A Hierarchical-based Greedy Algorithm for Echelon-Ferrers Construction

11/04/2019
by   Xianmang He, et al.
0

Echelon-Ferrers is one of important techniques to help researchers to improve lower bounds for subspace code. Unfortunately, exact computation of echelon ferrers construction is limited by the computation time. In this paper, we show how to attain codes of larger size for a given minimum distance d=4 or 6 by the hierarchical-based greedy algorithm for echelon-ferrers introduced in [6]. About 63 new constant-dimension subspace codes are better than previously best known codes.

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1 Introducation

Subspace coding was proposed by R.Koetter and F.R.Kschischang in [21] to correct errors and erasures in random network coding. The projective space of order over the finite field , denoted

, is the set of all subspaces of the vector space 

. The set of all -dimensional subspaces of an -vector space will be denoted by . For , its cardinality is given by the Gaussian binomial coefficient

Thus, .

A widely used distance measure for subspace codes (motivated by an information-theoretic analysis of the Kötter-Kschischang-Silva model, see e.g. [25]) are the subspace distance

where and are subspaces of .

A set of subspaces of is called a subspace code. The minimum distance of is given by . If the dimension of the codewords, is fixed as , we use the notation and call a constant dimension code(CDC for short). For fixed ambient parameters , , and , the main problem of subspace coding asks for the determination of the maximum possible size of an subspace code.

In this paper we give a greedy algorithm for the echelon-ferrers construction. About new constant-dimension subspace codes of larger size for a given minimum distance are illustrated in the table LABEL:tab:new-improvements.

The remaining part of this paper is structured as follows. The currently implemented lower bounds, constructions, are described in Section 2. The preliminaries are outlined in section 3. Constant dimension codes (CDC) by our algorithm are treated in Section LABEL:our_construction, Finally we draw a conclusion in Section 5.

2 Previous constructions

The lower and upper bounds on have been intensively studied in the last years, see e.g. [7]. The report [15] describes the underlying theoretical base of an on-line database, found at http://subspacecodes.uni-bayreuth.de and maintained by the research team in the University of Bayreuth that tries to collect up-to-date information on the best lower and upper bounds for subspace codes.

Lifted MRD codes, (we omit the details here, see subsection 3.2), are one type of building blocks of the Echelon- Ferrers construction, see subsection 3.3. The latter is a nice interplay between the subspace distance, the rank distance and the Hamming distance. Another construction based on similar ideas is the so-called coset construction [16]. The most effective general recursive construction is the linkage construction and its generalization. According the report [15], the lower bound with the highest score is the improved linkage construction, and it yields the best known lower bound in 69.1% of the constant dimension code parameters of the database currently. The linkage construction is to obtain large codes from the subspaces spanned by a given code and choices of an MRD code : rowspace{( are sampled from }. This resulting size of the constructed code is the size of times the size of the MRD code. By performing a tighter analysis of the occurring subspace distances, papers [24, 13, heinlein2017asymptotic] indicated that codes in a smaller ambient space can be further added.

The expurgation-augmentation method, which starts with a lifted MRD code and then adding and removing codewords, is invented by Thomas Honold. A starting point is possible a computer–free construction for the lower bound , see [22]. The subsequent studies contain for [19, Theorem 2], , [17], and [18, Theorem 4].

New subspace codes from two parallel versions of maximum rank distance codes was introduced by Xu and Chen [26]. The problem asks for the size of the construction const dimension codes was turned to find a suitable sufficient condition to restrict the number of roots of to , where and are -polynomials over the extension field :

Geometric concepts like the Segre variety and the Veronese variety where also used to obtain constructions for constant dimension codes :

Theorem 1 ([5, Theorem 3.11 and 3.8])

If

is odd, then

, using .

If is even, then , using if is odd and if is even.

In general, the exact determination of is a hard problem, whether in terms of theory or algorithms. The exact calculation for echelon ferrers construction is constrained by the computation time[8, 14, 9]. A greedy-type approach has been considered by Alexander Shishkin, see [23] and also [2]. It is implemented asgreedy_multicomponent. In [12, 11] the authors considered block designs as skeleton codes. [4] describes an algorithm to tackle the integer linear optimization problems representing the q-packing design construction by means of a metaheuristic approach, and gives some improvements on the size of . With a stochastic maximum weight clique algorithm and a systematic consideration of groups, authors in [3] gives some new lower bounds on for .

