Best Known (129, 129+127, s)-Nets in Base 2
(129, 129+127, 57)-Net over F2 — Constructive and digital
Digital (129, 256, 57)-net over F2, using
- t-expansion [i] based on digital (110, 256, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
(129, 129+127, 81)-Net over F2 — Digital
Digital (129, 256, 81)-net over F2, using
- t-expansion [i] based on digital (126, 256, 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
(129, 129+127, 273)-Net over F2 — Upper bound on s (digital)
There is no digital (129, 256, 274)-net over F2, because
- 1 times m-reduction [i] would yield digital (129, 255, 274)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2255, 274, F2, 126) (dual of [274, 19, 127]-code), but
- residual code [i] would yield OA(2129, 147, S2, 63), but
- 1 times truncation [i] would yield OA(2128, 146, S2, 62), but
- the linear programming bound shows that M ≥ 10 546031 115613 724859 656905 833525 360409 444352 / 30229 > 2128 [i]
- 1 times truncation [i] would yield OA(2128, 146, S2, 62), but
- residual code [i] would yield OA(2129, 147, S2, 63), but
- extracting embedded orthogonal array [i] would yield linear OA(2255, 274, F2, 126) (dual of [274, 19, 127]-code), but
(129, 129+127, 284)-Net in Base 2 — Upper bound on s
There is no (129, 256, 285)-net in base 2, because
- 19 times m-reduction [i] would yield (129, 237, 285)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2237, 285, S2, 108), but
- the linear programming bound shows that M ≥ 56343 399996 195265 169227 114420 218575 369487 305782 122681 969429 903696 483971 902670 896111 812608 / 204485 216198 046875 > 2237 [i]
- extracting embedded orthogonal array [i] would yield OA(2237, 285, S2, 108), but