Construction of Subspace Codes through Linkage
aa r X i v : . [ c s . I T ] M a y Construction of Subspace Codes through Linkage
Heide Gluesing-Luerssen ∗ and Carolyn Troha ∗ October 24, 2018
Abstract:
A construction is presented that allows to produce subspace codes of long lengthusing subspace codes of shorter length in combination with a rank metric code. The subspacedistance of the resulting code, called linkage code, is as good as the minimum subspace distanceof the constituent codes. As a special application, the construction of the best known partialspreads is reproduced. Finally, for a special case of linkage, a decoding algorithm is presentedwhich amounts to decoding with respect to the smaller constituent codes and which can beparallelized.
Keywords:
Random network coding, constant dimension subspace codes, partial spreads.
MSC (2010):
In [22] Koetter and Kschischang developed an approach to random network coding where theencoded information is represented as subspaces of a given ambient space. This accounts for theunknown network topology by assuming that any linear combination of packets may occur atthe nodes of the network.This approach has led to the area of subspace codes and specifically to intensive researchefforts on constructions of subspace codes with large subspace distance [23, 21, 25, 10, 14, 9, 16,12, 15, 11, 24, 29, 17]. Most of the research focuses on constant-dimension codes (CDC’s), thatis, codes where all subspaces have the same dimension.One direction for constructing CDC’s is based on so-called cyclic orbit codes [23, 9, 24,29, 17], which are orbits of a subspace in the F q -vector space F q n under the natural action of F ∗ q . While the resulting codes have very beneficial algebraic structure, they do not have largecardinality in general. Taking unions of such codes leads to cyclic subspace codes which stillhave nice structure, but it remains an open problem how to take unions of cyclic orbit codeswithout decreasing the distance.A second major research direction is based on rank-metric codes as introduced and studiedearlier by Delsarte [6] and Gabidulin [13]. Lifting rank-metric codes [26] is a very simple con-struction which results in subspace codes where the reduced row echelon form of each subspacehas its identity matrix in the leftmost position. While these codes are asymptotically good [21],they can still be improved upon. Through a careful study of general reduced row echelon forms, ∗ HGL was partially supported by the National Science Foundation Grant DMS-1210061. HGL and CT arewith the Department of Mathematics, University of Kentucky, Lexington KY 40506-0027, USA; { heide.gl, car-olyn.troha } @uky.edu. Let us first recall some basic facts about subspace codes and rank-metric codes. Throughoutwe fix a finite field F = F q . A subspace code of length n is simply a non-empty collection ofsubspaces in F n . The code is called a constant-dimension code if all subspaces have the samedimension. The subspace distance of a subspace code C is defined as d S ( C ) := min { d S ( V , W ) |V , W ∈ C , V 6 = W} , where the distance between two subspaces isd S ( V , W ) := dim V + dim W −
V ∩ W ) . It is a metric on the space of all subspaces, see [22, Lem. 1]. If dim V = dim W = k , thend S ( V , W ) = 2 (cid:0) dim( V + W ) − k (cid:1) . As a consequence, if C is a constant-dimension code of2imension k then d S ( C ) ≤ min { k, n − k ) } . (2.1)If the code C consist of a single subspace of dimension k , we define d S ( C ) := min { k, n − k ) } . Aconstant-dimension code of length n , dimension k , cardinality N will be called an ( n, N, k )-code,and it is a ( n, N, k, d )-code if its subspace distance is d .A k × m rank-metric code is a non-empty subset of F k × m , endowed with the rank metricd R ( A, B ) := rk( A − B ) (which is indeed a metric, see [13]). The rank distance of a rank-metriccode C is defined in the usual way as d R ( C ) := min { rk( A − B ) | A, B ∈ C , A = B } . If C consistsof single matrix, we define d R ( C ) := min { k, m } . It is well known (see [6, Thms. 5.4, 6.3] and [13,p. 2]) that if m ≥ k and C is a rank-metric code in F k × mq with rank distance d , then |C| ≤ q m ( k − d +1) . (2.2)Moreover, there exist rank-metric codes of distance d and size q m ( k − d +1) , and such codes caneven be constructed as linear subspaces of F k × mq . They are called MRD codes . The best knownclass of linear MRD codes are the Gabidulin codes, derived by Gabidulin in [13]. Just recently,other constructions of MRD codes were found by de la Cruz et al. [5] and Hernandez/Sison [20].However, as opposed to Gabidulin codes so far no decoding algorithm is known for the lattercodes.If
C ⊆ F k × m is an MRD code, then the subspace code b C = { im( I k | A ) | A ∈ C} is called a lifted MRD code . Here the notation im( M ) stands for the row space of the matrix M and I k denotes the k × k -identity matrix. If d R ( C ) = d , then d S ( b C ) = 2 d , see [26, Prop. 4], andtherefore b C is a ( k + m, q m ( k − d +1) , k, d )-code.The following specific class of MRD codes will be used in the next section when studyingpartial spreads. Remark 2.1.
There is a simple construction of MRD codes in F k × m of rank distance k . Let W ∈ F k × m be any matrix of rank k and M ∈ GL m ( F ) be the companion matrix of a primitivepolynomial in F [ x ] of degree m . Define C := { W M l | l = 0 , . . . , q m − } ∪ { } ⊆ F k × m . Since F [ M ] ∼ = F q m and | M | = q m −
1, the code C is a linear rank-metric code of size q m andrank distance k . Hence C is an MRD code. In fact, it can be shown that C is a Gabidulin code.In order to present our linkage construction we need to work with matrix representations ofsubspaces. The following terminology will be convenient. Definition 2.2.