3 Preliminaries

3.1 Basic Notation

Let be a -dimensional subspace of . We represent by the matrix

in reduced row echelon form, such that the rows of

form a basis of . The identifying vector of , denoted by , is the binary vector of length and weight , where the ones of are exactly in the positions where has the leading coefficients (the pivots).

In this section we give the definitions for two structures which are useful in describing a subspace in . The reduced row echelon form is a standard way to describe a linear subspace. The Ferrers diagram is a standard way to describe a partition of a given positive integer into positive integers.

A matrix is said to be in row echelon form if each nonzero row has more leading zeroes than the previous row.

A matrix with rank is in reduced row echelon form if the following conditions are satisfied.

  • The leading coefficient of a row is always to the right of the leading coefficient of the previous row.

  • All leading coefficients are ones.

  • Every leading coefficient is the only nonzero entry in its column.

A -dimensional subspace of can be represented by a generator matrix whose rows form a basis for . We usually represent a codeword of a projective space code by such a matrix. There is exactly one such matrix in reduced row echelon form and it will be denoted by .

A Ferrers diagram represents partitions as patterns of dots with the -th row having the same number of dots as the -th term in the partition. A Ferrers diagram satisfies the following conditions.

  • The number of dots in a row is at most the number of dots in the previous row.

  • All the dots are shifted to the right of the diagram.

The number of rows (columns) of the Ferrers diagram is the number of dots in the rightmost column (top row) of . If the number of rows in the Ferrers diagram is and the number of columns is we say that it is an Ferrers diagram.

Recall that the Hamming metric on is defined as , where denotes the number of nonzero entries in the vector . The following results are useful tools for constructions of subspace codes.

Proposition 1 ([9])

For we have

  • ,

  • if , then ,

3.2 Lifted MRD codes

A prominent code construction uses maximum rank distance (MRD) codes. For matrices the rank distance is defined via .

Theorem 2

(see [10]) Let be prime power, are positive integers, and be a rank-metric code with minimum rank distance . Then, .

Codes attaining this upper bound are called maximum rank distance (MRD) codes. They exist for all (suitable) choices of parameters. Using an identity matrix as a prefix one obtains the so-called lifted MRD codes. For any two MRD code and , the subspaces and spanned by rows of and are the same if and only if . The intersection is the set . Thus . The distance of this CDC is . A CDC constructed as above is called a lifted MRD code.

3.3 Echelon-Ferrers

In [9] presented the multi-level construction, which was based on lifted MRD codes. Let us briefly review the construction in the following theorem 3. Let be integers and a binary vector of weight . By we denote the set of all matrices over that are in row-reduced echelon form.

Theorem 3

(see [9]) For integers with and , let be a binary constant weight code of length , weight , and minimum hamming distance . For each let be a code in with minimum rank distance at least . Then, is a constant dimension code of dimension having a subspace distance of at least .

The code is also called skeleton code. For we have the following upper bound:

Theorem 4

(see [9]) Let be the Ferrers diagram of and be a subspace code having a subspace distance of at least , then

where is the number of dots in , which are neither contained in the first rows nor contained in the rightmost columns.

The authors of [9] conjecture that Theorem 4 is tight for all parameters , , and . Constructions settling the conjecture in several cases are given in [8].

Let denote the maximum size of a known MRD code over matching distance . The optimal Echelon-Ferrers construction can be modeled as an ILP:

This is implemented as echelon_ferrers. However, the evaluation of this ILP is only feasible for rather moderate sized parameters. The Echelon-Ferrers construction has even been fine-tuned to the pending dots [6].

Now, we are ready to give the formal definition about the problem that will be addressed in this paper.

Definition 1 (Problem Definition)

Given , there are total different identifying vectors, and each vector corresponding to a certain dimension. Among these vectors, we need to choose a binary vector to maximize the size of .

4 Greedy Algorithm

In this section, we will present the details of the construction: our greedy algorithm. We first briefly review the classic recursive backtracking procedure that exhaustively enumerates all maximal cliques in an undirected graph . Then we provide the greedy algorithm in the rest of the section.

4.1 Classic Maximum Clique Enumeration (MCE)

A classic Maximum Clique Enumeration (MCE) algorithm relies on recursive calls to procedure , which is illustrated in Algorithm 1. We denote the set of neighbors of a vertex by . The algorithm takes a graph as input and initially invokes . In Algorithm 1, the basic idea is to recursively backtrack to add a vertex from the set of candidate vertices in to grow the current clique . A vertex is a candidate to if and only if is a neighbor of all vertices in . Each time when is augmented by a vertex , we refine by keeping only the vertices that are also neighbors of . When becomes empty, cannot be further grown. At this point, we need to check whether is indeed maximal. Towards this, we maintain a set which keeps the set of vertices that are neighbors of all vertices in and have been outputted as part of some maximal clique earlier, i.e., the recursive procedure has outputted some maximal clique earlier, where . Thus, if is not empty, is not a maximal clique; otherwise, we output as a maximal clique.

