Best Known (138−57, 138, s)-Nets in Base 2
(138−57, 138, 56)-Net over F2 — Constructive and digital
Digital (81, 138, 56)-net over F2, using
- trace code for nets [i] based on digital (12, 69, 28)-net over F4, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 12 and N(F) ≥ 28, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
(138−57, 138, 63)-Net over F2 — Digital
Digital (81, 138, 63)-net over F2, using
(138−57, 138, 296)-Net in Base 2 — Upper bound on s
There is no (81, 138, 297)-net in base 2, because
- 1 times m-reduction [i] would yield (81, 137, 297)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 183898 103866 826789 071294 169691 268154 094600 > 2137 [i]
- extracting embedded orthogonal array [i] would yield OA(2137, 297, S2, 56), but
- 2 times code embedding in larger space [i] would yield OA(2139, 299, S2, 56), but
- adding a parity check bit [i] would yield OA(2140, 300, S2, 57), but
- the linear programming bound shows that M ≥ 17287 357861 569583 077318 309024 106544 972071 429625 844312 657293 607712 579667 261091 239239 041437 521982 336427 575753 277698 080494 799023 286191 473225 374823 732404 580505 838240 891058 119045 551642 664644 533323 198549 887134 939663 633343 530589 322320 488228 287453 007559 497089 024000 / 11287 720128 651672 768334 090202 144442 150913 657248 325975 812180 098496 802801 183480 625232 155290 452365 101656 876176 319675 627705 240155 801040 246225 018238 457982 786383 414127 839587 967171 659833 526569 677732 803664 291022 759570 928113 > 2140 [i]
- adding a parity check bit [i] would yield OA(2140, 300, S2, 57), but
- 2 times code embedding in larger space [i] would yield OA(2139, 299, S2, 56), but