A set of matrices
M ⊆ F k × n is called SC-representing if rk( M ) = k forall M ∈ M and im( M ) = im( M ′ ) for all M = M ′ . The induced constant-dimension code { im( M ) | M ∈ M} is denoted by C ( M ).For example, the rank-metric code C in Remark 2.1 forms an SC-representing set of a cyclicorbit code in the sense of [24, 29, 17]. Any set of full row rank matrices in reduced row echelon3orm is an SC-representing set. In general, an SC-representing set is simply a subset of orbitrepresentatives of the action of GL k ( F ) on F k × n via left multiplication.The linkage construction in the following theorem links subspace codes with the aid of arank-metric code and results in a subspace code of longer length without compromising thedistance. It makes use of representing matrices. The theorem generalizes a former constructionin [17, Thm. 5.1]. Theorem 2.3.
For i = 1 , let M i ⊆ F k × n i be SC-representing sets of cardinality N i . Thus C i = C ( M i ) is an ( n i , N i , k ) -code. Let d S ( C i ) = d i . Furthermore, let C R ⊆ F k × n be a linear rank-metric code with rank distance d R ( C R ) = d R and cardinality |C R | = N R . Define the subspacecode C of length n := n + n as C := ˜ C ∪ ˜ C ∪ ˜ C , where ˜ C = { im( U | k × n ) | U ∈ M } , ˜ C = { im(0 k × n | U ) | U ∈ M } , ˜ C = { im( U | M ) | U ∈ M , M ∈ C R \{ }} . Then C is a ( n, N, k, d ) -code, where N = N + N N R and d = min { d , d , d R } . We write C = C ∗ C R C for the resulting linkage code and call C the code obtained by linking C and C through C R . The notation C = C ∗ C R C has to be used with care because the code C depends on therepresenting sets M and M and not only on the codes C and C . Thus the notation M ∗ C R M is actually the accurate one, but we prefer the former because the properties of the linkage thatwe are interested in are associated to the subspace codes C , C . The sets M , M are meretechnicalities in our context, and the notation C ∗ C R C will not lead to any confusion.However, in order to illustrate the dependence on the SC-representing sets for C and C we will present, after the proof, an example showing that different choices lead in general todifferent distance distributions of the linkage code. Proof.
The cardinality of C is clear because the three sets ˜ C i are pairwise disjoint. Furthermore,it is obvious that d S ( ˜ C i ) = d S ( C i ) for i = 1 ,
2. Moreover, it is clear that each subspace in ˜ C intersects trivially with each subspace in ˜ C and ˜ C . Thus d S ( W , W ) = 2 k for all W ∈ ˜ C and W ∈ ˜ C ∪ ˜ C .Next, let U = im( U | ∈ ˜ C and V = im( U ′ | M ) ∈ ˜ C . Thus U, U ′ ∈ M and M ∈ C R \{ } .Then rk( M ) ≥ d R by linearity of the code C R and so dim ker M ≤ k − d R , where ker M = { x ∈ F k | xM = 0 } . Let now v ∈ U ∩ V . Then v = x ( U |
0) = y ( U ′ | M ) for some x, y ∈ F k and thus y ∈ ker M and xU = yU ′ ∈ im( U ) ∩ im( U ′ ). Since the maps from U ∩ V mapping x ( U |
0) to xU and y ( U ′ | M ) to y are both injective, this yieldsdim( U ∩ V ) ≤ min { dim(im( U ) ∩ im( U ′ )) , dim(ker( M )) } . Now one concludes d S ( U , V ) ≥ min { d , d R } , as desired.Lastly, let U = im( U | M ) , V = im( U ′ | M ′ ) ∈ ˜ C , and let U 6 = V , thus U = U ′ or M = M ′ .Let v ∈ U ∩ V . Then v = x ( U | M ) = y ( U ′ | M ′ ) for some x, y ∈ F k . Hence xU = yU ′ ∈ im( U ) ∩ im( U ′ ) and xM = yM ′ ∈ im( M ) ∩ im( M ′ ) . U = U ′ , then dim( U ∩ V ) ≤ dim(im( U ) ∩ im( U ′ )) and d S ( U , V ) ≥ d S (im( U ) , im( U ′ )) ≥ d .If U = U ′ , then x = y because U has full row rank. Moreover, M − M ′ ∈ C R \{ } andthus dim(ker( M − M ′ )) ≤ k − d R . Now x ∈ ker( M − M ′ ) along with the injectivity of the map x ( U | M ) x from U ∩V to ker( M − M ′ ) shows that dim( U ∩V ) ≤ dim(ker( M − M ′ )) ≤ k − d R ,and thus d S ( U , V ) ≥ d R . This concludes the proof.The following example shows that different choices of the SC-representing sets for C and C lead to different distance distributions of the linkage code. Since we will not further study thedistance distribution of linkage codes, we continue to use the notation C ∗ C R C for the linkage. Example 2.4.