In the worst case, the algorithm can be achieved in [1, 20] time complexity. The time taken to compute and output the set of all maximal cliques is acceptable when the is small. The following algorithm makes use of this feature. On a normal PC machine, when the size of V is under 80, the classic maximum clique enumeration algorithm can be calculated in a few minutes.

1:if  and  then
2:     output as a maximal clique;
3:     return;
4:end if
5:choose a pivot vertex from
6:
7:for each  do
8:     call MCE()
9:     
10:end for
Algorithm 1 MCE(

4.2 Algorithm

As mentioned in Section 3.3, the optimal Echelon-Ferrers construction of code

can be modeled as an Integer Linear Programming(ILP). Consider that the evaluation of this ILP is only feasible for rather moderate sized parameters, we present a hierarchical-based greedy algorithm

2 as illustrated in the following. The greedy algorithm iteratively maintains a set of identifying vectors. The algorithm starts by initializing a set of all the identifying vectors denoted by , and computing its corresponding dimension by Theorem 4, then we sort the into a descending order of their dimensions. We put the identifying vector with maximal dimension into the result set . At this point, we need to calculate from second maximal dimension, and eventually down to 0 dimension. Each dimension is treated as a layer. Then for each dimension , the algorithm constructs the vectors , which is compatible to . That is, for each vector , we have . Then, the MCE is called to generate all the maximal cliques. In the end, we choose the best clique into . In some cases, the is an empty set, due to the fact that the compatible condition is not satisfied.

1:,
2:Set of identifying vectors
3:generate all the identifying vectors denoted by .
4:compute its corresponding dimensions for
5:sort the by the dimension in descending order
6:put the first identifying vector into
7:for  do
8:      get all the identify vectors with dimension
9:      find the identify vectors set () which is compatible to
10:      call MCE to generate all the maximal cliques
11:      pick the largest clique into
12:end for
Algorithm 2 Greedy()

In the above algorithm, the way to choose the clique is critical for the resulting solution. Suppose that cliques were calculated from the previous step. We pick the click with largest codes into . If there exists serval clicks with same largest codes, we need to evaluate the impact on the subsequent selection after joining the result set . Towards this, suppose that was added to , we choose the vectors with dimensions from to , which were compatible to the new result set , we invoke Algorithm again to generate all the possible cliques. Among all the cliques, we pick the one that maximizes the total number of codes. The parameter makes the can be finished in acceptable time.

Example 1

Let be any prime power, be , we observe that total identifying vectors. After apply the greedy algorithm, we obtain 100 identifying vectors, 24 of which are illustrated in table 1. With this, the codes of have the cardinalities are . Table LABEL:A-q-5-4 gives some new lower bounds for codes .

identifying vector dimension identifying vector dimension
1 1111100000000 32 13 0110101010000 22
2 1110011000000 28 14 0110110001000 22
3 1101010100000 26 15 0111001001000 22
4 1011001100000 24 16 1010101001000 21
5 1001111000000 24 17 1110000001100 20
6 1100110010000 24 18 0101110000100 20
7 1010110100000 24 19 0101100110000 20
8 1110000110000 24 20 0111000100100 20
9 1100101100000 24 21 0011100101000 19
10 0111010010000 24 22 0011101000100 19
11 1101001010000 24 23 0011110000010 19
12 1011010001000 23 24 1011000010100 19
Table 1: Construction for

It has been proved that for general diagrams , the bound of Theorem 4 is attained for (see [9, 8] for more details). The improvements on CDC codes are given in Table 2-3, achieved by our greedy algorithm. All the codes are attached in the Supplementary material.

5 Discussion

The echelon-ferrers construction is an important method to construct the const dimension code. One of the outstanding advantages is that this method can be applied to various parameters. In this paper, we give a greedy algorithm for the echelon-ferrers construction. About improvements are given by our greedy algorithm. It is also interesting if the greedy algorithm of this paper can be improved to get larger codes.

