Best Known (190−93, 190, s)-Nets in Base 2
(190−93, 190, 54)-Net over F2 — Constructive and digital
Digital (97, 190, 54)-net over F2, using
- t-expansion [i] based on digital (95, 190, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-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 5 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 (95, 53)-sequence over F2, using
(190−93, 190, 65)-Net over F2 — Digital
Digital (97, 190, 65)-net over F2, using
- t-expansion [i] based on digital (95, 190, 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
(190−93, 190, 214)-Net over F2 — Upper bound on s (digital)
There is no digital (97, 190, 215)-net over F2, because
- 1 times m-reduction [i] would yield digital (97, 189, 215)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2189, 215, F2, 92) (dual of [215, 26, 93]-code), but
- 5 times code embedding in larger space [i] would yield linear OA(2194, 220, F2, 92) (dual of [220, 26, 93]-code), but
- adding a parity check bit [i] would yield linear OA(2195, 221, F2, 93) (dual of [221, 26, 94]-code), but
- 5 times code embedding in larger space [i] would yield linear OA(2194, 220, F2, 92) (dual of [220, 26, 93]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2189, 215, F2, 92) (dual of [215, 26, 93]-code), but
(190−93, 190, 248)-Net in Base 2 — Upper bound on s
There is no (97, 190, 249)-net in base 2, because
- 1 times m-reduction [i] would yield (97, 189, 249)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 887 628344 221521 528050 221672 162556 035654 076003 064839 605880 > 2189 [i]