Improving the Linkage Construction with Echelon-Ferrers for Constant-Dimension Codes
aa r X i v : . [ c s . I T ] J u l Improving the Linkage Construction withEchelon-Ferrers for Constant-Dimension Codes
Xianmang He, Yindong Chen, Zusheng Zhang
Abstract —Echelon-Ferrers is an important method to improvelower bounds for constant-dimension codes, which can be appliedon various parameters. Fagang Li [12] combined the linkageconstruction and echelon-Ferrers to obtain some new lowerbounds of constant-dimension codes. In this letter, we generalizethis linkage construction to obtain new lower bounds.keywords: Subspace Coding, Linkage Construction, Echelon-Ferrers Construction, Constant-Dimension Codes
I. I
NTRODUCTION
Let F q be the finite field with q elements. The set of all k -dimensional subspaces of an F q -vector space V will bedenoted by G q ( k, n ) . In general, the projective space of order n over the finite field F q , denoted by P q ( n ) , is the set of allsubspaces of the vector space F nq . All these subspaces form ametric space with respect to the subspace distance , which isdefined as d S ( U, W ) := dim( U + W ) − dim( U ∩ W )= 2 · dim( U + W ) − dim( U ) − dim( W ) , where U and W are subspaces of F nq .A set C of subspaces of V is called a subspace code . The minimum distance of C is given by d = min { d S ( U, W ) | U, W ∈ C , U = W } . If the dimension of the codewords is fixed as k , we use the notation ( n, C , d, k ) q and call C a constant dimension code (CDC for short). The maximalpossible size of an ( n, M, d, k ) q CDC is often denoted by A q ( n, d, k ) .Subspace coding was first proposed by R. K¨oetter and F. R.Kschischang in [10] to error control in random network cod-ing. The main problem in subspace coding is to determine themaximal possible size A q ( n, d, k ) , which makes the subspacedistance satisfies: for any two different subspaces U and W ,we have d ( U, W ) = 2 k − U ∩ W ) ≥ d .A plethora of results on the construction of CDCs areinvented in the literatures. The lower and upper bounds on A q ( n, d, k ) have been in-depth studied in the last decade, see[7]. The report [7] describes an on-line database, which werefer to the online tables http://subspacecodes.uni-bayreuth.de.These tables gather up-to-date information about the currentlower and upper bounds for subspace codes. Lifted maximumrank-distance (MRD for short) codes are one type of build-ing blocks of the echelon-Ferrers construction [3]. The ideaof multilevel construction is widely used, including parallelconstruction [14], coset construction [8], pending dot [13], etc.The most powerful construction is the linkage construction [4]and its improved construction [9]. The linkage construction isimproved by these new works [2], [5], [6]. Recently, Fagang Li combined the two methods of linkageconstruction and echelon-Ferrers to obtain some new lowerbounds of CDCs. In this letter we generalize this construction.Based on this, we further improved the construction by agreedy algorithm.II. THE C OMBING M ETHOD IN [12]Let X be a k -dimensional subspace of G q ( k, n ) . We canrepresent X by the matrix in reduced row echelon form E ( X ) ,whose k rows form a basis for X . The identifying vector of X , denoted by v ( X ) , is a binary vector of length n and weight k , where the k ones of v ( X ) are exactly the pivots of E ( X ) .The zeroes are removed from each row of E ( X ) , which lieon the left of the pivots. Then we delete the columns whichexactly having the pivots. After that, all the remaining entriesare shifted to the right. We finally get the Ferrers tableaux formof a subspace X , denoted by F ( X ) . The Ferrers diagram of X can be obtained from F ( X ) by replacing the entries of F ( X ) with dots.Let F m × ℓq be an m × ℓ matrices space over the field F q .For any two distinct matrices A, B ∈ F m × ℓq , the rank-metric isdefined as d R ( A, B ) := rank( A − B ) . A subset of F m × ℓq withthe rank-metric is called a rank-metric code. If a rank-metriccode is a linear subspace of F m × ℓq , we can call it a linear rank-metric code. It is clear that the rank-distance of a rank-metriccode C can be defined as d R ( C ) := min { d R ( A, B ) :