New Old
4796417559 4794061075
1880918023783990 1853306869495369
18525690479132333173 18447026753270989253
23322304248923865096456 23283124070485023029131
1104898620939789578683671514 1104427865906845441766065829
79247846163915655208442806985 79228167237804056983985067529
3434214279120353599762054717228 3433683900071278422100477868743
76745404672 76641774536
152354354408240436 150117856399907497
4742576757171205745408 4722438848816618554449
14576440154794852120820500 14551952544051476527718901
2652861588875282767080909163052 2651731306042334943557302058601
324599177887378338095324360943616 324518573006045405876299328553537
22531879885308389971852673089028988 22528400068367657625538198208974789
38325131657 38325127529
50782269101569336 50031831779643235
1185639430145591024577
1180591903396972741061
2915286427121720397974126
2910383105378468421466631
378980216844611802379704124332
378818692457327706478178392571
40574896910456482568427309053441
40564819212026880567284943689225
2503542202545027727509319406311282
2503155505073020367253014614865311
1227203232293
12340234566810426241
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9110270832108553578429954401
6369520523839151727821917674342793
1329558223045414729174409793499570241
147831663555770209444761739522908440329
39267675031563
2998676636295383433055
1243233943362040432057180581
28469596349440811610995019309681
10705253144414985109399576183764
9623043
43566963852751451455533741586485
379191945
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42905023381
1256703351587805
728678523483522880513165
1273071559584674249524907514705
8896748859492157909452997378522
6401
1799231895982079405217983712681
675184889205
1427602271526982761008273790284
715743116587585
5154571384601397730287591300333
30465293577720745
40210734642430233
177068857538981556600415147
1303625275416014562978042328889
121
2780234018500650513000010339634
26380051
3023969047576656762486652218945
3324986367024243
4677967123339541192988109893334
6105696388708610177
3043722856893271957593877642162
4397010040042569507531
313923840120169 313890227593181
80962387333738514962426 79766493108704289506596
79566863724904828874349525569
79228163694856240990516691397
3558699030750375431966367668488876
3552713681710884034734089759357256
36719018111669485366326051726236
511270134
36703368217672944133687538738511838581
372
22306285465490013824377105083555
006005322241
22300745198571187960747745153215646449
669641
63636683737167010339661770016731
80121197368344
63626854411384455139799017829057872455
05081632
59021591907098096238648717 58149848582764764023592067
325905875503966895183219736444928
324518573106546434155786844197589
55604672369921077974140644073486328125
55511151459100586488071778857407029781
431995576183099162028683927491197134367
9804101
43181145677085642767007330129704577796
55883719
584745889706780730982259844569699067757
1597238272
58460065494087063243496900390732227771
40820452937
338191428419664795870630989049561439689
8957540722053
33813919135347009334982427022052504229
15161727754729
1285780755925958656 1285807203784040529
43026740586030433477849264332 42391238041709312218583939797
1334910466075153545917778842397704192
1329228075013079567760048050797035973
86882300578011697800830006599426269531
2500
86736174154111722891684627543004312849
7881
50823847542365442865880653737635543921
0673571016516
50802186077632673353986114178590404051
0946363417373
15328762651129433020259724863226574216
92282703427141632
15324955408881896035569182151110569287
71954768633271817
17972879091077507906314355839720905589
86192090028734124332
17970102999207938958543182750235849415
14381414783356188033
31366481390109307710095048570592 30903212532481195126684665395596
546779321806040481999310568251301507
8912
544451819525358852196042787469513558
0885
13575359458996180137997725978493690490
722656250
13552527211579956884193746498946050250
229295406
59793748394835040265326746482178302037
689516829976915564
59768263898474063874498683542286352229
995607192086565368
40183431564146475997690442413946479412
8095421048885275000832
40173451107059357543489922940964034066
9623971222839029278281
95515248370399043860359211584572109956
7855543645243756034671222
95500495080020662880672540270791225462
2140389397258291564608570
1285780755925958656 1285807203784040529
43026740586030433477849264332 42391238041709312218583939797
1334910466075153545917778842397704192
1329228075013079567760048050797035973
86882300578011697800830006599426269531
2500
86736174154111722891684627543004312849
7881
50823847542365442865880653737635543921
0673571016516
50802186077632673353986114178590404051
0946363417373
15328762651129433020259724863226574216
92282703427141632
15324955408881896035569182151110569287
71954768633271817
17972879091077507906314355839720905589
86192090028734124332
17970102999207938958543182750235849415
14381414783356188033
Table 2: New constant subspace codes in the case
New Old
68897965009 68897963738
150102699182228692 150094917741002635
4722384855719339140737
