For constant dimension codes that contain a lifted MRD code, Theorem 10 gives an upper bound which is tighter than the Johnson bound of Theorem 7. In [5] two infinite series of constructions have been given where the code sizes exceed the MRD bound of Theorem 10 for q = 2,d= 4, andk= 3. Given the data available from [23] we mention that, besidesd= 4,k= 3, the only other case where the MRD bound was superseded is A2(8,4; 4)≥4801 >4797, see [10]. Next, we show that ford= 4 andk= 3 the MRD bound can be superseded for all field sizes q if v is large enough. For the limit of the achievable ratio we obtain:
Proposition 10. For q≥3 we have lim
v→∞
Aq(v,4;3) q2v−6+hv−3
2 i
q
≥1 +2q13. Proof. For q≥2, [25, Theorem 4] gives
Aq(7,4; 3)≥q8+q5+q4+q2−q ≥q8+q5+q4. With this, we conclude
Aq(v0,4; 3)≥Aq(7,4; 3)·q2v0−14+Aq(v0−6,4; 3)≥q2v0−10· q4+q+ 1 from Corollary 4 choosingm= 7. Applying Proposition 6 withs= 3 gives
Aq(v0+ 3l,4; 3)≥q6lAq(v0,4; 3) +q6l−1
q6−1 ≥q6lAq(v0,4; 3) forv0 ∈ {12,13,14}, so that Aq(v,4; 3)≥q2v−10· q4+q+ 1
for all v≥12.
From Lemma 5 we conclude
v→∞lim
q2v−6+v−3
2
q
q2v−10 =q4+ (1/q; 1/q)2= q3(q4−q3−q2+q+ 1) (q−1)2(q+ 1) . Since
q4+q+ 1
/q3(q4−q3−q2+q+ 1)
(q−1)2(q+ 1) = 1 + 1
q3 − q+ 1
q2(q4−q3−q2+q+ 1), the statement follows for q≥3.
For q = 2 the estimations of the proof of Proposition 10 are too crude in order to obtain a factor larger than one. However, for the binary case better codes with moderate dimensions of the ambient space have been found by computer searches – with the prescription of automorphisms as the main ingredient in order to reduce the computational complexity, see e.g. [31].
Proposition 11. For v≥19 we have A2(v,4;3)
22v−6+hv−3 2
i
2
≥1.3056.
Proof. Applying Proposition 6 withs= 3 and usingA2(4,4; 3)≥0 givesA2(v0+3l,4; 3)≥ A2(v0,4; 3)·26l for all v0 ≥6 and l≥0, so that
A2(v0+ 3l,4; 3) 22(v0+3l)−6+h
(v0+3l)−3 2
i
2
≥ A2(v0,4; 3)
7
3 ·22v0−7 . (9)
Using A2(7,4; 3) ≥ 333 [23], A2(8,4; 3) ≥ 1326 [10], A2(9,4; 3) ≥ 5986 [10], and A2(13,4; 3) = 1597245 [9] we apply Corollary 4 with m= 13 to obtain lower bounds for A2(v0,4; 3) with 19≤v0 ≤21. For these values of v0 the minimum of the right hand side of Inequality (9) is attained atv0 = 20 with value 1.3056442377.
Note that the application of Proposition 6 was used in a rather crude estimation in the proof of Proposition 11. Actually, we do not use the codewords generated by the codewords of cdcC2 in Theorem 18, so that we might have applied [38, Theorem 37]
directly for this part of the proof – similarly for Proposition 10, which then allows to consider just one instead ofs= 3starters. In the latter part of the proof of Proposition 11 the use of Corollary 4 is essential in order to obtain large codes for medium sized dimensions of the ambient space fromA2(13,4; 3) = 1597245 and relatively good lower bounds for small dimensions. This is a relative typical behavior of Corollary 4 and Proposition 6, i.e., the first few applications yield a significant improvement which quickly bottoms out – in a certain sense. As columnbklb of Table 3 suggests, we may slightly improve upon the value stated in Proposition 11 by some fine-tuning effecting the omitted less significant digits.
v bklb mrdb bkub lold lnew ea
6 77 71 77 65 65
7 333 291 381 257 265 301
8 1326 1179 1493 1033 1101 1117
9 5986 4747 6205 4929 4929 4852
10 23870 19051 24698 21313 21313 18924
11 97526 76331 99718 85249 85257 79306
12 385515 305579 398385 383105 383105 309667
13 1597245 1222827 1597245 1532417 1532425 1287958
14 6241665 4892331 6387029 6241665 6241665 4970117
15 24966665 19571371 25562941 24966657 24966665 20560924 16 102223681 78289579 102243962 102223681 102223681 79608330 17 408894729 313166507 409035142 408894721 408894729
18 1635578957 1252682411 1636109361 1635578889 1635578957
19 6542315853 5010762411 6544674621 6542315597 6542315853 5200895489 Table 1: Lower and upper bounds for A2(v,4; 3).
In Tables 1, 2, and 3 we compare the sizes of different constructions with the LMRD and the best known upper bound. Herebklb and bkub stand for best known lower and upper bound respectively. The values of Theorem 10 are given in columnmrdb. Applying Theorem 17 and Theorem 18 to the best known codes give the columns lold andlnew, respectively. The results obtained in [5] are stated in column ea. The achieved ratio between the mentioned constructions and the MRD bound can be found in Table 3. Since differences partially are beyond the given accuracy, we give absolute numbers in Table 1.
