Best Known (58, 58+49, s)-Nets in Base 2
(58, 58+49, 42)-Net over F2 — Constructive and digital
Digital (58, 107, 42)-net over F2, using
- t-expansion [i] based on digital (54, 107, 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
(58, 58+49, 170)-Net over F2 — Upper bound on s (digital)
There is no digital (58, 107, 171)-net over F2, because
- 1 times m-reduction [i] would yield digital (58, 106, 171)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2106, 171, F2, 48) (dual of [171, 65, 49]-code), but
- construction Y1 [i] would yield
- OA(2105, 145, S2, 48), but
- the linear programming bound shows that M ≥ 727 070369 678229 731686 401086 929362 121333 407744 / 17 521374 765895 > 2105 [i]
- OA(265, 171, S2, 26), but
- discarding factors would yield OA(265, 170, S2, 26), but
- the linear programming bound shows that M ≥ 15890 939443 762705 189763 180081 227518 381108 756480 / 413 412834 411090 648392 969149 > 265 [i]
- discarding factors would yield OA(265, 170, S2, 26), but
- OA(2105, 145, S2, 48), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2106, 171, F2, 48) (dual of [171, 65, 49]-code), but
(58, 58+49, 176)-Net in Base 2 — Upper bound on s
There is no (58, 107, 177)-net in base 2, because
- 1 times m-reduction [i] would yield (58, 106, 177)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 87 150791 387399 290799 706235 193068 > 2106 [i]