Let ( n , n , k, q ) = (4 , , ,
2) and M = M = n (cid:18) (cid:19) , (cid:18) (cid:19) o and M ′ = n (cid:18) (cid:19) , (cid:18) (cid:19) o and C R = n (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) o . Note that C ( M ) = C ( M ′ ). We find that in the linkage code M ∗ C R M (see the notation of theparagraph after Theorem 2.3) there exist 5 pairs of distinct subspaces with subspace distance 2and all other pairs have subspace distance 4, whereas in the linkage code M ′ ∗ C R M only 3pairs have subspace distance 2 and all others have subspace distance 4.The next two examples illustrate that we can easily construct very large codes of long lengthby suitable linkage. Example 2.5.
We aim at constructing a constant-dimension code over F of length 13, dimen-sion k = 3, and distance 4. Let n = 7 and n = 6. The largest known codes of dimension 3and length 7 (resp. 6) with distance 4 have cardinality 329 (resp. 77), see [3, Tables I and II] byBraun/Reichelt as well as [28] by Honold et al., where it is shown that 77 is actually the largestpossible size for length 6. We choose these codes for C and C , respectively, and an MRD code C R in F × with rank distance d R = 2 and thus cardinality N R = 2 − = 2 due to (2.2). Theresulting linkage code has therefore cardinality 77 + 2 ·
329 = 1 , , , , C and C have been found. The linkage code beats the codes that have been foundby Etzion/Silberstein with the aid of the multilevel construction [10] where the best such codehas cardinality 1 , , , , q = 2 , k = 3 , d S = 4and various lengths n . We make use of the best codes found in [3, Table II] for lengths 6 , . . . , C R ⊆ F × n with rank distance 2. Thus N R = 2 n thanks to (2.2). We alsoshow the largest size obtained via the modified multilevel (MML) construction [11, Thm. 17](which always beats the multilevel construction in [10]) as well as the largest size known so far.For n ≤
14 the latter has been found by computer search [3, Tables I and II], while for n = 15no such search has been conducted yet and linkage with n = 9 , n = 6 results in the largestknown code. Note that for all n shown in the table, every partition into n = n + n leads to5 linkage code that is larger than the MML construction. This is probably due to the fact thatthe MML construction leads to subspace codes that contain a lifted MRD code. This restrictionalso restricts the size of these codes. n n n N N Linkage MML Largest Known12 6 6 77 77 315 ,
469 305 ,
324 385 , , ,
897 1 , ,
296 1 , , , ,
661 1 , ,
296 1 , , , ,
665 4 , ,
184 5 , , ,
312 5 , ,
584 4 , ,
184 5 , , ,
312 77 5 , ,
029 4 , ,
184 5 , , ,
312 329 21 , ,
137 19 , ,
736 23 , , ,
312 21 , ,
656 19 , ,
736 23 , , ,
694 20 , ,
782 19 , ,
736 23 , , ,
694 77 23 , ,
701 19 , ,
736 23 , , n = 13 the largest known cardinality 1 , ,
245 isactually the optimum by the anticode bound, and the existence of a code with that size hasbeen established by Braun et al. in [2] via a 2-analogue of a Steiner triple system.
Example 2.6.
Let us consider Theorem 2.3 for the case where C is a lifted MRD code and C R is an MRD code. For C we may choose a lifted MRD code or an arbitrary subspace code. Letus consider the case where C is a lifted MRD code. Thus, let n = n + n , where n i ≥ k for i = 1 ,
2, and C i be the lifting of an MRD code in F k × ( n i − k ) with rank distance d . Then |C i | = q ( n i − k )( k − d +1) and d S ( C i ) = 2 d . Moreover, let C R ⊆ F k × n be an MRD code of rankdistance d . Thus, |C R | = q n ( k − d +1) . By Theorem 2.3 the linkage code C ∗ C R C has subspacedistance 2 d and cardinality q ( n − k )( k − d +1) + q ( n − k )( k − d +1) . Note that the second term is thecardinality of a lifted MRD code in F n with subspace distance 2 d . Thus, linkage always resultsin a better code than lifting. In fact, with our choice the code ˜ C ∪ ˜ C in Theorem 2.3 is alifted MRD code and thus the cardinality of the linkage code is clearly larger than that of alifted MRD code. Furthermore we observe that only the first term depends on the partition n = n + n , and that the cardinality of C ∗ C R C is largest when n is largest. The followingtable shows the size of the linkage construction for q = 2 , k = 3 , d S = 4 and various lengths. Ineach case, C is a lifted MRD code of distance 4 and C R is an MRD code of rank distance 2. Eachgiven length n is split into n = n + n such that n is maximal subject to n i ≥ k for i = 1 , largest ” we present the cardinality of the linkage code where weuse the largest known subspace code for C . In the column “Link MRD ” we use a lifted MRDcode for C . For comparison we also show the size, 2 n − , of a lifted MRD code of length n .It should be noted that the linkage codes are smaller than the codes obtained from the MMLconstruction; see the previous table. This is explained by the fact that the MML constructionis a careful design to create additional subspaces without compromising the distance. It may beregarded as a replacement of the code ˜ C in Theorem 2.3 by a larger set, where the zero blockmatrix is replaced by suitable matrices. In the column “Extended Lifted MRD” we illustratethat our codes are slightly smaller than those constructed in [27] by Skachek , which are also One can show that in our situation the optimal design choice for the parameter h n = h ℓ + m in [27] is h n = 0,and therefore the cardinality of the resulting code is as in Section IV.C of [27]. n n n Link largest
Link
MRD
Lifted MRD Extended Lifted MRD12 6 6 262 ,
221 262 ,
208 262 ,
144 266 , , ,
905 1 , ,
832 1 , ,
576 1 , , , ,
616 4 , ,
328 4 , ,
304 4 , , With the aid of Theorem 2.3 we can construct optimal partial spreads for certain cases. Recallthat a partial spread in F n is a collection of subspaces that pairwise intersect trivially. If allsubspaces have the same dimension, say k , then this is simply a constant-dimension code ofdimension k and distance 2 k , and we call the code a partial k -spread . It is well known that if k divides n , then an optimal partial k -spread (i.e., a partial k -spread of maximum cardinality) is a k - spread , i.e., the spaces intersect trivially and cover the entire F n . In this case a simple countingargument shows that the cardinality is ( q n − / ( q k − F = F q . Several constructionsof k -spreads are known. For later reference we provide the following two options. Remark 3.1.
