, k` columns, dimension ρand minimum distance
kthat satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen
as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length n−k.
The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the
union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code
with length n, cardinality qn−1
qk−1, minimum injection distance kand dimension kthat satisfies the
anticode upper bound can be constructed.
2010 MSC: 94B05, 94B60, 94B65
Keywords: Ferrers diagram, Rank-metric code, Grassmannian, Constant dimension, Anticode bound
1. Introduction
Let Fqbe the finite field of order q. The projective space of order nover Fq, denoted by Pq(n), is the
set of all subspaces of Fn
q. Given an integer ksuch that 0≤k≤n, the set of all k-dimensional subspaces
of Fn
qis known as a Grassmannian, denoted by Gq(n, k). A subspace code is a nonempty subset of Pq(n),
while a Grassmannian code is a nonempty subset of Gq(n, k). Subspace codes are used in network coding,
a method that is far more efficient than classical coding. This paper aims to generalize the results of [4],
i.e. to construct maximum rank distance (MRD) codes whose generalized lifts form an anticode-optimal
Grassmannian code.
The paper is organized as follows. The next section gives some preliminaries and the construction
of subspace codes in [2]. Section 3shows how to construct MRD codes from linear block codes. Given
Ben Paul Dela Cruz (Corresponding Author), John Mark Lampos, Herbert Palines, Virgilio Sison; Institute of
Mathematical Sciences and Physics, University of the Philippines, Los Baños, College, Laguna 4031, Philippines
(email: bbdelacruz2@up.edu.ph, jtlampos@up.edu.ph, hspalines@up.edu.ph, vpsison@up.edu.ph).
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
a specific skeleton code, these MRD codes turn out to be Ferrers diagram maximum rank distance
(FDMRD) codes. In Section 4, anticode-optimal Grassmannian code will be constructed using these
FDMRD codes and the multi-level construction in [2]. Instead of using pending dots, we will use the
multi-level construction as presented in [6]. Lastly, we give our conclusion in Section 5.
2. Preliminaries
A[k×`]matrix code over Fqis a nonempty subset of Mk×`(Fq). The rank distance between two
k×`matrices over Fq, say Aand B, is given by dR(A, B) = rank(A−B).
The minimum rank distance of a matrix code C, denoted by δ, is defined by δ= min{dR(A, B)|A, B ∈
C, A 6=B}. A [k×`, δ]rank-metric code is a [k×`]matrix code with minimum rank distance δ. It is
worth noting that a linear code in Mk×`(Fq)is a subspace of the vector space Mk×`(Fq). A [k×`, ρ, δ ]
rank-metric code Cis a linear code in Mk×`(Fq)with dimension ρand minimum distance δ.
The following theorem gives the relationship of the minimum distance of a rank-metric code with its
minimum nonzero rank.
Theorem 2.1. [4] Let Cbe a [k×`, ρ, δ]rank-metric code with minimum nonzero rank Ω. Then δ= Ω.
Theorem 2.2. [1] For a [k×`, ρ, δ]rank−metric code C,
ρ≤min{k(`−δ+ 1), `(k−δ+ 1)}.
A code that attains the bound in Theorem 2.2 is called a maximum rank distance code or an MRD
code.
Before we go to Grassmannian codes, we first give two definitions which are vital to our construction.
Let A∈Mk×`(Fq). The lift of A, denoted by L(A), is the standard matrix (IkA). We adapted this
definition from [4]. For a given matrix A, we denote the row space of Aby hAi.
Example 2.3. Let A=101
011∈M2×3(F2). Then
L(A) = 10101
01011.
Moreover, the rowspace generated by L(A)is the linear block code of length 5, rate 2/5over F2.
hL(A)i={(0,0,0,0,0),(1,0,1,0,1),(0,1,0,1,1),(1,1,1,1,0)}.
Definition 2.4. [4] Let Cbe a [k×`]rank-metric code. The set
Λ(C) = {hL(A)i|A∈C}
is called the lift of C.
