Best Known (135, 135+115, s)-Nets in Base 2
(135, 135+115, 66)-Net over F2 — Constructive and digital
Digital (135, 250, 66)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (39, 96, 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, 154, 33)-net over F2, using
- net from sequence [i] based on digital (39, 32)-sequence over F2 (see above)
- digital (39, 96, 33)-net over F2, using
(135, 135+115, 81)-Net over F2 — Digital
Digital (135, 250, 81)-net over F2, using
- t-expansion [i] based on digital (126, 250, 81)-net over F2, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
(135, 135+115, 331)-Net over F2 — Upper bound on s (digital)
There is no digital (135, 250, 332)-net over F2, because
- 5 times m-reduction [i] would yield digital (135, 245, 332)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(2245, 332, F2, 110) (dual of [332, 87, 111]-code), but
(135, 135+115, 377)-Net in Base 2 — Upper bound on s
There is no (135, 250, 378)-net in base 2, because
- 1 times m-reduction [i] would yield (135, 249, 378)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 946 326381 850217 295409 358273 619544 929745 401997 010016 095961 916883 818430 979569 > 2249 [i]