Let k divide n .(a) [24, Thm. 11] The orbit of the subfield F q k in the field F q n under the natural action of thegroup F ∗ q n is a k -spread in F q n .(b) [18, Thm. 6, Rem. 8] Let m = n/k and M ∈ GL k ( F ) be the companion matrix of a primitivepolynomial of degree k . Then the set { im( A , . . . , A m ) | A i ∈ F q [ M ] , not all A i are zero } isa k -spread in F n . It is called a Desarguesian spread .If k does not divide n , then the maximum size of a partial k -spread in F nq is in general notknown – with one exception which will be considered below in further detail. The followingresult can be found in [1, Thms. 4.1, 4.2] and [7, Thm. 7]; see also [8, Thm. 3]. Theorem 3.2.
Let n (mod k ) = c , where c ∈ { , . . . , k − } . Denote the largest possiblecardinality of a partial k -spread in F nq by µ ( n, k ) . Then µ ( n, k ) ≥ q n − q c q k − − q c + 1 with equality if c ∈ { , } . Furthermore, if c > then µ ( n, k ) ≤ q n − q c q k − − ⌊ θ ⌋ − , where θ = p q k ( q k − q c ) − (2 q k − q c + 1)2 . k -spreads and cardinality m ( n, k ) := q n − q c q k − − q c + 1 , where n (mod k ) = c, (3.1)were presented in [1, Thms. 4.2] as well as [12, Thm. 11] and [19, Thm. 13]. Hence for c ∈ { , } these partial spreads have maximum possible cardinality. The latter two constructions arespecial cases of our linkage and will be described in our terminology in the following examples. Example 3.3.
We describe the construction of partial k -spreads by Etzion/Vardy [12, Thm. 11].Let n ≥ k and write n = lk + c , where c ∈ { , . . . , k − } . Set n = k ( l −
1) and n = k + c .Consider the F q -vector space F q n × F q n . In F q n choose the k -spread C given by the orbitof the subfield F q k under the action of the cyclic group F ∗ q n ; see Remark 3.1(a). Furthermore,let β be a primitive element of F q n and set C = (cid:8) span F { , β, . . . , β k − } (cid:9) . Note that this istrivially a partial k -spread in F q n of maximal possible cardinality because k > n /
2. Considerthe coordinate map w.r.t. the basis { , β, . . . , β n − } of F q n , that is, ϕ : F q n −→ F n q , n − X i =0 f i β i ( f , . . . , f n − ) . Using the identification ϕ , the code C simply translates into C = { im( I k | k × c ) } . Finally,in F k × n choose the rank-metric code C R = { ( I k | k × c ) M j | j = 0 , . . . , q n − } ∪ { } , where M is the companion matrix of the minimal polynomial of β over F q . Note that the matrix( I k | k × c ) M j consist exactly of the rows ϕ ( β j ) , . . . , ϕ ( β j + k − ). By Remark 2.1, C R is an MRDcode of rank distance k . Identifying F q n with F n q , the linkage code C ∗ C R C is exactly thepartial k -spread constructed in [12, Thm. 11]. It has cardinality 1 + q n ( q n − / ( q k − m ( n, k ). Example 3.4.