It is well known that the cardinality of Gq(n, k)is given by the q-ary Gaussian coefficient
|Gq(n, k)|=n
kq
=
k−1
Y
i=0
qn−i−1
qk−i−1.(1)
Consequently, |Pq(n)|=| ∪n
k=0 Gq(n, k)|.
Furthermore, the subspace distance and the injection distance, defined as
dS(A, B) = dim(A+B)−dim(A∩B)
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
and
dI(A, B) = max{dim A, dim B} − dim(A∩B)
respectively, for any A, B in Pq(n)are metrics on Pq(n), and so in Gq(n, k ). It is clear that if Aand B
have the same dimension, then
dI(A, B) = 1
2dS(A, B).
We say that C ⊆ Gq(n, k)is an (n, M , d, k)qcode in the Grassmannian, or a constant-dimension code,
if |C| =Mand the minimum injection distance of Cis d= min{dI(A, B)|A, B ∈ C, A 6=B}. Since
dI(A, B) = 1
2dS(A, B), then the minimum subspace distance of a Grassmannian code is twice of its
minimum injection distance. Equivalently, one may opt to use the subspace distance instead of injection
distance.
The next theorem gives the parameters of the resulting Grassmannian code from a lifted rank-metric
code.
Theorem 2.5. [6] Let Cbe a [k×`, ρ, δ]rank-metric code. Then Λ(C)is a (k+`, qρ, δ, k)qGrassmannian
code.
The maximum number of codewords in an (n, M, d, k)qcode is denoted by Aq(n, d, k). Bounds
for Aq(n, d, k)were given in [7], [3] and [11]. The lift of maximum rank distance (MRD) codes in [10]
asymptotically attains the bounds given in [3] and [7]. The next theorem gives the bound that were used
to check the optimality of our constructed code.
Theorem 2.6. [11] Anticode Bound.
Aq(n, d, k)≤n
kq
n−k+d−1
d−1q
=n
k−d+ 1 q
k
k−d+ 1 q
(2)
Etzion and Silberstein provided a multi-level construction of Grassmannian codes in [2]. The codes
constructed using multi-level construction are called lifted Ferrers Diagram (FD) codes in [6]. Further-
more, an alternative form of the construction of lifted FD codes which uses matrices instead of the usual
pending dots were presented in [6]. In this paper, we opt to use the alternative form of the said con-
struction. As defined in [6], for a nonzero X∈Mk×`(Fq), there corresponds a vector prof(X)∈ {0,1}n,
called the profile vector of X, in which supp(prof(X)) is the set of column positions of the leading ones
in the rows of the row reduced echelon form of X. If X= 0, then we set prof(X)to be the zero vector.
Associated with a vector space U∈ Gq(n, k), k > 0, is a unique k×nmatrix XUin row reduced echelon
form such that U=hXUi. Now define the profile vector of U, denoted by prof(U), given by
prof(U) = prof(XU).
In addition, prof(U)=0∈Fn
2if dim(U)=0. Let b∈Fn
2, the Schubert cell in Pq(n)corresponding to b
is the set
Sq(b) = {U∈ Pq(n)|prof(U) = b}.
Some papers denote this set as prof−1(b).
Given any binary vector bof length nand Hamming weight k, the permutation matrix with respect
to b, denoted by P(b), is the n×npermutation matrix whose rows indexed by supp(b)form P(b)supp(b)=
Ik0k×(n−k)and the remaining rows of P(b)form P(b)supp(¯
b)=0(n−k)×kI(n−k), where b+¯
bis
the all one vector of length n.
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
Example 2.7. Let b= (1,0,0,1,0). So ¯
b= (0,1,1,0,1).
P(b) =
10000
00100
00010
01000
00001
We changed some notations in the next definition to be consistent with our earlier notations but the
essence is exactly the same.
Definition 2.8. [6] Let bbe a binary vector of length nand Hamming weight k. For X∈Mk×(n−k)(Fq),
define the generalized lifting of Xwith respect to b, denoted by Λb(X), as
Λb(X) = hL(X)iP(b)−1=hL(X)P(b)−1i.