A, B ∈C , A = B } . It is well-known that the number of codewordsin C is upper bounded by q max { m,ℓ }· (min { m,ℓ }− d +1) . A codeattaining this bound is called a maximum rank-distance (MRD)code.Let F be a Ferrers diagram with ℓ dots in the top row and m dots in the rightmost column. If for any codeword M of C F , all entries of M not in F are zeroes, a linear rank-metriccode C F of F m × ℓq is called a Ferrers diagram rank-metric(FDRM) code. An FDRM code C F is denoted an [ F, d, δ ] FDRM code, if rank( A ) ≥ d for any nonzero codeword A ,and dim( C F ) = δ .The following theorem determines an upper bound on thesize of dim( C F ) . Theorem 1. (see [3]) Let F be the Ferrers diagram of ℓ inthe top row and m dots in the rightmost column. Let C F ⊆ F m × ℓq be the corresponding FDRM code fulfilling ∀ A, B ∈C F , rank( A − B ) ≥ δ . Then |C F | ≤ q min i { w i } , where w i isthe number of dots in F , which are neither contained in therightmost δ − − i columns nor contained in the first i rowsfor ≤ i ≤ δ − . Furthermore, the authors of [3] proved that the upper boundcan be attained when d = 2 , , and then conjectured that theupper bound is also tight for other cases.For simplify, for any given matrix M ∈ F k × ℓq over F q , therow space of M is denoted by im( M ) .We recall some basic notations of linkage in [4]. A set U ⊂ F k × nq with the size k × n matrices over F q is called an SC-representation set if rank( U ) = k for all U ∈ U and im( U ) =im( U ) for all U = U in U . Proposition 1. (see [4]). Let U be an SC-representation setof a ( n , N , d , k ) q constant dimension subspace code and M ⊂ F k × n q be a linear rank-metric code with distance d and N elements. Consider the set of k dimension subspaces in F n + n q defined by C = { im( U | M ) : U ∈ U , M ∈ M} . Thisis an ( n + n , N N , min { d , d } , k ) q constant dimensioncode. Here ( U | M ) is a k × ( n + n ) matrix concatenatedfrom U and Q . We quote the following theorem (Theorem 3.1 in [12]) tobriefly describe the construction method, see the paper [12]for details.
Theorem 2. [12] Let n > k, n > k, k ≥ d . For i = 1 , , let U i ⊆ F k × n i q be SC-representing sets with cardinality N i , and d S ( U i ) = d . Assume that C R ⊆ F k × n q is a linear rank-metriccode with |C R | = N R and d R ( C R ) = d .Let the identifying vectors v j with length n := n + n andweight k satisfy the following properties for j = 1 , , · · · .(a) For each identifying vector v j , the count of ’s in thefirst n positions and the last n positions are both greaterthan or equal to d .(b) For any two distinct identifying vectors v j and v j , theHamming distance H ( v j , v j ) ≥ d. Let C F j ⊆ F k × ( n − k ) q be an FDRM code and d R ( C F j ) = d ,where F j is a Ferrers diagram corresponding to the identifyingvector v j .Denote by C the subspace code of length n = n + n as C = C ∪ C ∪ C , where C = { im( U | M ) | U ∈ U , M ∈ C R } ; C = { im( k × n | U ) | U ∈ U } ; C = ∪ j C F j , C F j is the lifted FDRM code of C F j .Thus, C is an ( n, N, d, k ) q CDC with N = N + N · N R + P j | C F j | . This construction modifies the echelon-Ferrers construction,which replaces the lifted MRD code im( I k , C R ) with thelinkage construction im( U, C R ) , where C R and C R arelinear rank-metric codes with size k × ( n − k ) and k × n ,respectively.III. C ONSTRUCTION A ND A LGORITHM
In this section, we’re going to give the details of ourconstruction and an algorithm with greedy strategy.
A. General Construction
We now generalize the multilevel construction and linkageconstruction.
Theorem 3.