Essentially the same construction as in Example 3.3 but with different specifica-tions of the constituent codes is used by Gorla/Ravagnani in [19, Thm. 13]. Again, let n = lk + c and set n = k ( l −
1) and n = k + c . Then the code constructed in [19, Thm. 13] is the linkage C ∗ C R C with the following specifications: C is a Desarguesian k -spread in F n (see Remark 3.1)while C is the subspace code { im(0 k × c | I k ) } and C R is an MRD code in F k × n as in Remark 2.1with matrix W = (0 | I k ), thus the nonzero matrices in C R are the last k rows of the matrices M l .In addition to the construction, the authors also present a decoding algorithm for their partialspreads by making explicit use of the structure of the Desarguesian spread; see [19, Sec. 5]. Incontrast, no decoding algorithm is given in [12] for the partial spreads constructed therein.Instead of partitioning n into n + n with the specific choice of n = k + c as in the previousexamples, we may use any other splitting n = lk + n . This will be summarized in the nextresult where we also address maximality of the partial spread. A partial k -spread in F n is called maximal if it is maximal with respect to inclusion, that is, it is not properly contained in anyother partial k -spread. The following result shows, among other things, that linking a k -spreadand a maximal partial k -spread through an MRD code leads to a maximal partial k -spread.8 heorem 3.5. Let n = lk + n , where l ≥ and n ≥ k . Let C be a k -spread in F lk and C be a partial k -spread in F n . Furthermore, let C R be a linear MRD code in F k × n with rankdistance k and thus cardinality q n . Finally, let C = C ∗ C R C be the resulting linkage code as inTheorem 2.3.(a) If |C | = m ( n , k ) , then |C| = m ( n, k ) .(b) If C is a maximal partial k -spread then so is C .Proof. (a) Theorem 2.3 tells us that |C| = q n m ( lk, k ) + m ( n , k ). But this is easily seen tobe m ( n, k ).(b) Let ( W | W ) ∈ F k × ( n + n ) be of rank k and set W := im( W | W ). We have to showthat there exists a subspace V ∈ C such that
W ∩ V 6 = { } , for then C is a maximal partial k -spread. Assume first that W = 0. Then there exists ( x, y ) ∈ W such that x = 0. Since C isa spread of F lk , the vector x is in exactly one subspace of C , say im( U ). Let x = αU , where α ∈ F k \{ } . Since C R is a linear rank-metric code with rank distance k , we have αM = αM ′ forall distinct M, M ′ ∈ C R . This shows that the set { αM | M ∈ C R } has cardinality |C R | = q n andtherefore equals F n . As a consequence, y = αM for some M ∈ C R . Hence ( x, y ) = α ( U | M )and W ∩ V 6 = { } for the subspace V = im( U | M ) ∈ C . Assume now W = im(0 | W ). Thenrk( W ) = k and the maximality of C implies W ∩V 6 = { } for some subspace V = im(0 | U ) ∈ C .All of this shows that C is a maximal partial k -spread in F n .The following is an immediate consequence of Examples 3.3 and 3.4 because in both casesthe chosen code C is trivially a maximal partial k -spread. Corollary 3.6.
The partial spreads constructed in [12, Thm. 11] and in [19, Thm. 13] aremaximal.
Maximality of the partial spreads in [19] has also been established by Gorla/Ravagnaniin [19, Prop. 20].In Theorem 3.2 we have seen that the maximum cardinality of a partial k -spread in F n isknown whenever n (mod k ) ∈ { , } . There is one more case where the cardinality is known,and that is if q = 2 and k = 3. The following result covers all remainders of n modulo 3. Theorem 3.7 ([8, Thm. 5]) . Let k = 3 and n ≥ . Let n (mod 3) = c . Then the maximumcardinality of a partial 3-spread in F n is n − c − c. We call a partial -spread with this cardinality a maximum partial -spread. Note that for c ∈ { , } the result is simply a special case of Theorem 3.5, whereas for c = 2the cardinality m ( n, k ) in (3.1) is one below the maximum. As a consequence, the constructionsin [12, Thm. 11] and [19, Thm. 13] are just one subspace short of being maximum.The proof of Theorem 3.7 is based on a concrete example for n = 8 and an extensionconstruction for n >
8. It makes use of a result in [4, Lem. 4], which establishes a partitionof F nq into subspaces of two distinct dimensions. Below we will provide an alternative extension,where we will also make use of the maximum 3-spread in F .9 xample 3.8 ([8, Ex. 2]) . There exists a partial 3-spread in F with cardinality 34. Hence thespread is maximum. It has been found by computer search and is explicitly given in [8].Now we can provide a simple construction of maximum partial 3-spreads in F n for any n ≥
10. Note that, due to the previous example and earlier discussions, a maximum partial 3-spreadin F n is available for the values n ∈ { , , , } . Corollary 3.9.
Let n ≥ and write n = 3 l + n for some l ≥ and n ∈ { , , } . Choose a3-spread C in F l and a maximum partial 3-spread C in F n . Finally, let C R be an MRD codewith rank distance in F × n . Then C ∗ C R C is a maximum partial -spread in F n .Proof. The resulting code is certainly a partial spread. Let n (mod 3) = c , thus n (mod 3) = c .By Theorems 3.7 and 2.3 the cardinality of C ∗ C R C is given by2 n l −
17 + 2 n − c − c = 2 n − c − c, and this is the maximum value due to Theorem 3.7. In this section we turn to decodability of the linkage codes from Theorem 2.3. Of course, oneaims at reducing decoding of C ∗ C R C to decoding of the smaller codes C , C , C R . We will showfirst that this strategy does not work if one utilizes the rank metric for the code C R . Insteadone has to employ the subspace distance for all codes involved. We will show that if we usesuitable MRD codes and liftings thereof, then decoding can indeed be reduced to decoding ofthe constituent codes. Since lifted Gabidulin codes can be efficiently decoded, as proven by Silvaet al. [26], this leads to an efficient decoding algorithm for a particular instance of linkage codes.The following terminology is standard. Definition 4.1.
Let C be a subspace code in F n with subspace distance d . A subspace V ⊆ F n is called decodable w.r.t. C if there exists a subspace U ∈ C such that d S ( U , V ) ≤ d − .Since the subspace distance is a metric on the set of all subspaces in F n (see [22, Lem. 1]),a decodable subspace has a unique closest codeword in C .The following simple fact will be useful later. Remark 4.2.
