Best Known (250−139, 250, s)-Nets in Base 2
(250−139, 250, 57)-Net over F2 — Constructive and digital
Digital (111, 250, 57)-net over F2, using
- t-expansion [i] based on digital (110, 250, 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
(250−139, 250, 72)-Net over F2 — Digital
Digital (111, 250, 72)-net over F2, using
- t-expansion [i] based on digital (110, 250, 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
(250−139, 250, 232)-Net over F2 — Upper bound on s (digital)
There is no digital (111, 250, 233)-net over F2, because
- 27 times m-reduction [i] would yield digital (111, 223, 233)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2223, 233, F2, 112) (dual of [233, 10, 113]-code), but
- residual code [i] would yield linear OA(2111, 120, F2, 56) (dual of [120, 9, 57]-code), but
- residual code [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]
- residual code [i] would yield linear OA(255, 63, F2, 28) (dual of [63, 8, 29]-code), but
- residual code [i] would yield linear OA(2111, 120, F2, 56) (dual of [120, 9, 57]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2223, 233, F2, 112) (dual of [233, 10, 113]-code), but
(250−139, 250, 233)-Net in Base 2 — Upper bound on s
There is no (111, 250, 234)-net in base 2, because
- 1 times m-reduction [i] would yield (111, 249, 234)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1028 254436 159413 895767 107354 166344 336325 268823 449871 617708 179820 350503 208770 > 2249 [i]