Definition 2.9. [6] Let bbe a binary vector of length n, Hamming weight kand let Cbbe a [k×(n−k)]
rank-metric code. The set
Λb(Cb) = {Λb(C)|C∈ Cb}
is called the generalized lift of Cb.
We now present some necessary conditions for the definition of lifted FD codes which were established
in [6]. Let Q= [aij ]be the n×nupper triangular matrix with aij = 1 if j≥iand aij = 0 otherwise.
Given a binary profile vector bof length nand Hamming weight k, regarded as an element of Z1×n, define
the vector c(b)∈Z1×nvia
c(b) = bQP (b).
Let X= [xij ]∈Mk×(n−k)(Fq). According to [6], Λb(X)is guaranteed to be in the Schubert cell
corresponding to bprovided that for 1≤i≤kand 1≤j≤n−k,
i>c(b)j+kimplies that xij = 0.(3)
An FD(b) code is a rank-metric code Cb⊆Mk×(n−k)(Fq)in which each codeword satisfies (3) while the
code Λb(Cb)is referred to as lifted FD(b) code.
We now consider the minimum distance between elements in distinct Schubert cells. Let u, v ∈Fn
2and
u6=v. Now define the logical AND of uand v, denoted by u∧v, as (u∧v)i=uivi. Also the asymmetric
distance between uand v, denoted by da(u, v), is given by da(u, v) = max{wtH(u), wtH(v)} − wtH(u∧v).
Theorem 2.10. [5] Let u, v ∈Fn
2,u6=v,U∈Sq(u)and V∈Sq(v)and d(U, V )be the injection distance
of Uand V. Then d(U, V )≥da(u, v).
The following steps to construct an (n, M, d, k)qcode Care from [5].
1. Choose a binary constant weight code Bof length n, Hamming weight k, and minimum asymmetric
distance d.
2. For each b∈ B, consider an FD(b) code with minimum rank distance d.
3. Construct the lifted FD(b) code Λb(Cb)for each b∈ B.
4. Set C=[
b∈B
Λb(Cb)
The cardinality Mof Cgreatly depends on the choice of B.
Theorem 2.11. [2]Cis an (n, M, d, k)qconstant-dimension code, where M=Pb∈B |Λb(Cb)|.
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
3. Rank-metric codes and Grassmannian codes
The following well-known theorem will be the cornerstone of our construction.
Theorem 3.1. [9] Let f(x) =
k
P
i=0
aixi∈Fq[x]be a monic irreducible polynomial and Xbe its companion
matrix. Then the mapping
π:Fq[x]→Mk(Fq), g(x)7→ g(X)(4)
induces a unital embedding
τ:Fq[x]/(f) = Fqk→Mk(Fq).(5)
Let nbe a positive integer. Note that τcan be extended to the following monomorphism φdefined
by φ:Fn
qk→Mk×kn (Fq)where
φ(α1, α2, ..., αn) = τ(α1)τ(α2)... τ(αn).
Lemma 3.2. If Cis a linear block code of length nover Fqkthen C∼
=φ(C)as Fq−vector spaces.
The following theorem is a generalization of Theorem 3.7 of [4].
Theorem 3.3. Let Cbe a linear block code of length nover Fqkand ρits dimension as an Fq-vector
space. Then
i. φ(C)is a [k×kn, ρ, k]rank-metric code,
ii. Λ (φ(C)) is a (k+kn, qρ, k , k)qcode,
iii. the pairwise intersection of codewords of Λ (φ(C)) is trivial.
Proof. Let Cbe a linear block code of length nover Fqkand ρits dimension as an Fq-vector space.
By Lemma 3.2,Cand φ(C)are isomorphic as Fq−vector spaces. Hence, the dimension of φ(C)is ρ.
Consider the case n= 1. Since Fqkis a field, then ∀α∈Fqk− {0}, φ (α) = τ(α)∈Mk(Fq)is a unit and
thus has rank k, where τis as defined in (5). Thus, τ(α1)τ(α2)... τ (αn)has rank kfor any positive
integer nwhere αi∈Fqk− {0}for some i, 1≤i≤n. Therefore, by Theorem 2.1, the minimum distance
of φ(C)is k.