Let C be a constant-dimension code in F n with dimension k and subspace dis-tance d . Suppose V ⊆ F n is a decodable K -dimensional subspace and U ∈ C is the unique closestcodeword. Then dim(
U ∩ V ) > K/ U + V ) < k + K/ . The first inequality follows from d S ( U , V ) ≤ ( d − / < k , see (2.1), which then reads as K + k − U ∩ V ) < k . The second inequality is obtained by using dim( U + V ) = k + K − dim( U ∩ V ).We start with an example illustrating that the rank distance of the code C R cannot be usedin the natural way for decoding the linkage code C ∗ C R C .Throughout this section we call a matrix V ∈ F K × n a matrix representation of the sub-space V ⊆ F n if V = im( M ) := { xM | x ∈ F K } . We explicitly allow dim( V ) < K .10 xample 4.3. Let n = n = 8 , k = 4 and F = F . Define C = C = { im( I | I ) , im( I | } . where I and 0 are the identity and the zero matrix in F × , respectively. Moreover, let C R = (cid:8) × , ( I | , ( M | , ( I + M | (cid:9) , where M = . Then d S ( C ) = d S ( C ) = 8 and d R := d R ( C R ) = 4. Thus the linkage code C = C ∗ C R C haslength 16, subspace distance d = 8 and cardinality N = 10. Consider the received word V = im (cid:18) (cid:19) . Then d S ( V , U ) = 2 for the codeword U = im( U | U ) ∈ C , where U = ( I | I ) and U = ( M | . In particular, V is decodable. Note that V ⊆ U (i.e., only erasures occurred during the trans-mission). One can check straightforwardly that there exists no matrix representation ( V | V )of V in F × such that d R ( V , U ) ≤ d R − = . Even worse, for all matrix representations( V | V ) ∈ F × of V for which the matrix V has a unique closest matrix in C R with re-spect to the rank distance, this unique closest matrix is the zero matrix and therefore doesnot lead to the correct decoding U . The fact that the “obvious decoding” does not work maybe explained by the fact that the subspaces represented by the nonzero matrices in C R , i.e.,im( I | , im( M | , im( I + M | C R to be oflittle help with decoding.The example can be generalized. We introduce the following notation. Define the projections π : F n + n −→ F n , ( a, b ) a and π : F n + n −→ F n , ( a, b ) b. (4.1)For a subspace A ⊆ F n we define A i = π i ( A ). Thus, if A = im( A | A ), then A i = im( A i ) for i = 1 , Proposition 4.4.
Let C be as in Theorem 2.3 and assume d ≥ d R + 2 . Then there exists asubspace U = im( U | U ) ∈ C and a received word V ⊆ F n such that(a) d S ( U , V ) ≤ d − (that is, V is decodable),(b) V ⊆ U (hence only erasures occurred during transmission),(c) for any V ∈ F k × n such that im( V ) = π ( V ) we have rk( V − U ) > d R − . In other words, it is not possible to decode V by making use of the rank metric for the code C R . roof. Let d ≥ d R + 2. Since d ≤ d ≤ k , we have r := k − ⌈ d R ⌉ ≥ , and thus r ≥ U and V as stated in the proposition. First choosea subspace U = im( U | U ) ∈ C with ( U | U ) ∈ F k × n such that rk( U ) ≥ k − r = ⌈ d R ⌉ . Bydefinition of C such an element does indeed exist. Next, there exists a matrix X ∈ GL k ( F ) suchthat X ( U | U ) = (cid:18) U | U U | U (cid:19) and where U ∈ F ( k − r ) × n has rank k − r and and im( U ) ∩ im( U ) = { } . Put V = im( U | U ) . Then dim( V ) = r because the rows of ( U | U ) are linearly independent. Moreover, V ⊆ U andtherefore d S ( V , U ) = k + r − r = (cid:6) d R (cid:7) ≤ d − . This establishes (a) and (b). For (c) consider now all matrices V in F k × n whose row space is π ( V ). These matrices can be written as V := X − (cid:18) M M (cid:19) U , where (cid:18) M M (cid:19) ∈ F k × r is any matrix of rank r. The matrix X − does not change the row space, and we include it only to simplify the nextstep. Indeed, for each such matrix V we haverk( V − U ) = rk( XV − XU ) = rk (cid:18) M U − U M U − U (cid:19) ≥ rk( M U − U ) . The rightmost matrix has full row rank, k − r . Indeed, suppose u ( M U − U ) = 0. Then uM U = uU ∈ im( U ) ∩ im( U ). Since this intersection is trivial, we obtain uU = 0,which in turn implies u = 0. All of this shows that rk( U − V ) ≥ ⌈ d R ⌉ > d R − for all matrixrepresentations of π ( V ).The last observation suggests to modify the linkage construction by simply replacing therank-metric code C R by matrix representations of a subspace code. This results in a code that isdecodable if its constituent codes are decodable. But since these codes are considerably smallerthan the original linkage codes, we will not follow that path.Instead, we will show now how to decode linkage codes C ∗ C R C for the case where C and C R are (lifted) MRD codes. We need the following lemma. Lemma 4.5.