Let n ≥ k, n ≥ k, U ⊆ F k × n q beSC-representing sets with cardinality N , and d S ( U ) = d .Assume that C R ⊆ F k × n q is a linear rank-metric code with |C R | = N R and d R ( C R ) = d . Let the identifying vector v j with length n := n + n and weight k satisfy the followingproperties for j = 1 , , · · · .(a) For any identifying vector v j , the count of ’s in the last n positions is at least d .(b) For any two distinct identifying vectors v j and v j , theHamming distance H ( v j , v j ) ≥ d. Let C F j ⊆ F k × ( n − k ) q be an FDRM code and d R ( C F j ) = d ,where F j is a Ferrers diagram corresponding to the identifyingvector v j .Denote by C the subspace code of length n = n + n as C = C ∪ C , where C = { im( U | M ) | U ∈ U , M ∈ C R } ; C = ∪ j C F j , C F j is the lifted FDRM code of C F j .Thus, C is an ( n, N, d, k ) q CDC with N = N · N R + P j | C F j | .Proof. We note that C is an ( n + n , N N R , d, k ) q constantdimension code, and C is the set of the lifted FDRM code.Consider that the pivots of C and C are pairwise disjoint,therefore, the cardinality of the code is N · N R + P j | C F j | .According to the definition, C F j is a CDC with d S ( C F j ) ≥ d . Hence, it is sufficient to prove that for any w ∈ C , w ∈ C , d S ( w , w ) ≥ d . In light of the definition of subspacedistance d S as mentioned before, it is equivalent to prove that dim(im( w ) + im( w )) ≥ k + d .For any identifying vector v j ( j = 1 , , · · · ) , we note thatthis vector has d ones in the last n positions, and can beillustrated in reduced row echelon form as follows: Z := (cid:18) Z Z Z Z (cid:19) k × n , where Z is a matrix with the size of ( k − d ) × n , Z is amatrix with the size of ( k − d ) × n , Z is a zero matrix withthe size of d × n , Z is a matrix with the size of d × n ,and Z contains at least d pivots.It is clear that im( w ) = im( U | M ) , U ∈ U , M ∈ M , then Z := U k × n M k × n Z Z Z Z k × n . Notice that the rank of U is k , rank( Z ) ≥ d , and Z = d × n . Hence, the rank of the matrix Z is at least k + d .Here we finish the proof of the theorem.Remark: Under careful comparison, we can find that theconstruction in Theorem 3 differs from the one in Theorem 2in that the restriction of k ≥ d is removed, and the condition( a ) is relaxed, sacrificing the code C in Theorem 2. The sizeof code C contains only small codewords, and the candidateset of identifying vectors increases greatly, which will finallyadd more code into ∪ j C F j . These findings are verified by thegreedy algorithm. B. Greedy Algorithm
Due to the limitation of data scale, the optimal echelon-Ferrers is difficult to operate effectively. Therefore, we em-ploy an algorithm with greedy strategy, which is illustrateddetailedly in Algorithm 1.
Algorithm 1
Greedy()
Input: n , n , d, k Output: target identifying vector set S v construct V set : a number of P k − d ∆=0 (cid:0) n k − d − ∆ (cid:1) × (cid:0) n d +∆ (cid:1) identifying vectors compute corresponding dimensions for each vector in V set sort V set in descending order by the dimension values pick up the first vector of the sorted V set and put it to S v for i = maxdimension − down to do iSet : compatible vectors with dimension i in V set choose v from iSet under the greedy criteria: it hasminimum distance to the latest vector in S v pick v out from iSet and put it to S v repeat Step and until there’s no more such v end for The greedy algorithm operates by selecting identifyingvectors and adding them to the target set S v . Firstly, a totalnumber of P k − d ∆=0 (cid:0) n k − d − ∆ (cid:1) × (cid:0) n d +∆ (cid:1) identifying vectors areadded to V set . For all the vectors in V set , we compute theircorresponding dimensions by Theorem 1, and sort them indescending order according to the value of dimensions. Thetarget set S v is empty initially, and the first vector (withmaximum dimension) of the sorted V set is put into S v . Thenthe loop step runs from the second maximum dimension downto dimension . For each round with dimension i , we denoteby iSet as the set of vectors with dimension i in