Clearly, ii. follows directly from Theorem 2.5
If Λ (φ(C)) is a (k+kn, qρ, k , k)qcode, the minimum injection distance of Λ (φ(C)) is k. Let
A, B ∈Λ (φ(C)). We have k≤d(A, B) = max{dim A, dim B}−dim (A∩B).Hence, k≤k−dim (A∩B)
which makes dim (A∩B)to be equal to 0. Therefore, the pairwise intersection of codewords of Λ (φ(C))
is trivial.
Note that for any positive integer n, the rank-metric code φ(Fn
qk)meets the Singleton bound as given
in Theorem 9 of [7].
Example 3.4. Consider the irreducible polynomial x2+x+ 1 over F2and its companion matrix X=
1 1
1 0 , and the linear block code F2
4. Then φ(F2
4)is a [2 ×4,4,2] rank-metric code which satisfies the
Singleton bound. Moreover, Λ(φ(F2
4)) is a (6,16,2,2)2−code.
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
4. Anticode-optimal Grassmannian code
Now we construct Grassmannian codes using the multi-level construction in [2] and the result in
Theorem 3.3. Let nbe a multiple of kand n>k. For the skeleton code B, choose
B={b`= (0,· · · ,0
| {z }
n−k−k`
,1,· · · ,1
| {z }
k
,0,· · · ,0
| {z }
k`
)|0≤`≤n−k
k}.
Clearly, Bis a binary constant weight code of length n, weight k, and minimum asymmetric distance k.
Then for 0≤`≤n−k
k
P(b`) =
In−k−k`
Ik
Ik`
.
Let Q= [aij ]be the n×nupper triangular matrix with aij = 1 if j≥iand aij = 0 otherwise. Then,
c(b`) = b`QP (b`)
= (0,· · · ,0
| {z }
n−k−k`
,1,· · · ,1
| {z }
k
,0,· · · ,0
| {z }
k`
)QP (b`)
= (0,· · · ,0
| {z }
n−k−k`
,1,2,· · · , k, k, · · · , k
| {z }
k`
)P(b`)
= (1,2,· · · , k, 0,· · · ,0
| {z }
n−k−k`
, k, · · · , k
| {z }
k`
)
.
Lemma 4.1. Let nbe a multiple of k,n>kand
Cb`=n0k×(n−k−k`)φ(v)v∈F`
qko
where 1≤`≤n−k
k. Then, for 1≤`≤n−k
k,Cb`is an FD(b`) code and has minimum rank distance k.
Proof. Recall that for a rank-metric code Cto be an FD(b) code, Cmust be a subset of Mk×(n−k)(Fq)
and all of its codewords must satisfy (3). Let 1≤`≤n−k
k. Clearly, Cb`⊆Mk×(n−k)(Fq). Now let
X= [xij ]∈ Cb`. By (3), for 1≤i≤k,1≤j≤n−k,i>c(b`)j+kimplies that xij = 0. Notice that in
(3), we only check the last n−kcomponents of c(b`). Now
c(b`)j+k=(0if 1≤j≤n−k−k`
kif n−k−k` < j ≤n−k.
Hence the entries in the first n−k−k` columns of the elements of Cb`must be all zero. Clearly,
Cb`satisfies this. Lastly, by Theorem 3.3,φ(v)has minimum rank distance k. Clearly, Cb`has minimum
rank distance k.
Theorem 4.2. The rank-metric code Cb0={(0k×(n−k))}is an FD(b0)code.
Although Cb0does not have a minimum distance kas required in [2] and [6], it will not pose any
problem since the resulting Grassmannian will still have a minimum distance k. Ironically, b0is included
in Example 10 of [2]. Note also that Cb0is the only rank-metric code that will satisfy b0.