Let C be the code from Theorem 2.3 and V ⊆ F n be a decodable K -dimensionalsubspace. Let U ∈ C be the closest codeword, thus d S ( U , V ) ≤ d − . Then U = 0 ⇐⇒ dim V ≤ K/ . Proof. “ ⇐ =” Assume U = 0. Then rk( U ) = k by definition of C . Thus dim( U ) = k = dim( U ),and the map π | U is injective, where π is the projection from (4.1). We computedim( U ∩ V ) = dim( π ( U ∩ V )) ≤ dim( π ( U ) ∩ π ( V )) ≤ dim( π ( V )) = dim V ≤ K/ , U = 0.“= ⇒ ” Let U = 0, thus U = im(0 | U ) and rk( U ) = k by definition of C . Write V = im( V | V )for some ( V | V ) ∈ F K × n . With the aid of Remark 4.2 we obtainrk( V ) + k ≤ rk (cid:18) U V V (cid:19) = dim( U + V ) < k + K . Hence dim V = rk( V ) < K/ ⇐ =” of the last lemma is in general not true for U and V because the matrices in C R may not have rank k .Now we are in the position to discuss decoding of the linkage codes from Theorem 2.3. Weconsider the following situation which is a special case of the general linkage construction. Itmay also be regarded as an extension of the codes considered in Example 2.6. Theorem 4.6.
For i = 1 , let n i ≥ k and let M i ⊆ F k × n i be a linear MRD code withrank distance d , thus |M i | = q n i ( k − d +1) . Moreover, let M ⊆ F k × n and M ⊆ F k × n beSC-representing sets of constant-dimension codes with subspace distance d . Consider the code C = C ′ ∪ C ′′ ∪ C ′′′ , where C ′ = { im( I k | M | M ) | M ∈ M , M ∈ M } , C ′′ = { im(0 k × k | M | k × n ) | M ∈ M } , C ′′′ = { im(0 k × k | k × n | M ) | M ∈ M } . Then C is an ( n, N, k, d ) -code, where n = k + n + n and N = q ( n + n )( k − d +1) + |M | + |M | .Proof. This is a simple application of Theorem 2.3 : C = ˜ C ∗ ˜ C R ˜ C with the codes ˜ C = { im( I k ) } ,˜ C = { im( M | | M ∈ M } ∪ { im(0 | M ) | M ∈ M } , and ˜ C R = { ( M | M ) | M i ∈ M i } .Note that if M represents a lifted MRD code, then C ′ ∪ C ′′′ = C ∗ M C , where C = { im( I | M ) | M ∈ M } , C = { im( M ) | M ∈ M } . Hence the code is of the form as discussedin Example 2.6, and in Theorem 4.6 we improve upon the codes in that example by the sizeof M . In the column “Link MRD ” of the table in Example 2.6 we listed, for a specific choiceof parameters, the largest codes of the form C ′ ∪ C ′′′ above. As we saw already, the largest sizeis attained when n is largest subject to n = k + n + n with n + k ≥ k and n ≥ k (now n + k takes the role of n from that example), thus for n = k and n = n − k . But in thatcase |M | = 1 and thus we improve upon the codes in that table by exactly one subspace.We now turn to decoding of the codes in Theorem 4.6. As we show next this can be reducedto decoding of the constituent codes. Theorem 4.7.
Consider the setting of Theorem 4.6. To ease notation we set n := k . For i = 1 , define the lifted MRD codes C i := { im( I k | M ) | M ∈ M i } . Furthermore, set C := C ( M ) and C := C ( M ) . Then, if C , . . . , C are decodable then so is the linkage code C . Moreprecisely, let V = im( V | V | V ) ⊆ F n , V i ∈ F K × n i , be a K -dimensional received word such that d S ( V , C ) ≤ (2 d − / . Then exactly one of the following situations occurs.(a) rk( V | V ) < K/ . In this case the unique closest codeword in C is in C ′′′ and given by U =im(0 | | M ) , where M ∈ M is the unique matrix such that d S (im( M ) , im( V )) ≤ (2 d − / . b) rk( V | V ) < K/ . In this case the unique closest codeword in C is in C ′′ and given by U =im(0 | M | , where M ∈ M is the unique matrix such that d S (im( M ) , im( V )) ≤ (2 d − / .(c) rk( V ) > K/ . In this case the unique closest codeword in C is in C ′ and given by U =im( I | M | M ) , where M i ∈ M i are the unique matrices such that d S (cid:0) im( I | M i ) , im( V | V i ) (cid:1) ≤ (2 d − / for i = 1 , .Proof. First of all, the uniqueness of the matrices M and M i in (a) – (c) is guaranteed since thesubspace codes C ( M ) , C ( M ) and the lifted MRD codes C i all have subspace distance 2 d .Let us denote by U = im( U | U | U ) the unique codeword in C closest to V . We will usethe notation U i = im( U i ) and V i = im( V i ) for i = 0 , , V ) > K/ V ) ≤ K/ r := rk( V | V ) < K/ V | V ) ≥ K/
2. After suitable row operations we may assume( V | V | V ) = (cid:18) V V V V (cid:19) , where the first block row has r rows. Then rk( V | V | V ) = K implies that rk( V ) > K/
2. Thisimplies rk( V | V ) > K/
