Best Known (223−123, 223, s)-Nets in Base 2
(223−123, 223, 55)-Net over F2 — Constructive and digital
Digital (100, 223, 55)-net over F2, using
- net from sequence [i] based on digital (100, 54)-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 6 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]
(223−123, 223, 65)-Net over F2 — Digital
Digital (100, 223, 65)-net over F2, using
- t-expansion [i] based on digital (95, 223, 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
(223−123, 223, 210)-Net over F2 — Upper bound on s (digital)
There is no digital (100, 223, 211)-net over F2, because
- 19 times m-reduction [i] would yield digital (100, 204, 211)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2204, 211, F2, 104) (dual of [211, 7, 105]-code), but
(223−123, 223, 211)-Net in Base 2 — Upper bound on s
There is no (100, 223, 212)-net in base 2, because
- 23 times m-reduction [i] would yield (100, 200, 212)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2200, 212, S2, 100), but
- the linear programming bound shows that M ≥ 53067 521973 608894 859497 756137 474553 785693 231666 682950 592992 641024 / 29087 > 2200 [i]
- extracting embedded orthogonal array [i] would yield OA(2200, 212, S2, 100), but