V set andcompatible to S v , i.e., iSet = { v ∈ V set | dim( v ) = i, d H ( v, s j ) ≥ d, ∀ s j ∈ S v } . Now, we select vectors from iSet and add them to S v oneby one. In order to add vectors as more as possible, a greedystrategy is employed: the vector v has the minimum Hammingdistance to the latest vector in S v . Then vector v is picked outfrom iSet and added to S v . This process will continue untilno more such vector v can be found to add to S v . In somecase of i , maybe the iSet is an empty set. The total cost ofthe algorithm is bounded by O ( m · log m ) , where m equals P k − d ∆=0 (cid:0) n k − d − ∆ (cid:1) × (cid:0) n d +∆ (cid:1) . Example In order to apply Theorem 3 for A q (12 , , ,we can choose n = 8 and n = 4 . By applying the echelon-Ferrers construction, a number of identifying vectors areobtained. We list all the obtained identifying vectors in de-scending order according to their dimension values in Table I.It is known that A q (8 , , ≥ q + q ( q +1)2( q + q +1)+1 .Then we have A q (12 , , ≥ q ( q + ( q + q + 1)( q +1) ( q + 1) + ( q + 2 q + 3 q + 5 q + q + 2 q + q + 7 q + q + q +1) . When q = 2 , we obtain A (12 , , ≥ ,which is an improvement of the corresponding results in [12],[14]. However, it is still weaker than the paper [2], [5], [6]. TABLE IC
ONSTRUCTION FOR A q (12 , , Identifying Vector Dim Identifying Vector Dim1 110000001100 12 14 001100000011 62 101000001010 10 15 000001101001 53 001100001100 10 16 000010011001 44 011000001001 9 17 000000111100 45 011000000110 9 18 000011000011 46 010100001010 9 19 000010010110 47 110000000011 8 20 000010100101 48 101000000101 8 21 000001100110 49 100100001001 8 22 000001011010 410 100100000110 8 23 000001010101 311 000011001100 8 24 000000110011 212 010100000101 7 25 000000001111 013 000010101010 6
C. Examples
In this section, we give several examples constructed byour methods, and in the meanwhile the expressions of thesebounds are also given.1) d ≥ k Let n = 8 , n = 5 , apply the algorithm, we have A q (13 , , ≥ A q (8 , , × q + q + 2 q + 3 q + 5 q + q + 3 q + 6 q + 7 q + 5 q + 3 q + 3 q + q + 1 . When q = 2 , there’s A (13 , , ≥ . This bound isstrictly improves upon the corresponding results in [2], [1],[9], [14], [6], [5].Let n = 8 , n = 6 , we have A q (14 , , ≥ A q (8 , , × q + q +2 q +3 q +5 q + q +4 q +6 q +10 q +8 q +8 q + 4 q + 2 q + q + 2 q + q + 1 . When q = 2 , there’s A (14 , , ≥ , which exceeds the current bestbound 1258757174.Let n = 8 , n = 7 , we have A q (15 , , ≥ A q (8 , , × q + q + 2 q + 3 q + 5 q + q + 4 q + 6 q + 11 q +10 q + 12 q + 9 q + 8 q + 4 q + 3 q + 2 q + q + q + q + q +2 q +1 . When q = 2 , there’s A (15 , , ≥ ,which exceeds the current best bound 10071464646.Let n = 8 , n = 8 , we have A q (16 , , ≥ A q (8 , , × q + q + 2 q + 3 q + 5 q + q + 4 q + 6 q + 11 q +11 q + 11 q + 11 q + 15 q + 6 q + 5 q + 4 q + 5 q + q + 2 q + q + 7 q + q + q + 1 . When q = 2 , there’s A (16 , , ≥ , while the current best bound is80590267742 in the paper [5], [6], [2].Let n = 8 , n = 9 , we have A q (17 , , ≥ A q (8 , , × q + q + 2 q + 3 q + 5 q + q + 4 q + 6 q + 11 q +11 q + 15 q + 13 q + 12 q + 8 q + 6 q + 6 q + 7 q + q + 2 q + 5 q + 4 q + q + q + q . When q = 2 , there’s A (17 , , ≥ , while the current best boundis 644711939518.Let n = 8 , n = 10 , we have A q (18 , , ≥ A q (8 , , × q + q + 2 q + 3 q + 5 q + q + 4 q + 6 q + 11 q +11 q + 14 q + 15 q + 14 q + 6 q + 5 q + 5 q + 9 q +2 q +6 q +5 q +5 q +2 q +3 q +2 q + q +2 q + q +1 .When q = 2 , there’s A (18 , , ≥ , while thecurrent best bound is 5157723124262.Let n = 8 , n = 11 , we have A q (19 , , ≥ A q (8 , , × q + q + 2 q + 3 q + 5 q + q + 4 q + 6 q + 11 q +11 q + 14 q + 13 q + 14 q + 5 q + 6 q + 6 q + 8 q + TABLE IIN
