Best Known (77, 77+97, s)-Nets in Base 2
(77, 77+97, 50)-Net over F2 — Constructive and digital
Digital (77, 174, 50)-net over F2, using
- t-expansion [i] based on digital (75, 174, 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+97, 52)-Net over F2 — Digital
Digital (77, 174, 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+97, 163)-Net over F2 — Upper bound on s (digital)
There is no digital (77, 174, 164)-net over F2, because
- 17 times m-reduction [i] would yield digital (77, 157, 164)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2157, 164, F2, 80) (dual of [164, 7, 81]-code), but
(77, 77+97, 165)-Net in Base 2 — Upper bound on s
There is no (77, 174, 166)-net in base 2, because
- 1 times m-reduction [i] would yield (77, 173, 166)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14265 196485 743482 395641 900505 895473 124449 124341 994248 > 2173 [i]