Best Known (77, 77+73, s)-Nets in Base 2
(77, 77+73, 50)-Net over F2 — Constructive and digital
Digital (77, 150, 50)-net over F2, using
- t-expansion [i] based on digital (75, 150, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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 (75, 49)-sequence over F2, using
(77, 77+73, 52)-Net over F2 — Digital
Digital (77, 150, 52)-net over F2, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 77 and N(F) ≥ 52, using
(77, 77+73, 174)-Net over F2 — Upper bound on s (digital)
There is no digital (77, 150, 175)-net over F2, because
- 3 times m-reduction [i] would yield digital (77, 147, 175)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2147, 175, F2, 70) (dual of [175, 28, 71]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2148, 176, F2, 70) (dual of [176, 28, 71]-code), but
- adding a parity check bit [i] would yield linear OA(2149, 177, F2, 71) (dual of [177, 28, 72]-code), but
- 1 times code embedding in larger space [i] would yield linear OA(2148, 176, F2, 70) (dual of [176, 28, 71]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2147, 175, F2, 70) (dual of [175, 28, 71]-code), but
(77, 77+73, 202)-Net in Base 2 — Upper bound on s
There is no (77, 150, 203)-net in base 2, because
- 1 times m-reduction [i] would yield (77, 149, 203)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 741 417933 816110 775554 976257 612570 779333 006740 > 2149 [i]