2, as desired. Using symmetry, all of this shows that if rk( V ) ≤ K/ V ) ≤ K/ C = ˜ C ∗ ˜ C R ˜ C with ˜ C , ˜ C , ˜ C R as in the proof of Theorem 4.6. Therefore, Lemma 4.5 along with rk( V ) ≤ K/ U = 0.Thus, U ∈ C ′′ ∪ C ′′′ . Let us assume U ∈ C ′′ , say U = im(0 | M | M ) = k wederive rk( V | V ) + k ≤ rk (cid:18) V V V M (cid:19) = dim( U + V ) < k + K/ , (4.2)where the last inequality is due to Remark 4.2. As a consequence, rk( V | V ) < K/
2. Simi-larly,
U ∈ C ′′′ implies rk( V | V ) < K/ V | V ) < K/
2. Suppose the closest codeword U is in C ′′ , say U = im(0 | M | V | V ) < K/
2. But this means that also case (b) occurs, a contradic-tion. Hence
U ∈ C ′ ∪ C ′′′ . But this code is a linkage code. Indeed, C ′ ∪ C ′′′ = C ∗ C R C , where C = { im( I k | M ) | M ∈ M } , C = C ( M ), and C R = M . Hence Lemma 4.5implies ( U | U ) = (0 | U = im(0 | | M ) ∈ C ′′′ with M ∈ M . In particular, rk( M ) = k .Consider the projection π of F k + n + n onto F n . Then ( π ) | U is injective and thus dim( U ∩ V ) ≤ dim( U ∩ V ). This implies d S ( V , U ) ≤ K + k − V ∩ U ) ≤ d S ( V , U ) ≤ (2 d − / V to its closest codeword in C ( M ) results in U . Using its unique matrixrepresentation M ∈ M , i.e., U = im( M ), we arrive at the correct decoding U = im(0 | | M )of the received space V .(b) The case rk( V | V ) < K/ V ) > K/
2. Then Lemma 4.5 applied to ˜ C ∗ C R ˜ C (see proof of Theorem 4.6)implies ( U ) = 0. Thus U ∈ C ′ . In particular, we may assume U = I k . For i = 1 , V ′ i :=14m( V | V i ) and U ′ i := im( I k | U i ). Then U ′ i ∈ C i for i = 1 ,
2, where C i = { im( I k | M ) | M ∈ M i } is the lifting of the MRD code M i for i = 1 ,
2. Consider the projections ψ i : F k + n + n −→ F k + n i , ( a , a , a ) ( a , a i )Then ( ψ i ) | U is injective and thus dim( U ∩ V ) ≤ dim( U ′ i ∩ V ′ i ). As in (a) this implies d S ( U ′ i , V ′ i ) ≤ d S ( U , V ) ≤ (2 d − / i = 1 ,
2. Hence V ′ i can be uniquely decoded w.r.t. C i and the closestcodeword is given by U ′ i . Using the unique matrix representations ( I | U i ) , U i ∈ M i , of thespaces U i , we arrive at the correct decoding of V .We summarize the result in the following algorithm. Algorithm 4.8:
Decoding algorithm for the codes in Theorem 4.6
Data : a decodable K -dimensional subspace V = im( V | V | V ) with ( V | V | V ) ∈ F K × nq Result : the unique
U ∈ C ′ ∪ C ′′ ∪ C ′′′ such that d s ( V , U ) ≤ d − . if rk( V | V ) < K then decode im( V ) in C ( M ) to im( U ); return U = im(0 | | U ) . elseif rk( V | V ) < K then decode im( V ) in C ( M ) to im( U ); return U = im(0 | U | . else decode im( V | V ) in C to im( I k | U );decode im( V | V ) in C to im( I k | U ); return U = im( I k | U | U ) . One should observe that in the last case of the algorithm, the two decoding steps can beperformed in parallel. A similar, but not identical, form of parallelizing decoding is also usedfor the spread codes in [19]; recall Example 3.4 for the relation to our linkage codes.Clearly, the construction in Theorem 4.6 and its decoding can easily be generalized to morethan 3 blocks.
Remark 4.9.
A very efficient decoding is obtained when we use Gabidulin codes for M and M and lifted Gabidulin codes for M and M (thus n i > k ). In this case, all codes relevant fordecoding in the previous proof are lifted Gabidulin codes, and the decoding algorithm derivedby Silva et al. [26] may be employed. If n i >> k , then even better efficiency is obtained by usingdirect products of Gabidulin codes as the MRD codes and the lifting of such a code for M and M ; see [26, Sec. VI.E]. In fact, our code C ′ in Theorem 4.6 (or rather its generalizationto more than 3 blocks) is of the form proposed in [26] and with the above we have shown howto enlarge the code without compromising its properties. In this sense, our results put theconsiderations in [26, Sec. VI.E] in a broader context.15 eferences [1] A. Beutelspacher. Partial spreads in finite projective spaces and partial designs. Math.Zeit. , 145:211–229, 1975.[2] M. Braun, T. Etzion, P. ¨Osterg˚ard, A. Vardy, and A. Wassermann. Existence of q -analogsof Steiner systems. Preprint 2013. arXiv: 1304.1462.[3] M. Braun and J. Reichelt. q -analogs of packing designs. J. Comb. Designs , 22:306–321,2014.[4] T. Bu. Partitions of a vector space.
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