Best Known (139−67, 139, s)-Nets in Base 2
(139−67, 139, 49)-Net over F2 — Constructive and digital
Digital (72, 139, 49)-net over F2, using
- t-expansion [i] based on digital (70, 139, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [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 (70, 48)-sequence over F2, using
(139−67, 139, 184)-Net over F2 — Upper bound on s (digital)
There is no digital (72, 139, 185)-net over F2, because
- 1 times m-reduction [i] would yield digital (72, 138, 185)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2138, 185, F2, 66) (dual of [185, 47, 67]-code), but
- residual code [i] would yield OA(272, 118, S2, 33), but
- 1 times truncation [i] would yield OA(271, 117, S2, 32), but
- the linear programming bound shows that M ≥ 436299 229870 540803 344540 378506 426337 722368 / 173 222223 038567 257555 > 271 [i]
- 1 times truncation [i] would yield OA(271, 117, S2, 32), but
- residual code [i] would yield OA(272, 118, S2, 33), but
- extracting embedded orthogonal array [i] would yield linear OA(2138, 185, F2, 66) (dual of [185, 47, 67]-code), but
(139−67, 139, 194)-Net in Base 2 — Upper bound on s
There is no (72, 139, 195)-net in base 2, because
- 1 times m-reduction [i] would yield (72, 138, 195)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 399310 439134 380329 233605 272723 139288 909000 > 2138 [i]