Best Known (112−95, 112, s)-Nets in Base 8
(112−95, 112, 65)-Net over F8 — Constructive and digital
Digital (17, 112, 65)-net over F8, using
- t-expansion [i] based on digital (14, 112, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(112−95, 112, 269)-Net over F8 — Upper bound on s (digital)
There is no digital (17, 112, 270)-net over F8, because
- extracting embedded orthogonal array [i] would yield linear OA(8112, 270, F8, 95) (dual of [270, 158, 96]-code), but
- construction Y1 [i] would yield
- OA(8111, 134, S8, 95), but
- the linear programming bound shows that M ≥ 3213 718744 248367 464211 966828 793200 685277 723323 269317 408810 044111 174024 201867 677763 903619 161246 026196 671187 301423 896682 561536 / 165729 471662 280587 916075 > 8111 [i]
- linear OA(8158, 270, F8, 136) (dual of [270, 112, 137]-code), but
- discarding factors / shortening the dual code would yield linear OA(8158, 266, F8, 136) (dual of [266, 108, 137]-code), but
- residual code [i] would yield OA(822, 129, S8, 17), but
- 1 times truncation [i] would yield OA(821, 128, S8, 16), but
- the linear programming bound shows that M ≥ 1 008968 613742 170487 095115 575000 563712 / 108734 057573 457133 > 821 [i]
- 1 times truncation [i] would yield OA(821, 128, S8, 16), but
- residual code [i] would yield OA(822, 129, S8, 17), but
- discarding factors / shortening the dual code would yield linear OA(8158, 266, F8, 136) (dual of [266, 108, 137]-code), but
- OA(8111, 134, S8, 95), but
- construction Y1 [i] would yield
(112−95, 112, 295)-Net in Base 8 — Upper bound on s
There is no (17, 112, 296)-net in base 8, because
- 3 times m-reduction [i] would yield (17, 109, 296)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8109, 296, S8, 92), but
- 4 times code embedding in larger space [i] would yield OA(8113, 300, S8, 92), but
- the linear programming bound shows that M ≥ 1 781634 836527 846280 733573 421279 205384 148048 302886 393164 147084 545703 578857 792296 708926 378665 174216 714878 248812 357380 788360 776709 613868 283479 330140 356855 376940 510352 173783 047454 883068 768858 845190 073739 610843 855277 585166 090308 718583 204304 152985 628748 961216 296825 766248 404045 242811 530403 857093 553525 312462 943689 435291 288979 601906 367806 540226 502956 367018 438382 241646 041249 912858 419617 092054 244579 079789 485744 230527 344764 266035 095072 145408 / 1 431647 292130 476616 342789 490195 130245 054128 800987 997812 153536 145708 177007 663987 018731 157475 368375 301128 714467 353204 662653 623489 766988 263547 589353 697685 188907 676992 490829 989948 048151 483180 039627 647686 805042 253494 989595 037194 083600 003258 218464 394506 012821 219350 377748 097865 570578 694639 578900 064124 989028 724642 242417 098660 352437 927643 > 8113 [i]
- 4 times code embedding in larger space [i] would yield OA(8113, 300, S8, 92), but
- extracting embedded orthogonal array [i] would yield OA(8109, 296, S8, 92), but