Best Known (179−96, 179, s)-Nets in Base 2
(179−96, 179, 51)-Net over F2 — Constructive and digital
Digital (83, 179, 51)-net over F2, using
- t-expansion [i] based on digital (80, 179, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(179−96, 179, 57)-Net over F2 — Digital
Digital (83, 179, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
(179−96, 179, 176)-Net over F2 — Upper bound on s (digital)
There is no digital (83, 179, 177)-net over F2, because
- 8 times m-reduction [i] would yield digital (83, 171, 177)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2171, 177, F2, 88) (dual of [177, 6, 89]-code), but
(179−96, 179, 178)-Net in Base 2 — Upper bound on s
There is no (83, 179, 179)-net in base 2, because
- 6 times m-reduction [i] would yield (83, 173, 179)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2173, 179, S2, 90), but
- the (dual) Plotkin bound shows that M ≥ 1 149371 655649 416643 768760 270362 731887 714054 341637 701632 / 91 > 2173 [i]
- extracting embedded orthogonal array [i] would yield OA(2173, 179, S2, 90), but