Best Known (257−122, 257, s)-Nets in Base 2
(257−122, 257, 62)-Net over F2 — Constructive and digital
Digital (135, 257, 62)-net over F2, using
- 2 times m-reduction [i] based on digital (135, 259, 62)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (19, 81, 20)-net over F2, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 19 and N(F) ≥ 20, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- digital (54, 178, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- digital (19, 81, 20)-net over F2, using
- (u, u+v)-construction [i] based on
(257−122, 257, 81)-Net over F2 — Digital
Digital (135, 257, 81)-net over F2, using
- t-expansion [i] based on digital (126, 257, 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
(257−122, 257, 331)-Net over F2 — Upper bound on s (digital)
There is no digital (135, 257, 332)-net over F2, because
- 12 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
(257−122, 257, 353)-Net in Base 2 — Upper bound on s
There is no (135, 257, 354)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 237973 689605 476295 675951 920820 837563 585738 817159 387818 458393 384144 089684 112560 > 2257 [i]