Best Known (248−112, 248, s)-Nets in Base 2
(248−112, 248, 66)-Net over F2 — Constructive and digital
Digital (136, 248, 66)-net over F2, using
- 4 times m-reduction [i] based on digital (136, 252, 66)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (39, 97, 33)-net over F2, using
- net from sequence [i] based on digital (39, 32)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 39 and N(F) ≥ 33, using
- net from sequence [i] based on digital (39, 32)-sequence over F2, using
- digital (39, 155, 33)-net over F2, using
- net from sequence [i] based on digital (39, 32)-sequence over F2 (see above)
- digital (39, 97, 33)-net over F2, using
- (u, u+v)-construction [i] based on
(248−112, 248, 84)-Net over F2 — Digital
Digital (136, 248, 84)-net over F2, using
(248−112, 248, 383)-Net over F2 — Upper bound on s (digital)
There is no digital (136, 248, 384)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2248, 384, F2, 112) (dual of [384, 136, 113]-code), but
- construction Y1 [i] would yield
- linear OA(2247, 332, F2, 112) (dual of [332, 85, 113]-code), but
- construction Y1 [i] would yield
- OA(2246, 302, S2, 112), but
- 2 times truncation [i] would yield OA(2244, 300, S2, 110), but
- the linear programming bound shows that M ≥ 727209 932038 995964 963694 786156 506719 769259 430085 807852 102717 045283 079365 167115 254635 995296 956416 / 21378 491075 472167 578125 > 2244 [i]
- 2 times truncation [i] would yield OA(2244, 300, S2, 110), but
- linear OA(285, 332, F2, 30) (dual of [332, 247, 31]-code), but
- the Johnson bound shows that N ≤ 220 247871 551638 926977 652794 793575 888633 238636 124356 543746 403131 692235 320146 < 2247 [i]
- OA(2246, 302, S2, 112), but
- construction Y1 [i] would yield
- linear OA(2136, 384, F2, 52) (dual of [384, 248, 53]-code), but
- discarding factors / shortening the dual code would yield linear OA(2136, 374, F2, 52) (dual of [374, 238, 53]-code), but
- the improved Johnson bound shows that N ≤ 2 360702 811474 389063 749944 041391 364709 858504 534718 295956 994191 765553 370834 < 2238 [i]
- discarding factors / shortening the dual code would yield linear OA(2136, 374, F2, 52) (dual of [374, 238, 53]-code), but
- linear OA(2247, 332, F2, 112) (dual of [332, 85, 113]-code), but
- construction Y1 [i] would yield
(248−112, 248, 390)-Net in Base 2 — Upper bound on s
There is no (136, 248, 391)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 497 525865 789534 489378 343894 003975 301600 958575 316161 191467 030070 152573 895120 > 2248 [i]