Note that the values in columnbklb of Table 3 show that Proposition 11 is also valid forv≥16, while we have a smaller ratio forv <16. The relative advantage over lifted MRD codes is displayed in Table 2.
v bklb mrdb bkub lold lnew ea
6 1.203125 1.109375 1.203125 1.015625 1.015625
7 1.300781 1.136719 1.488281 1.003906 1.035156 1.175781 8 1.294922 1.151367 1.458008 1.008789 1.075195 1.090820 9 1.461426 1.158936 1.514893 1.203369 1.203369 1.184570 10 1.456909 1.162781 1.507446 1.300842 1.300842 1.155029 11 1.488129 1.164719 1.521576 1.300797 1.300919 1.210114 12 1.470623 1.165691 1.519718 1.461430 1.461430 1.181286 13 1.523252 1.166179 1.523252 1.461427 1.461434 1.228292 14 1.488129 1.166423 1.522786 1.488129 1.488129 1.184968 15 1.488129 1.166545 1.52367 1.488129 1.488129 1.225527 16 1.523252 1.166606 1.523554 1.523252 1.523252 1.186257 17 1.523252 1.166636 1.523775 1.523252 1.523252
18 1.523252 1.166651 1.523746 1.523252 1.523252
19 1.523252 1.166659 1.523801 1.523252 1.523252 1.210928
Table 2: Lower and upper bounds for A2(v,4; 3) divided by the size of a corresponding lifted MRD code.
To conclude this section, we remark that an application of Corollary 4 with 2k≤m≤ v−k using a lifted MRD in the cdcC1 cannot generate a code that exceeds the MRD bound of Theorem 10.
Lemma 6. Using the notation of Theorem 18, letk≤min{v1−k, v2−k+d/2}, Cr a linear MRD code,dr=d1/2, andC1 contains a lifted MRD code (in
h
Fvq1
k
i
). Then, the codes constructed in Theorem 18 contain a lifted MRD code (in
h
Fvq1+v2−k+d/2
k
i ).
Proof. Let {τ−1(Ik×k|M) :M ∈R} ⊆C1 be the lifted MRD code in C1. Since R is a (k×(v1−k), d1/2)q MRD code, we have #R = q(v1−k)(k−d1/2+1). The first set of the
construction contains
{τ−1(Ik×k|M |A) :M ∈R, A∈Cr}
in which {(M |A) : M ∈ R, A ∈ Cr} forms a (k×(v1+v2 −2k+d/2), N, dr)q rank metric code of sizeN =q(v1+v2−2k+d/2)(k−dr+1), hence it is a maximum rank metric code.
v bklb mrdb bkub lold lnew ea
6 1.084507 1.0 1.084507 0.915493 0.915493
7 1.144330 1.0 1.309278 0.883162 0.910653 1.034364 8 1.124682 1.0 1.266327 0.876166 0.933842 0.947413 9 1.261007 1.0 1.307141 1.038340 1.038340 1.022119 10 1.252953 1.0 1.296415 1.118734 1.118734 0.993334 11 1.277672 1.0 1.306389 1.116833 1.116938 1.038975 12 1.261589 1.0 1.303705 1.253702 1.253702 1.013378 13 1.306190 1.0 1.306190 1.253176 1.253182 1.053263 14 1.275806 1.0 1.305519 1.275806 1.275806 1.015900 15 1.275673 1.0 1.306140 1.275672 1.275673 1.050561 16 1.305712 1.0 1.305972 1.305712 1.305712 1.016845 17 1.305678 1.0 1.306127 1.305678 1.305678
18 1.305661 1.0 1.306085 1.305661 1.305661
19 1.305653 1.0 1.306124 1.305653 1.305653 1.037945
Table 3: Lower and upper bounds for A2(v,4; 3) divided by the corresponding MRD bound.
7 Conclusion
In this paper we have considered the maximal sizes of constant dimension codes. With respect to constructive lower bounds we have improved the so-called linkage construction, which then yields the best known codes for many parameters, see Footnote 2. With respect to upper bounds there is a rather clear picture. The explicit Corollary 2, which refers back to bounds for partial spreads, is the best known parametric bound in the case ofd6= 2 min{k, v−k}, while Theorem 8 or the linear programming method may possibly yield improvements. Since Theorem 8 implies the Johnson bound and so Corollary 2, it would be worthwhile to study whether it can be strictly sharper than Theorem 7 for d6= 2 min{k, v−k} at all. Compared to Corollary 2, the only two known improvements are given for the specific parameters from Theorem 9 and Proposition 5. In the case of partial spreads we have reported the current state-of-the-art mentioning that further improvements are far from being unlikely.
In general we have shown that the ratio between the best-known lower and upper bound is strictly larger than 0.577576 for all parameters. The bottleneck is formed by the parametersq = 2,d= 4, andk=bv/2c, where no known method can properly improve that factor, see Footnote 2 for the linkage construction. Ford= 4,k= 3 and general
field sizes q we have applied the improved linkage construction in order to show that Aq(v, d;k) is by a factor, depending on q, larger than the MRD bound for sufficiently large dimensions v.
Acknowledgement
The authors would like to thank Harout Aydinian for providing an enlarged proof of Theorem 8, Natalia Silberstein for explaining the restriction 3k≤v in [38, Corollary 39], Heide Gluesing-Luerssen for clarifying the independent origin of the linkage construction, and Alfred Wassermann for discussions about the asymptotic results of Frankl and R¨odl.
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