Best Known (89, 89+67, s)-Nets in Base 2
(89, 89+67, 54)-Net over F2 — Constructive and digital
Digital (89, 156, 54)-net over F2, using
- 2 times m-reduction [i] based on digital (89, 158, 54)-net over F2, using
- trace code for nets [i] based on digital (10, 79, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- trace code for nets [i] based on digital (10, 79, 27)-net over F4, using
(89, 89+67, 64)-Net over F2 — Digital
Digital (89, 156, 64)-net over F2, using
(89, 89+67, 294)-Net in Base 2 — Upper bound on s
There is no (89, 156, 295)-net in base 2, because
- 1 times m-reduction [i] would yield (89, 155, 295)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2155, 295, S2, 66), but
- 4 times code embedding in larger space [i] would yield OA(2159, 299, S2, 66), but
- adding a parity check bit [i] would yield OA(2160, 300, S2, 67), but
- the linear programming bound shows that M ≥ 701 516166 771767 578631 389541 059374 879498 756917 597323 515318 103726 158201 862769 375666 092660 644487 866691 091245 590510 558941 249169 584877 888911 941253 045004 269526 631295 838223 413666 262195 245212 606196 574872 647598 351317 178082 754498 547628 422526 425779 666944 / 278 590792 616933 114400 625776 990499 879037 950603 614926 623064 302635 400334 417896 878232 008869 967932 569057 360914 836377 348264 529703 893096 684295 586927 692720 993041 624357 470341 874289 877459 739017 007121 574375 > 2160 [i]
- adding a parity check bit [i] would yield OA(2160, 300, S2, 67), but
- 4 times code embedding in larger space [i] would yield OA(2159, 299, S2, 66), but
- extracting embedded orthogonal array [i] would yield OA(2155, 295, S2, 66), but