EW SUBSPACE CODES ON A q ( n, , A q ( n, d, k ) New Old A (13 , , A (13 , , A (13 , , A (13 , , A (13 , , A (13 , , A (13 , , A (14 , , A (14 , , A (14 , , A (14 , , A (14 , , A (14 , , A (14 , , A (15 , , A (15 , , A (15 , , A (15 , , A (15 , , A (15 , , A (15 , , q + 8 q + 8 q + 7 q + 3 q + 3 q + 4 q + 2 q + 3 q +2 q + 2 q + q + q + q + q + 2 q + 1 . When q = 2 , there’s A (19 , , ≥ , while the current best boundis 41261547000158.All the upon improvements are listed and compared inTable II.2) k > d Let n = 7 , n = 3 , we have A q (10 , , ≥ A q (7 , , × q + q + q + 1 . When q = 2 , A (10 , , ≥ , while thecurrent best bound is 21319 [11], [6].Let n = 7 , n = 4 , we have A q (11 , , ≥ A q (7 , , × q + q + q + 2 q + q + 1 . When q = 2 , A (11 , , ≥ ,while the current best bound is 85283 [11], [6].Let n = 7 , n = 5 , we have A q (12 , , ≥ A q (7 , , × q + q + q + 2 q + 2 q + 2 q + q + 1 . When q = 2 , A (12 , , ≥ , while the current best bound is 383111[11], [6]. When set n = 9 , n = 3 , q = 2 , we have the samebound 38311.When n varies from 13 to 16, and q in the set { } , we have the similar bounds to the results inthe paper [11], [6]. IV. C ONCLUSION
A construction for constant dimension code is presentedin this letter, and new lower bounds of the sizes of constantdimension codes A q ( n, d, k ) are also given. This construc-tion gives an improved bounds for the linkage construc-tion with echelon-Ferrers. The results of these lower boundsin [12] are not the best, and our construction generalizethe construction. With the help of the greedy algorithm,we have improved at least the following lower bounds: A q (13 , , , A q (14 , , , A q (15 , , (listed in Table II), theexpression of these bounds are also given. All these boundsexceeds the bounds presented in [12]. Moreover, the identify-ing vectors underlying the improved bounds are listed in theappendix. R EFERENCES[1] Hao Chen, Xianmang He, Jian Weng, and Liqing Xu. New constructionsof subspace codes using subsets of mrd codes in several blocks.
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Here, we list the identifying vectors underlying the im-proved lower bounds of A q (13 , , , A q (14 , , , A q (15 , , in Table III, Table IV and Table V, respectively. TABLE IIII
DENTIFYING V ECTORS FOR C ONSTRUCTION OF A q (13 , , Identifying Vector Dim Identifying Vector Dim1 1100000011000 15 22 1000010001001 72 0011000011000 13 23 0100100000101 73 1010000010100 13 24 0100001010001 74 0110000001100 12 25 0000110000110 75 0101000010100 12 26 0000100101100 76 0110000010010 12 27 0000101001010 77 1100000000110 11 28 0000100110010 78 0000110011000 11 29 0001010000101 69 1010000001010 11 30 0010100000011 610 1001000010010 11 31 0001000110001 611 1001000001100 11 32 0010001001001 612 0101000001010 10 33 0000010101010 613 1000100010001 9 34 1000001000101 514 0011000000110 9 35 0100010000011 515 0000101010100 9 36 0100000101001 516 0001100001001 8 37 0001001000011 417 0000001111000 8 38 0000001100110 418 0000011001100 8 39 0010000100101 419 0010010010001 8 40 1000000100011 320 0000011010010 8 41 0000000011110 021 0000010110100 8 TABLE IVI
DENTIFYING V ECTORS FOR C ONSTRUCTION OF A q (14 , , Identifying Vector Dim Identifying Vector Dim1 11000000110000 18 31 00001100001100 102 00110000110000 16 32 00000110010100 103 01100000101000 16 33 10000100010001 94 10010000101000 15 34 10000100001010 95 10100000100100 15 35 00100010100001 96 10100000011000 15 36 00100010010010 97 11000000001100 14 37 00101000001001 98 01010000100100 14 38 00001001010100 99 01010000011000 14 39 10000001100010 910 00001100110000 14 40 01001000000110 911 01100000010100 14 41 00010010010001 812 10010000010100 13 42 00010001100001 813 00101000100010 12 43 00010001010010 814 00000110101000 12 44 00110000000011 815 00000011110000 12 45 00100100000110 816 00110000001100 12 46 00000011001100 817 00010100100010 11 47 00011000000101 818 10001000100001 11 48 00010100001001 819 10001000010010 11 49 00100001010001 720 00001010011000 11 50 01000001001010 721 00001001101000 11 51 10000010000110 722 00001010100100 11 52 10000010001001 723 11000000000011 10 53 00001100000011 624 01000010100010 10 54 01000010000101 625 00000101100100 10 55 10000001000101 526 00000101011000 10 56 00000011000011 427 01001000010001 10 57 00000000111100 428 01000100100001 10 58 00000000110011 229 01000100010010 10 59 00000000001111 030 00011000001010 10 TABLE VI
DENTIFYING V ECTORS FOR C ONSTRUCTION OF A q (15 , ,4)