4722366764345697328193
14551922748241775734567126
14551915287971527113290751
2651730911778210523130685893904
2651730846051234707907501127187
324518556081269363814110817985537
324518553663149093301211322745345
22528399603172190076386709129674082
22528399545018940855122915585089383
1102367740720 1102367739809
12158318633936434911 12157688337037220737
1208930523064167881368832
1208925891672499656134657
9094951717651110434914331875
9094947054982204477500406251
6366805919179483466167202128673975
6366805761369014533690749456337541
13292280057088793141837029829242
10176
13292279958042586861617972816403
49697
14780882979641273909118047876478
0440907
14780882941486927095046165744617
4515967
17637885117217 17637884567985
984823809349898855629 984772755302827825304
309486213904426891598430721
309485028268160807488274522
568434482353194399432024062
6251
568434190936387787802784421
8912
152867010119499398022264943
22203613977
152867006330470038954609367
88322109304
544451791138356967089577848
7274901086209
544451787081424357851974081
6189670031954
969773732294263981177227448
612171403472391
969773729790957286705989818
733440915565152
282206174721269 282206169223861
79770728557541262831049 79770528994296955194991
79228470759533498436223522065
79228465213535437618551984193
35527155147074650443812457129881
51
35527154986053780317306518554845
01
36703369129691805465310215681701
031147309
36703369126904824755396790969987
924701979
22300745365027101371993243436762
479445971009
2230074536469019022543282877208
1255730905601
63626854575826659805038602216023
11031952447275
6362685457559470145240699283992
755392451222033
4515298847779284 4515298730748862
6461429013171995258781459 6461417369472937542117973
20282488514440621893559548060816
20282487415579548140041494597697
22204471966921656798893547819953
46525
22204471885174522176980972290039
22001
88124789280390024922586787041904
923299423119
88124789274625593704300569118173
789808207533
91343853015151007219700686510908
916105199764544
91343853013939625003475366792854
319199699599873
41745579287199871498086279962806
949211299041687641
41745579287064298367037383503578
317354686374220297
34432185344 34432090228
50034101937449940 50031545103789355
1180596129533618814976
1180591620717679804753
2910384536314002757812500
2910383045673376465235151
378818701653726539723645188684
378818692265664782360946466387
40564819509544287726043985346560
40564819207303340852292565865025
2503155511454435281834458579675756
2503155504993241601338448821594903
2954463963046489945809 2954312729428813990579
1237944768674801424294477824
1237940039357438145380552705
2842172398866423035583618164
0625
2842170943044126035694332071
8751
1070069070755212446299883649
67293623929
1070069044235984933429042496
99254898347
4355614329041038258627293015
4239328518144
4355614296588014266612520774
1487691071489
8727963590616552703971769530
289198197347289
8727963568087712949239031857
923967131259815
2954463963046489945809 2954312729428813990579
1237944768674801424294477824
1237940039357438145380552705
2842172398866423035583618164
0625
2842170943044126035694332071
8751
1070069070755212446299883649
67293623929
1070069044235984933429042496
99254898347
4355614329041038258627293015
4239328518144
4355614296588014266612520774
1487691071489
8727963590616552703971769530
289198197347289
8727963568087712949239031857
923967131259815
717934859275509204177834 717898037723397692559604
1267655447232483285394310299648
1267650601408821022214383010129
8881788747791253696319580078
1250000
8881784197292290627956390991
2500776
1798465087220824484230198643
417896607747734
1798465042647790965312546005
352010295717636
1427247703340481388090275458
439930355797458944
1427247692706000445877493272
790347428948611649
5153775220623387717818557063
15044039145435664748
5153775207320138341919661230
07222611150929116904
174458170399445062802792208 174449223166785648589189287
1298079177949694792623525109
694464
1298074215842632726751651324
815360
2775558983684134809957028579
71191406250
2775557561653840821236376762
39308865625
3022680272092038061709235593
3967722099693638844
3022680197178142275400796071
2498383927949957043
4676805274306089179379511219
1631637439410138513408
4676805239459022261051369956
2803183808164427005952
3043252730025904195116850962
5919602689266747460249974
3043252722170468489520140759
7453596033986399175840325
42393335521752447820465426254 42391161229528932943108423780
1329233078240589709748976035
041378304
1329227997022855912261245205
660897281
8673621824015261177754879617
69104003906250
8673617380168252566364035010
52896689859376
5080218733305091542512879685
57612011225792825830766
5080218607397303722266117961
45559406022133779734348
1532495552284619388500421614
873920544243085745181425664
1532495540865932414501312907
292975308826480609874673665
1797010304552996180275709594
710174698404463023145829263594
1797010299914439938376747917
222554278763333551504982366216
Table 3: New constant subspace codes in the case

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