Best Known (205−99, 205, s)-Nets in Base 2
(205−99, 205, 56)-Net over F2 — Constructive and digital
Digital (106, 205, 56)-net over F2, using
- t-expansion [i] based on digital (105, 205, 56)-net over F2, using
- net from sequence [i] based on digital (105, 55)-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 7 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 (105, 55)-sequence over F2, using
(205−99, 205, 65)-Net over F2 — Digital
Digital (106, 205, 65)-net over F2, using
- t-expansion [i] based on digital (95, 205, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(205−99, 205, 244)-Net over F2 — Upper bound on s (digital)
There is no digital (106, 205, 245)-net over F2, because
- 1 times m-reduction [i] would yield digital (106, 204, 245)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2204, 245, F2, 98) (dual of [245, 41, 99]-code), but
- residual code [i] would yield OA(2106, 146, S2, 49), but
- 1 times truncation [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]
- 1 times truncation [i] would yield OA(2105, 145, S2, 48), but
- residual code [i] would yield OA(2106, 146, S2, 49), but
- extracting embedded orthogonal array [i] would yield linear OA(2204, 245, F2, 98) (dual of [245, 41, 99]-code), but
(205−99, 205, 275)-Net in Base 2 — Upper bound on s
There is no (106, 205, 276)-net in base 2, because
- 1 times m-reduction [i] would yield (106, 204, 276)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 26 409279 278453 742270 806865 588111 753019 435001 358335 118763 909608 > 2204 [i]