Best Known (235−121, 235, s)-Nets in Base 2
(235−121, 235, 57)-Net over F2 — Constructive and digital
Digital (114, 235, 57)-net over F2, using
- t-expansion [i] based on digital (110, 235, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-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 8 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 (110, 56)-sequence over F2, using
(235−121, 235, 73)-Net over F2 — Digital
Digital (114, 235, 73)-net over F2, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 114 and N(F) ≥ 73, using
(235−121, 235, 239)-Net over F2 — Upper bound on s (digital)
There is no digital (114, 235, 240)-net over F2, because
- 5 times m-reduction [i] would yield digital (114, 230, 240)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2230, 240, F2, 116) (dual of [240, 10, 117]-code), but
- residual code [i] would yield linear OA(2114, 123, F2, 58) (dual of [123, 9, 59]-code), but
- residual code [i] would yield linear OA(256, 64, F2, 29) (dual of [64, 8, 30]-code), but
- 1 times truncation [i] would yield linear OA(255, 63, F2, 28) (dual of [63, 8, 29]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- 1 times truncation [i] would yield linear OA(255, 63, F2, 28) (dual of [63, 8, 29]-code), but
- residual code [i] would yield linear OA(256, 64, F2, 29) (dual of [64, 8, 30]-code), but
- residual code [i] would yield linear OA(2114, 123, F2, 58) (dual of [123, 9, 59]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2230, 240, F2, 116) (dual of [240, 10, 117]-code), but
(235−121, 235, 240)-Net in Base 2 — Upper bound on s
There is no (114, 235, 241)-net in base 2, because
- 13 times m-reduction [i] would yield (114, 222, 241)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2222, 241, S2, 108), but
- the linear programming bound shows that M ≥ 65 939284 597237 046425 151657 508194 701172 717895 120115 873148 022756 434178 473984 / 7 611275 > 2222 [i]
- extracting embedded orthogonal array [i] would yield OA(2222, 241, S2, 108), but