Best Known (62, 62+78, s)-Nets in Base 2
(62, 62+78, 43)-Net over F2 — Constructive and digital
Digital (62, 140, 43)-net over F2, using
- t-expansion [i] based on digital (59, 140, 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
(62, 62+78, 44)-Net over F2 — Digital
Digital (62, 140, 44)-net over F2, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 62 and N(F) ≥ 44, using
(62, 62+78, 132)-Net over F2 — Upper bound on s (digital)
There is no digital (62, 140, 133)-net over F2, because
- 14 times m-reduction [i] would yield digital (62, 126, 133)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2126, 133, F2, 64) (dual of [133, 7, 65]-code), but
(62, 62+78, 134)-Net in Base 2 — Upper bound on s
There is no (62, 140, 135)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 559146 665284 616895 727458 727029 995675 610056 > 2140 [i]