Best Known (61, 61+77, s)-Nets in Base 2
(61, 61+77, 43)-Net over F2 — Constructive and digital
Digital (61, 138, 43)-net over F2, using
- t-expansion [i] based on digital (59, 138, 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
(61, 61+77, 130)-Net over F2 — Upper bound on s (digital)
There is no digital (61, 138, 131)-net over F2, because
- 13 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
(61, 61+77, 133)-Net in Base 2 — Upper bound on s
There is no (61, 138, 134)-net in base 2, because
- 1 times m-reduction [i] would yield (61, 137, 134)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 206686 450147 570256 609483 195053 285951 291488 > 2137 [i]