Best Known (139−78, 139, s)-Nets in Base 2
(139−78, 139, 43)-Net over F2 — Constructive and digital
Digital (61, 139, 43)-net over F2, using
- t-expansion [i] based on digital (59, 139, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(139−78, 139, 130)-Net over F2 — Upper bound on s (digital)
There is no digital (61, 139, 131)-net over F2, because
- 14 times m-reduction [i] would yield digital (61, 125, 131)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2125, 131, F2, 64) (dual of [131, 6, 65]-code), but
(139−78, 139, 131)-Net in Base 2 — Upper bound on s
There is no (61, 139, 132)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 808520 052622 853459 402492 415433 383418 439328 > 2139 [i]