Best Known (59, 59+74, s)-Nets in Base 2
(59, 59+74, 43)-Net over F2 — Constructive and digital
Digital (59, 133, 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]
(59, 59+74, 127)-Net over F2 — Upper bound on s (digital)
There is no digital (59, 133, 128)-net over F2, because
- 10 times m-reduction [i] would yield digital (59, 123, 128)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2123, 128, F2, 64) (dual of [128, 5, 65]-code), but
(59, 59+74, 128)-Net in Base 2 — Upper bound on s
There is no (59, 133, 129)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 11591 990908 523476 569916 531721 205640 339392 > 2133 [i]