Best Known (78, 158, s)-Nets in Base 2
(78, 158, 50)-Net over F2 — Constructive and digital
Digital (78, 158, 50)-net over F2, using
- t-expansion [i] based on digital (75, 158, 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
(78, 158, 52)-Net over F2 — Digital
Digital (78, 158, 52)-net over F2, using
- t-expansion [i] based on digital (77, 158, 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
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
(78, 158, 165)-Net over F2 — Upper bound on s (digital)
There is no digital (78, 158, 166)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2158, 166, F2, 80) (dual of [166, 8, 81]-code), but
- residual code [i] would yield linear OA(278, 85, F2, 40) (dual of [85, 7, 41]-code), but
- “Hel†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(278, 85, F2, 40) (dual of [85, 7, 41]-code), but
(78, 158, 190)-Net in Base 2 — Upper bound on s
There is no (78, 158, 191)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 426820 235734 647395 613452 496196 435739 049849 715047 > 2158 [i]