Now we are ready to construct the lifted FD(b) code Λb(Cb). By Definition 2.8 and Definition 2.9,
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
Λb`(Cb`) =
nD0k×(n−k−k`)Ikφ(v)Ev∈F`
qko,1≤`≤n−k
k
nD0k×(n−k)IkEo, ` = 0.
The lifted FD code, denoted by Ω(B), is given by
Ω(B) = [
b∈B
Λb(Cb).
As an immediate consequence of Lemma 4.1 and Theorem 4.2, we have the following theorem.
Theorem 4.3. Ω(B)is an n, qn−1
qk−1, k, kqcode.
Since Cb0does not have a minimum distance kand b0appeared only as an example in [2], we will
separate the case of b0in our proof.
Proof. For 1≤`≤n−k
k,Cb`is an FD(b`) code with minimum rank distance kby Lemma 4.1. Now,
Ω(B) = [
b∈B
Λb(Cb) =
[
b∈B,b6=b0
Λb(Cb)
∪Λb(Cb0).
By Theorem 2.11,Sb∈B,b6=b0Λb(Cb)is an (n, M, k , k)qcode where M=Pb∈B,b6=b0|Λb(Cb)|. Clearly,
Λb0(Cb0)⊆ Gq(n, k). Now we compute for the distance of codewords U∈Sb∈B,b6=b0Λb(Cb)and V∈
Λb0(Cb0). Since U∈Sb∈B,b6=b0Λb(Cb), then U∈Λb`(Cb`)for some b`∈ B − {b0}. By Theorem 2.10,
d(U, V )≥da(b`, b0)≥k−0 = k. Note that Λb0(Cb0)has only one codeword. Therefore, Ω(B)has
minimum distance k.
Observe that Λb`(Cb`)∩Λbj(Cbj) = ∅, where `6=j. Hence,
|Ω(B)|=X
b∈B
|Λb(Cb)|=
n−k
k
X
i=0
qki =qn−1
qk−1.
Note that Ω(B)attains the Anticode bound. The following illustrates the construction of an
Anticode-optimal Grassmannian code.
Example 4.4. Consider the irreducible polynomial x3+x+ 1 over F2, its companion matrix X=
011
100
010
,
φ1:F23→M3×3(F2), α17→ τ(α1),
φ2:F2
23→M3×6(F2),(α1, α2)7→ τ(α1)τ(α2),
and the skeleton code
B={b`= (0,· · · ,0
| {z }
6−3`
,1,· · · ,1
| {z }
3
,0,· · · ,0
| {z }
3`
)|0≤`≤2}.
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B. P. Dela Cruz et al. / J. Algebra Comb. Discrete Appl. 8(1) (2021) 31–39
Now,
Cb0={03×6}
Cb1={(03×3|φ1(v)) : v∈F23}
Cb2={φ2(v) : v∈F2
23}.
So
Λb0(Cb0) = {h03×6|I3i}
Λb1(Cb1) = {h03×3|I3|φ1(v)i:v∈F23}
Λb2(Cb2) = {hI3|φ2(v)i:v∈F2
23}.
Finally, Ω(B)=Λb0(Cb0)∪Λb1(Cb1)∪Λb2(Cb2)with cardinality 1 + 8 + 64 = 73 = A2(9,3,3).
Theorem 4.3 is a generalization of Theorem 3.16 in [4] and is similar with the construction of spread
codes in [8]. Note that by choosing vto be in a different linear block code Cof length `over Fqkinstead
of F`
qkin Lemma 4.1, one may construct a Grassmannian code that is not a spread code.
5. Summary and conclusion
We presented two constructions of Grassmannian codes for any length, over any finite field and whose
dimension is equal to its minimum injection distance. These two constructions are generalizations of some
constructions in [4]. The first construction uses a linear block code to construct a rank-metric code. The
resulting rank-metric code was then used to create a Grassmannian code that meets the Singleton bound.
The second construction uses the results in the first construction together with the concept of Ferrers
diagram to get an anticode-optimal Grassmannian code. The resulting code from the second construction
is similar to the construction of spread codes found in [8].
Acknowledgment: The authors would like to thank the reviewer for his/her valuable comments.
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