Best Known (247−111, 247, s)-Nets in Base 2
(247−111, 247, 66)-Net over F2 — Constructive and digital
Digital (136, 247, 66)-net over F2, using
- 5 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
(247−111, 247, 85)-Net over F2 — Digital
Digital (136, 247, 85)-net over F2, using
(247−111, 247, 385)-Net over F2 — Upper bound on s (digital)
There is no digital (136, 247, 386)-net over F2, because
- 1 times m-reduction [i] would yield digital (136, 246, 386)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2246, 386, F2, 110) (dual of [386, 140, 111]-code), but
- construction Y1 [i] would yield
- linear OA(2245, 332, F2, 110) (dual of [332, 87, 111]-code), but
- construction Y1 [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]
- OA(287, 332, S2, 32), but
- discarding factors would yield OA(287, 302, S2, 32), but
- the Rao or (dual) Hamming bound shows that M ≥ 161 699122 225452 699910 750634 > 287 [i]
- discarding factors would yield OA(287, 302, S2, 32), but
- OA(2244, 300, S2, 110), but
- construction Y1 [i] would yield
- linear OA(2140, 386, F2, 54) (dual of [386, 246, 55]-code), but
- discarding factors / shortening the dual code would yield linear OA(2140, 376, F2, 54) (dual of [376, 236, 55]-code), but
- the improved Johnson bound shows that N ≤ 638271 859848 635367 775356 676031 149307 988937 824926 330314 504342 114737 011555 < 2236 [i]
- discarding factors / shortening the dual code would yield linear OA(2140, 376, F2, 54) (dual of [376, 236, 55]-code), but
- linear OA(2245, 332, F2, 110) (dual of [332, 87, 111]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2246, 386, F2, 110) (dual of [386, 140, 111]-code), but
(247−111, 247, 397)-Net in Base 2 — Upper bound on s
There is no (136, 247, 398)-net in base 2, because
- 1 times m-reduction [i] would yield (136, 246, 398)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 116 802458 369082 565080 610928 802808 790949 765327 321581 785467 213660 046172 205504 > 2246 [i]