Best Known (246−134, 246, s)-Nets in Base 2
(246−134, 246, 57)-Net over F2 — Constructive and digital
Digital (112, 246, 57)-net over F2, using
- t-expansion [i] based on digital (110, 246, 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
(246−134, 246, 72)-Net over F2 — Digital
Digital (112, 246, 72)-net over F2, using
- t-expansion [i] based on digital (110, 246, 72)-net over F2, using
- net from sequence [i] based on digital (110, 71)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 110 and N(F) ≥ 72, using
- net from sequence [i] based on digital (110, 71)-sequence over F2, using
(246−134, 246, 235)-Net over F2 — Upper bound on s (digital)
There is no digital (112, 246, 236)-net over F2, because
- 22 times m-reduction [i] would yield digital (112, 224, 236)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2224, 236, F2, 112) (dual of [236, 12, 113]-code), but
- residual code [i] would yield linear OA(2112, 123, F2, 56) (dual of [123, 11, 57]-code), but
- residual code [i] would yield linear OA(256, 66, F2, 28) (dual of [66, 10, 29]-code), but
- adding a parity check bit [i] would yield linear OA(257, 67, F2, 29) (dual of [67, 10, 30]-code), but
- residual code [i] would yield linear OA(256, 66, F2, 28) (dual of [66, 10, 29]-code), but
- residual code [i] would yield linear OA(2112, 123, F2, 56) (dual of [123, 11, 57]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2224, 236, F2, 112) (dual of [236, 12, 113]-code), but
(246−134, 246, 236)-Net in Base 2 — Upper bound on s
There is no (112, 246, 237)-net in base 2, because
- 28 times m-reduction [i] would yield (112, 218, 237)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2218, 237, S2, 106), but
- the linear programming bound shows that M ≥ 59311 882667 731407 548267 433191 443187 482001 717729 589259 458427 942548 275200 / 110349 > 2218 [i]
- extracting embedded orthogonal array [i] would yield OA(2218, 237, S2, 106), but